Projective representations of groups using Hilbert right C*-modules
aa r X i v : . [ m a t h . OA ] N ov PROJECTIVE REPRESENTATIONS OF GROUPSUSING HILBERT RIGHT C*-MODULES
CORNELIU CONSTANTINESCUJune 20, 2018
AbstractThe projective representation of groups was introduced in 1904 by IssaiSchur (1875-1941) in his paper [S]. It differs from the normal representationof groups (introduced by his tutor Ferdinand Georg Frobenius (1849-1917) atthe suggestion of Richard Dedekind (1831-1916)) by a twisting factor, whichwe call Schur function in this paper and which is called sometimes multipliersor normalized factor set in the literature (other names are also used). It startswith a group T and a Schur function f for T . This is a scalar valued functionon T × T satisfying the conditions f (1 ,
1) = 1 and | f ( s, t ) | = 1 , f ( r, s ) f ( rs, t ) = f ( r, st ) f ( s, t )for all r, s, t ∈ T . The projective representation of T twisted by f is a unitalC*-subalgebra of the C*-algebra L ( l ( T )) of operators on the Hilbert space l ( T ). This representation can be used in order to construct many examples ofC*-algebras (see e.g. [C1] Chapter 7). By replacing the scalars IR or IC withan arbitrary unital (real or complex) C*-algebra E the field of applications isenhanced in an essential way. In this case l ( T ) is replaced by the Hilbert right E -module (cid:13)| t ∈ T E ≈ E ⊗ l ( T ) and L ( l ( T )) is replaced by L E ( E ⊗ l ( T )), theC*-algebra of adjointable operators of L ( E ⊗ l ( T )). The projective represen-tation of groups, which we present in this paper, has some similarities with theconstruction of cross products with discrete groups. It opens the way to createmany K-theories. 1n a first section we introduce some results which are needed for this con-struction, which is developed in the second section. In the third section wepresent examples of C*-algebras obtained by this method. Examples of a spe-cial kind (the Clifford algebras) are presented in the last section. AMS Subject Classification: 22D25 (Primary) 20C25, 46L08 (Secondary)Key Words: Hilbert right C ∗ -modules, Projective groups representations Contents E -C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Some topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 T := ZZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2 T := ZZ × ZZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.3 T := ( ZZ ) n with n ∈ IN . . . . . . . . . . . . . . . . . . . . . . . 1283.4 T := ZZ n with n ∈ IN . . . . . . . . . . . . . . . . . . . . . . . . . 1333.5 T := ZZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Throughout this paper we use the following notation: T is a group, 1 isits neutral element, K := l ( T ), 1 K := id K := identity map of K , E isa unital C*-algebra (resp. a W*-algebra), 1 E is its unit, ˘ E denotes theset E endowed with its canonical structure of a Hilbert right E -module([C1] Proposition 5.6.1.5), H := ˘ E ⊗ K ≈ (cid:13)| t ∈ T ˘ E , (resp. H := ˘ E ¯ ⊗ K ≈ W (cid:13)| t ∈ T ˘ E )([C3] Proposition 2.1, (resp. [C3] Corollary 2.2)). In some examples, inwhich T is additive, 1 will be replaced by 0.The map L E ( ˘ E ) −→ E, u u E | E i = u E is an isomorphism of C*-algebras with inverse E −→ L E ( ˘ E ) , x x · . We identify E with L E ( ˘ E ) using these isomorphisms.3n general we use the notation of [C1]. For tensor products of C*-algebras weuse [W], for W*-tensor products of W*-algebras we use [T], for tensor productsof Hilbert right C ∗ -modules we use [L], and for the exterior W*-tensor productsof selfdual Hilbert right W ∗ -modules we use [C2] and [C3].In the sequel we give a list of notation used in this paper.1) IK denotes the field of real numbers (:= IR) or the field of complex numbers(:= IC). In general the C*-algebras will be complex or real. IH denotesthe field of quaternions, IN denotes the set of natural numbers (0 IN),and for every n ∈ IN ∪ { } we putIN n := { m ∈ IN | m ≤ n } . ZZ denotes the group of integers and for every n ∈ IN we put ZZ n :=ZZ / ( n ZZ ) .
2) For every set A , P ( A ) denotes the set of subsets of A , P f ( A ) the set offinite subsets of A , and Card A denotes the cardinal number of A . If f is a function defined on A and B is a subset of A then f | B denotes therestriction of f to B .3) If A, B are sets then A B denotes the set of maps of B in A .4) For all i, j we denote by δ i,j Kronecker’s symbol: δ i,j := (cid:26) i = j i = j .
5) If
A, B are topological spaces then C ( A, B ) denotes the set of continuousmaps of A into B . If A is locally compact space and E is a C*-algebrathen C ( A, E ) (resp. C ( A, E )) denotes the C*-algebra of continuous maps A → E , which are bounded (resp. which converge to 0 at the infinity).6) For every set I and for every J ⊂ I we denote by e J := e IJ the character-istic function of J i.e. the function on I equal to 1 on J and equal to 0on I \ J . For i ∈ I we put e i := ( δ i,j ) j ∈ I ∈ l ( I ).7) If F is an additive group and S is a set then F ( S ) := (cid:8) x ∈ F S (cid:12)(cid:12) { s ∈ S | x s = 0 } is finite (cid:9) .
4) If
E, F are vector spaces in duality then E F denotes the vector space E endowed with the locally convex topology of pointwise convergence on F ,i.e. with the weak topology σ ( E, F ).9) If E is a normed vector space then E ′ denotes its dual and E denotesits unit ball: E := { x ∈ E | k x k ≤ } . Moreover if E is an ordered Banach space then E + denotes the convexcone of its positive elements. If E has a unique predual (up to isomor-phisms), then we denote by ¨ E this predual and so by E ¨ E the vector space E endowed with the locally convex topology of pointwise convergence on¨ E .10) The expressions of the form ”... C*-... (resp. ... W*-...)”, which appearoften in this paper, will be replaced by expressions of the form ”... C**-...”.11) If F is a unital C*-algebra and A is a subset of F then we denote by 1 F the unit of F , by P r F the set of orthogonal projections of F , by A c := { x ∈ F | y ∈ A ⇒ xy = yx } , Re F := { x ∈ F | x = x ∗ } , and by U n F the set of unitary elements of F . If F is a real C*-algebrathen ◦ F denotes its complexification.12) If F is a C*-algebra then we denote for every n ∈ IN by F n,n the C*-algebra of n × n matrices with entries in F . If T is finite then F T,T has acorresponding signification.13) Let F be a C*-algebra and H, K
Hilbert right F -modules. We denote by L F ( H, K ) the Banach subspace of L ( H, K ) of adjointable operators, by1 H the identity map H → H which belongs to L F ( H ) := L F ( H, H ) . For ( ξ, η ) ∈ H × K we put η h · | ξ i : H −→ K , ζ η h ζ | ξ i and denote by K F ( H ) the closed vector subspace of L F ( H ) generated by { η h · | ξ i | ξ, η ∈ H } . 54) Let F be a W*-algebra and H, K
Hilbert right F -modules. We put for a ∈ ¨ F and ( ξ, η ) ∈ H × K , ] ( a, ξ ) : H −→ IK , ζ ζ | ξ i , a i , ^ ( a, ξ, η ) : L F ( H, K ) −→ IK , u uξ | η i , a i and denote by ¨ H the closed vector subspace of the dual H ′ of H generatedby n ] ( a, ξ ) (cid:12)(cid:12)(cid:12) a ∈ ¨ F , ξ ∈ H o and by ... H the closed vector subspace of L F ( H, K ) ′ generated by n ^ ( a, ξ, η ) (cid:12)(cid:12)(cid:12) ( a, ξ, η ) ∈ ¨ F × H × K o . If H is selfdual then ... H is the predual of L F ( H ) ([C1] Theorem 5.6.3.5b)) and ¨ H is the predual of H ([C1] Proposition 5.6.3.3). Moreover amap defined on F is called W*-continuous if it is continuous on F ¨ F . If G is a W*-algebra a C*-homomorphism ϕ : F → G is called a W*-homomorphism if the map ϕ : F ¨ F → G ¨ G is continuous; in this case ¨ ϕ denotes the pretranspose of ϕ .15) If F is a C**-algebra and ( H i ) i ∈ I a family of Hilbert right F -modulesthen we put (cid:13)| i ∈ I H i := ( ξ ∈ Y i ∈ I H i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) the family h ξ i | ξ i i i ∈ I is summable in F ) respectively W (cid:13)| i ∈ I H i := ( ξ ∈ Y i ∈ I H i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) the family h ξ i | ξ i i i ∈ I is summable in F ¨ F ) . ⊙ denotes the algebraic tensor product of vector spaces.17) If F, G are W*-algebras and H (resp. K ) is a selfdual Hilbert right F -module (resp. G-module) then we denote by H ¯ ⊗ K the W*-tensor productof H and K , which is a selfdual Hilbert right F ¯ ⊗ G -module ([C2] Definition2.3).18) ≈ denotes isomorphic. 6f T is finite then (by [C1] Theorem 5.6.6.1 f)) L E ( H ) = E T,T = IK
T,T ⊗ E = K E ( H ) . DEFINITION 1.1.1 A Schur E -function for T is a map f : T × T −→ U n E c such that f (1 ,
1) = 1 E and f ( r, s ) f ( rs, t ) = f ( r, st ) f ( s, t ) for all r, s, t ∈ T . We denote by F ( T, E ) the set of Schur E -functions for T and put ˜ f : T −→ U n E c , t f ( t, t − ) ∗ , ˆ f : T × T −→ U n E c , ( s, t ) f ( t − , s − ) for every f ∈ F ( T, E ) . Schur functions are also called normalized factor set or multiplier or two-co-cycle (for T with values in U n E c ) in the literature. We present in thissubsection only some elementary properties (which will be used in the sequel)in order to fix the notation and the terminology. By the way, U n E c can bereplaced in this subsection by an arbitrary commutative multiplicative group. PROPOSITION 1.1.2
Let f ∈ F ( T, E ) .a) For every t ∈ T , f ( t,
1) = f (1 , t ) = 1 E , f ( t, t − ) = f ( t − , t ) , ˜ f ( t ) = ˜ f ( t − ) . ) For all s, t ∈ T , f ( s, t ) ˜ f ( s ) = f ( s − , st ) ∗ , f ( s, t ) ˜ f ( t ) = f ( st, t − ) ∗ . a) Putting s = 1 in the equation of f we obtain f ( r, f ( r, t ) = f ( r, t ) f (1 , t )so f ( r,
1) = f (1 , t )for all r, t ∈ T . Hence f ( t,
1) = f (1 , t ) = f (1 ,
1) = 1 E . Putting r = t and s = t − in the equation of f we get f ( t, t − ) f (1 , t ) = f ( t, f ( t − , t ) . By the above, f ( t, t − ) = f ( t − , t ) , ˜ f ( t ) = ˜ f ( t − ) . b) Putting r = s − in the equation of f , by a), f ( s, t ) f ( s − , st ) = f ( s − , s ) f (1 , t ) = ˜ f ( s ) ∗ ,f ( s, t ) ˜ f ( s ) = f ( s − , st ) ∗ . Putting now t = s − in the equation of f , by a) again, f ( r, s ) f ( rs, s − ) = f ( r, f ( s, s − ) = ˜ f ( s ) ∗ ,f ( r, s ) ˜ f ( s ) = f ( rs, s − ) ∗ , f ( s, t ) ˜ f ( t ) = f ( st, t − ) ∗ . DEFINITION 1.1.3
We put Λ( T, E ) := { λ : T −→ U n E c | λ (1) = 1 E } and ˆ λ : T −→ U n E c , t λ ( t − ) ,δλ : T × T −→ U n E c , ( s, t ) λ ( s ) λ ( t ) λ ( st ) ∗ for every λ ∈ Λ( T, E ) . ROPOSITION 1.1.4 a) F ( T, E ) is a subgroup of the commutative multiplicative group ( U n E c ) T × T such that f ∗ is the inverse of f for every f ∈ F ( T, E ) .b) ˆ f ∈ F ( T, E ) for every f ∈ F ( T, E ) and the map F ( T, E ) −→ F ( T, E ) , f ˆ f is an involutive group automorphism.c) Λ( T, E ) is a subgroup of the commutative multiplicative group ( U n E c ) T , δλ ∈ F ( T, E ) for every λ ∈ Λ( T, E ) , and the map δ : Λ( T, E ) −→ F ( T, E ) , λ δλ is a group homomorphism with kernel { λ ∈ Λ( T, E ) | λ is a group homomorphism } such that c δλ = δ ˆ λ for every λ ∈ Λ( T, E ) . a) is obvious.b) For r, s, t ∈ T ,ˆ f ( r, s ) ˆ f ( rs, t ) = f ( s − , r − ) f ( t − , s − r − ) == f ( t − , s − ) f ( t − s − , r − ) = ˆ f ( r, st ) ˆ f ( s, t ) , so ˆ f ∈ F ( T, E ). For f, g ∈ F ( T, E ), c f g ( s, t ) = ( f g )( t − , s − ) = f ( t − , s − ) g ( t − , s − ) = ˆ f ( s, t )ˆ g ( s, t ) = ( ˆ f ˆ g )( s, t ) , c f g = ˆ f ˆ g , ˆ f ∗ ( s, t ) = ˆ f ( s, t ) ∗ = f ( t − , s − ) ∗ = f ∗ ( t − , s − ) = c f ∗ ( s, t ) , ( ˆ f ) ∗ = c f ∗ . c) For r, s, t ∈ T , δλ ( r, s ) δλ ( rs, t ) = λ ( r ) λ ( s ) λ ( rs ) ∗ λ ( rs ) λ ( t ) λ ( rst ) ∗ = λ ( r ) λ ( s ) λ ( t ) λ ( rst ) ∗ , λ ( r, st ) δλ ( s, t ) = λ ( r ) λ ( st ) λ ( rst ) ∗ λ ( s ) λ ( t ) λ ( st ) ∗ = λ ( r ) λ ( s ) λ ( t ) λ ( rst ) ∗ so δλ ∈ F ( T, E ). For λ, µ ∈ F ( T, E ) and s, t ∈ T , δλ ( s, t ) δµ ( s, t ) = λ ( s ) λ ( t ) λ ( st ) ∗ µ ( s ) µ ( t ) µ ( st ) ∗ == ( λµ )( s )( λµ )( t )( λµ )( st ) ∗ = δ ( λµ )( s, t ) , ( δλ )( δµ ) = δ ( λµ ) ,δλ ∗ ( s, t ) = λ ∗ ( s ) λ ∗ ( t ) λ ( st ) = ( δλ ( s, t )) ∗ = ( δλ ) ∗ ( s, t ) , δλ ∗ = ( δλ ) ∗ , so δ is a group homomorphism. The other assertions are obvious. PROPOSITION 1.1.5
Let t ∈ T , m, n ∈ ZZ , and f ∈ F ( T, E ) .a) f ( t m , t n ) = f ( t n , t m ) . b) m ∈ IN = ⇒ f ( t m , t n ) = m − Q j =0 f ( t n + j , t ) ! (cid:18) m − Q k =1 f ( t k , t ) ∗ (cid:19) .c) We define λ : ZZ −→ U n E c , n n − Q j =1 f ( t j , t ) ∗ if n ∈ IN − n Q j =1 f ( t − j , t ) if n IN . If t p = 1 for every p ∈ IN then f ( t m , t n ) = λ ( m ) λ ( n ) λ ( m + n ) ∗ for all m, n ∈ ZZ . a) We may assume m ∈ IN because otherwise we can replace t by t − . Put P ( m, n ) : ⇐⇒ f ( t m , t n ) = f ( t n , t m ) ,Q ( m ) : ⇐⇒ P ( m, n ) holds for all n ∈ ZZ . From f ( t m , t n − m ) f ( t n , t m ) = f ( t m , t n ) f ( t n − m , t m )10t follows P ( m, n ) ⇐⇒ P ( m, n − m ) ⇐⇒ P ( m, n − km )for all k ∈ ZZ .We prove the assertion by induction. P ( m,
0) follows from Proposition 1.1.2a). By the above P (1 , ⇐⇒ P (1 , k )for all k ∈ ZZ . Thus Q (1) holds.Assume Q ( p ) holds for all p ∈ IN m − . Then P ( m, p ) holds for all p ∈ IN m − ∪ { } . Let n ∈ ZZ . There is a k ∈ ZZ such that p := n − km ∈ IN m − ∪ { } . By the above P ( m, n ) holds. Thus Q ( m ) holds and this finishes the inductiveproof.b) We prove the formula by induction with respect to m . By a), the formulaholds for m = 1. Assume the formula holds for an m ∈ IN. Since f ( t m , t ) f ( t m +1 , t n ) = f ( t m , t n +1 ) f ( t, t n )we get by a), f ( t m +1 , t n ) = f ( t m , t n +1 ) f ( t, t n ) f ( t m , t ) ∗ == m − Y j =0 f ( t n +1+ j , t ) m − Y k =1 f ( t k , t ) ∗ ! f ( t n , t ) f ( t m , t ) ∗ == m Y j =0 f ( t n + j , t ) m Y k =1 f ( t k , t ) ∗ ! . Thus the formula holds also for m + 1.c) If m, n ∈ IN then by b), λ ( m ) λ ( n ) λ ( m + n ) ∗ == m − Y k =1 f ( t k , t ) ∗ ! n − Y j =1 f ( t j , t ) ∗ m + n − Y j =1 f ( t j , t ) =11 m − Y j =0 f ( t n + j , t ) m − Y k =1 f ( t k , t ) ∗ ! = f ( t m , t n ) . If m, n ∈ IN , n ≤ m − λ ( m ) λ ( − n ) λ ( m − n ) ∗ == m − Y j =1 f ( t j , t ) ∗ n Y j =1 f ( t − j , t ) m − n − Y j =1 f ( t j , t ) == m − Y j =0 f ( t − n + j , t ) m − Y k =1 f ( t k , t ) ∗ ! = f ( t m , t − n ) . If m, n ∈ IN , n ≥ m then by b), λ ( m ) λ ( − n ) λ ( m − n ) ∗ == m − Y k =1 f ( t k , t ) ∗ ! n Y j =1 f ( t − j , t ) n − m Y j =1 f ( t − j , t ) ∗ == n Y j = n − m +1 f ( t − j , t ) m − Y k =1 f ( t k , t ) ∗ ! = f ( t m , t − n ) . For all m, n ∈ IN put R ( m, n ) : ⇐⇒ f ( t − m , t − n ) = λ ( − m ) λ ( − n ) λ ( − m − n ) ∗ . By the above and by Proposition 1.1.2 a),b), λ ( − λ ( − λ ( − ∗ = f ( t − , t ) f ( t − , t ) ∗ = ˜ f ( t − ) ∗ f ( t, t − ) ∗ = f ( t − , t − ) , so R (1 ,
1) holds. Let now m, n ∈ IN and assume R ( m, n ) holds. Then λ ( − m ) λ ( − n − λ ( − m − n − ∗ == m Y j =1 f ( t − j , t ) n +1 Y j =1 f ( t − j , t ) m + n +1 Y j =1 f ( t − j , t ) ∗ == f ( t − m , t − n ) f ( t − n − , t ) f ( t − m − n − , t ) ∗ = f ( t − m , t − n − ) , so R ( m, n ) ⇒ R ( m, n +1). By symmetry and a), R ( m, n ) holds for all m, n ∈ IN.12
OROLLARY 1.1.6
The map
Λ( ZZ , E ) −→ F ( ZZ , E ) , λ δλ is a surjective group homomorphism with kernel { λ ∈ Λ( ZZ , E ) | n ∈ ZZ = ⇒ λ ( n ) = λ (1) n } . By Proposition 1.1.4 c), only the surjectivity of the above map has to beproved and this follows from Proposition 1.1.5 c). E -C*-algebras By replacing the scalars with the unital C*-algebra E we restrict the cate-gory of C*-algebras to the subcategory of those C*-algebras which are connectedin a certain way with E . The category of unital C*-algebras is replaced by thecategory of E -C*-algebras, while the general category of C*-algebras is replacedby the category of adapted E -modules. DEFINITION 1.2.1
We call in this paper E -module a C*-algebra F en-dowed with the bilinear maps E × F −→ F, ( α, x ) αx ,F × E −→ F, ( x, α ) xα such that for all α, β ∈ E and x, y ∈ F , ( αβ ) x = α ( βx ) , α ( xβ ) = ( αx ) β , x ( αβ ) = ( xα ) β ,α ( xy ) = ( αx ) y , ( xy ) α = x ( yα ) , α ∈ E c = ⇒ αx = xα , ( αx ) ∗ = x ∗ α ∗ , ( xα ) ∗ = α ∗ x ∗ , E x = x E = x . If F, G are E -modules then a C*-homomorphism ϕ : F → G is called E -linear if for all ( α, x ) ∈ E × F , ϕ ( αx ) = α ( ϕx ) , ϕ ( xα ) = ( ϕx ) α . α, x ) ∈ E × F , k αx k = k x ∗ α ∗ αx k ≤ k x k k α k , k xα k = k α ∗ x ∗ xα k ≤ k α k k x k so k αx k ≤ k α k k x k , k xα k ≤ k x k k α k . DEFINITION 1.2.2 An E -C**-algebra is a unital C**-algebra F for which E is a canonical unital C**-subalgebra such that E c defined with respect to E coincides with E c defined with respect to F i.e. for every x ∈ E , if xy = yx forall y ∈ E then xy = yx for all y ∈ F . Every closed ideal of an E -C*-algebra iscanonically an E -module.Let F, G be E -C**-algebras. A map ϕ : F −→ G is called an E -C**-homomorphism if it is an E -linear C**-homomorphism . If in addition ϕ is a C*-isomorphism then we say that ϕ is an E -C*-isomorphism and weuse in this case the notation ≈ E . A C**-subalgebra F of F is called E -C**-subalgebra of F if E ⊂ F . With the notation of the above Definition ( α − ϕα ) ϕx = 0 for all α ∈ E and x ∈ F . Thus ϕ is unital iff ϕα = α for every α ∈ E . The exampleIK −→ IK × IK , x ( x, E -C*-homomorphism need not be unital.If we put IT := { z ∈ IC | | z | = 1 } , E := C ( IT , IC), and x : IT −→ IC , z z and if we denote by λ the Lebesgue measure on IT then L ∞ ( λ ) is an E -C*-algebra, x ∈ U n E , and x is homotopic to 1 E in U n L ∞ ( λ ) but not in U n C ( IT , IC).
DEFINITION 1.2.3
We denote by C E (resp. by C E ) the category of E -C*-algebras for which the morphisms are the E -C*-homomorphisms (resp. theunital E -C*-homomorphisms). PROPOSITION 1.2.4
Let F be an E -module. ) We denote by ˇ F the vector space E × F endowed with the bilinear map ( E × F ) × ( E × F ) −→ E × F, (( α, x ) , ( β, y )) ( αβ, αy + xβ + xy ) and with the conjugate linear map E × F −→ E × F, ( α, x ) ( α ∗ , x ∗ ) . ˇ F is an involutive unital algebra with (1 E , as unit.b) The maps π : ˇ F −→ E , ( α, x ) α ,λ : E −→ ˇ F , α ( α, ,ι : F −→ ˇ F , x (0 , x ) are involutive algebra homomorphisms such that π ◦ λ is the identitymap of E , λ and ι are injective, and λ and π are unital. If there is anorm on ˇ F with respect to which it is a C*-algebra (in which case sucha norm is unique), then we call F adapted . We denote by M E thecategory of adapted E -modules for which the morphism are the E -linearC*-homomorphisms.c) If F is adapted then ˇ F is an E -C*-algebra by using canonically the injec-tion λ and for all α ∈ E and x ∈ F , k α k ≤ k ( α, x ) k ≤ k α k + k x k , k (0 , x ) k = k x k ≤ k ( α, x ) k , k ( α, , x ) k ≤ k α k k x k , k (0 , x )( α, k ≤ k x k k α k . In particular F (identified with ι ( F ) ) is a closed ideal of ˇ F .d) If E and F are C*-subalgebras of a C*-algebra G in such a way that thestructure of E -module of F is inherited from G then ϕ : ˇ F −→ E × G , ( α, x ) ( α, α + x ) is an injective involutive algebra homomorphism, ϕ ( ˇ F ) is closed, F isadapted, and for all α ∈ E and x ∈ F , k ( α, x ) k E × F = sup {k α k , k α + x k} . In particular every closed ideal of an E -C*-algebra is adapted and C E isa full subcategory of M E . ) A closed ideal G of an adapted E -module F , which is at the same timean E -submodule of F , is adapted.f ) If F is unital then it is adapted and ˇ F −→ IR + , ( α, x ) sup {k α k , k α F + x k} is the C*-norm of ˇ F .g) If lim y, F k αy − yα k = 0 for all α ∈ E + , where F denotes the canonical approximate unit of F ,then F is adapted and ˇ F −→ IR + , ( α, x ) sup ( k α k , lim sup y, F k αy + x k ) is the C*-norm of ˇ F . In particular F is adapted if E is commutative.h) If F is an adapted E -module then (with the notation of b)) −→ F ι −→ ˇ F π −→ λ ←− E −→ is a split exact sequence in the category M E . a) and b) are easy to see.c) Since λ and ι are injective and , π ( α, x ) = α , ( α, x ) = ( α,
0) + (0 , x ) , ( α, , x ) = (0 , αx ) , (0 , x )( α,
0) = (0 , xα )we get the first and the last two inequalities as well as the identity k (0 , x ) k = k x k . It follows k (0 , x ) k ≤ k ( α, x ) k + k ( α, k = k ( α, x ) k + k λπ ( α, x ) k ≤≤ k ( α, x ) k + k ( α, x ) k = 2 k ( α, x ) k .
16) It is easy to see that ϕ is an injective involutive algebra homomorphism.Let ( α, x ) ∈ ϕ ( ˇ F ). There are sequences ( α n ) n ∈ IN and ( x n ) n ∈ IN in E and F ,respectively, such that lim n →∞ ( α n , α n + x n ) = ( α, x ) . It follows α = lim n →∞ α n ∈ E , x − α = lim n →∞ x n ∈ F , ( α, x ) = ϕ ( α, x − α ) ∈ ϕ ( ˇ F ) . Thus ϕ ( ˇ F ) is closed, which proves the assertion by pulling back the norm of E × G .e) By c), F is a closed ideal of ˇ F so G is a closed ideal of ˇ F (use anapproximate unit of F ). Since G is an E -submodule of F its structure of E -module is inherited from ˇ F . By d), G is adapted.f) The map ˇ F −→ E × F, ( α, x ) ( α, α F + x )is an isomorphism of involutive algebras and so we can pull back the norm of E × F .g) It is easy to see that the above map is a norm. Sincesup {k α k , k x k} ≤ k ( α, x ) k ≤ k α k + k x k for all ( α, x ) ∈ E × F , ˇ F endowed with this norm is complete. For ( α, x ) ∈ E × F ,( α, x ) ∗ ( α, x ) = ( α ∗ α, α ∗ x + x ∗ α + x ∗ x ) , k ( α, x ) ∗ ( α, x ) k = sup {k α k , lim sup y, F k α ∗ αy + α ∗ x + x ∗ α + x ∗ x k} . For y ∈ F , (cid:13)(cid:13)(cid:13) ( αy + x ) ∗ ( αy + x ) − ( α ∗ αy + α ∗ x + x ∗ α + x ∗ x ) (cid:13)(cid:13)(cid:13) ≤≤ (cid:13)(cid:13)(cid:13) y α ∗ α − α ∗ αy (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) y α ∗ x − α ∗ x (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) x ∗ αy − x ∗ α (cid:13)(cid:13)(cid:13)
17o lim y, F (cid:13)(cid:13)(cid:13) ( αy + x ) ∗ ( αy + x ) − ( α ∗ αy + α ∗ x + x ∗ α + x ∗ x ) (cid:13)(cid:13)(cid:13) = 0 . Since the map F + → F + , y y maps F into itself and k αy + x k = k yα ∗ αy + yα ∗ x + x ∗ αy + x ∗ x k we have by the above, k ( α, x ) k = sup ( k α k , lim sup y, F (cid:13)(cid:13)(cid:13) αy + x (cid:13)(cid:13)(cid:13) ) == sup ( k α k , lim sup y, F (cid:13)(cid:13)(cid:13) ( αy + x ) ∗ ( αy + x ) (cid:13)(cid:13)(cid:13)) == sup {k α k , lim sup y, F k α ∗ αy + α ∗ x + x ∗ α + x ∗ x k} = k ( α, x ) ∗ ( α, x ) k . Thus the above norm is a C*-norm and F is adapted.h) ι is an injective E -C*-homomorphism and its image is equal to Ker π . COROLLARY 1.2.5
Let F an E -module, G a C*-algebra, and ⊗ σ the spatialtensor product.a) F ⊗ σ G is in a natural way an E -module the multiplication being given by α ( x ⊗ y ) = ( αx ) ⊗ y , ( x ⊗ y ) α = ( xα ) ⊗ y for all α ∈ E , x ∈ F , and y ∈ G .b) If F is an E -C*-algebra and G is unital then the map E −→ F ⊗ σ G, α α ⊗ G is an injective C*-homomorphism. In particular, the E -module F ⊗ σ G isan E -C*-algebra.c) If F is an adapted E -module then the E -module F ⊗ σ G is adapted and k ( α, z ) k = sup {k α k , k α + z k} for all ( α, z ) ∈ E × ( F ⊗ σ G ) . ) If F is an adapted E -module and G := C (Ω) for a locally compact space Ω then C (Ω , F ) is adapted and k ( α, x ) k = sup {k α k , k αe Ω + x k} for all ( α, x ) ∈ E × C (Ω , F ) . a) and b) are easy to see.c) If ˜ G denotes the unitization of G then by b), ˇ F ⊗ σ ˜ G is an E -C*-algebraand F ⊗ σ G is a closed ideal of it, so the assertion follows from Proposition1.2.4 d),e).d) follows from c). PROPOSITION 1.2.6 a) If
F, G are E -modules and ϕ : F → G is an E -linear C*-homomorphismthen the map ˇ ϕ : ˇ F −→ ˇ G , ( α, x ) ( α, ϕx ) is an involutive unital algebra homomorphism, injective or surjective if ϕ is so. If F = G and if ϕ is the identity map then ˇ ϕ is also the identitymap.b) Let F , F , F be E -modules and let ϕ : F → F and ψ : F → F be E -linear C*-homomorphisms. Then ˇ z }| { ψ ◦ ϕ = ˇ ψ ◦ ˇ ϕ . PROPOSITION 1.2.7
Let G be an E -module, F an E -submodule of G whichis at the same time an ideal of G , and ϕ : G → G/F the quotient map.a)
G/F has a natural structure of E -module and ϕ is E -linear.b) If G is adapted then G/F is also adapted. Moreover if ψ : ˇ G → ˇ G/F denotes the quotient map (where F is identified to { (0 , x ) | x ∈ F } ) thenthere is an E -C*-isomorphism θ : ˇ z }| { G/F → ˇ G/F such that ψ = θ ◦ ˇ ϕ .
19) is easy to see.b) Let ( α, z ) ∈ ˇ z }| { G/F and let x, y ∈ − ϕ ( z ). Then ψ ( α, x ) = ψ ( α, y ) and weput θ ( α, z ) := ψ ( α, x ). It is straightforward to show that θ is an isomorphismof involutive algebras. By pulling back the norm of ˇ G/F with respect to θ wesee that G/F is adapted.
LEMMA 1.2.8
Let { ( F i ) i ∈ I , ( ϕ ij ) i,j ∈ I ) } be an inductive system in the cate-gory of C*-algebras, { F, ( ϕ i ) i ∈ I } its inductive limit, G a C*-algebra, for ev-ery i ∈ I , ψ i : F i → G a C*-homomorphism such that ψ j ◦ ϕ ji = ψ i forall i, j ∈ I, i ≤ j , and ψ : F → G the resulting C*-homomorphism. If Ker ψ i ⊂ Ker ϕ i for every i ∈ I then ψ is injective. Let i ∈ I . Since Ker ϕ i ⊂ Ker ψ i is obvious, we have Ker ϕ i = Ker ψ i . Let ρ : F i → F i /Ker ψ i be the quotient map and ϕ ′ i : F i /Ker ψ i −→ F , ψ ′ i : F i /Ker ψ i −→ G the injective C*-homomorphisms with ϕ i = ϕ ′ i ◦ ρ , ψ i = ψ ′ i ◦ ρ . Then ψ ′ i ◦ ρ = ψ i = ψ ◦ ϕ i = ψ ◦ ϕ ′ i ◦ ρ . For x ∈ F i , since ψ ′ i and ϕ ′ i are norm-preserving, k ρx k = (cid:13)(cid:13) ψ ′ i ρx (cid:13)(cid:13) = (cid:13)(cid:13) ψϕ ′ i ρx (cid:13)(cid:13) ≤ (cid:13)(cid:13) ϕ ′ i ρx (cid:13)(cid:13) = k ρx k , k ψϕ i x k = (cid:13)(cid:13) ψϕ ′ i ρx (cid:13)(cid:13) = (cid:13)(cid:13) ϕ ′ i ρx (cid:13)(cid:13) = k ϕ i x k . Thus ψ preserves the norms on ∪ i ∈ I ϕ i ( F i ). Since this set is dense in F , ψ isinjective. PROPOSITION 1.2.9
Let { ( F i ) i ∈ I , ( ϕ ij ) i,j ∈ I } be an inductive system in thecategory M E and let ( F, ( ϕ i ) i ∈ I ) be its inductive limit in the category of E -modules (Proposition 1.2.4 c)) .a) F is adapted. ) Let ( G, ( ψ i ) i ∈ I ) be the inductive limit in the category C E of the inductivesystem { ( ˇ F i ) i ∈ I , ( ˇ ϕ ij ) i,j ∈ I } (Proposition 1.2.6 a),b)) and let ψ : G → ˇ F be the unital C*-homomorphism such that ψ ◦ ψ i = ˇ ϕ i for every i ∈ I .Then ψ is an E -C*-isomorphism. a) Put F := ( ( α, x ) ∈ ˇ F (cid:12)(cid:12) α ∈ E, x ∈ [ i ∈ I ϕ i ( F i ) ) ,p : F −→ IR + , ( α, x ) inf { k ( α, x i ) k | i ∈ I, x i ∈ F i , ϕ i x i = x } .F is an involutive unital subalgebra of ˇ F . p is a norm and by Proposition 1.2.4c), q ( α, x ) := lim ( α,y ) ∈ F y → x p ( α, y )exists and k α k ≤ q ( α, x ) ≤ k α k + k x k , k x k ≤ q ( α, x )for every ( α, x ) ∈ ˇ F .Let ( α, x ) ∈ F . Let further i ∈ I , x i , y i ∈ F i with ϕ i x i = x, ϕ i y i = α ∗ x + x ∗ α + x ∗ x . Then(0 , ϕ i ( α ∗ x i + x ∗ i α + x ∗ i x i − y i )) = ˇ ϕ i (( α, x i ) ∗ ( α, x i ) − ( α ∗ α, y i )) = 0so inf i ≤ j k ϕ ji ( α ∗ x i + x ∗ i α + x ∗ i x i − y i ) k = 0 . For ǫ > j ∈ I, i ≤ j, with k ϕ ji ( α ∗ x i + x ∗ i α + x ∗ i x i − y i ) k < ǫ . We get p ( α, x ) ≤ k ( α, ϕ ji x i ) k = k ( α, ϕ ji x i ) ∗ ( α, ϕ ji x i ) k == k ( α ∗ α, α ∗ ϕ ji x i + ( ϕ ji x ∗ i ) α + ϕ ji ( x ∗ i x i )) k == k ( α ∗ α, ϕ ji ( α ∗ x i + x ∗ i α + x ∗ i x i )) k ≤≤ k ( α ∗ α, ϕ ji y i ) k + k (0 , ϕ ji ( α ∗ x i + x ∗ i α + x ∗ i x i − y i )) k < k ( α ∗ α, ϕ ji y i ) k + ǫ . By taking the infimum on the right side it follows, since ǫ is arbitrary, p ( α, x ) ≤ p ( α ∗ α, α ∗ x + x ∗ α + x ∗ x ) = p (( α, x ) ∗ ( α, x ))21nd this shows that p is a C*-norm. It is easy to see that q is a C*-norms.By the above inequalities, ˇ F endowed with the norm q is complete, i.e. ˇ F is aC*-algebra and F is adapted.b) Let i ∈ I and let ( α, x ) ∈ Ker ˇ ϕ i . Then0 = ˇ ϕ i ( α, x ) = ( α, ϕ i x )so α = 0 , ϕ i x = 0 , inf j ∈ I, j ≥ i k ϕ ji x k = 0 , k ˇ ϕ ji (0 , x ) k = k (0 , ϕ ji x ) k = k ϕ ji x k , k ψ i ( α, x ) k = inf j ∈ I, j ≥ i k ˇ ϕ ji (0 , x ) k = 0 , ( α, x ) ∈ Ker ψ i . By Lemma 1.2.8, ψ is injective.Let ( β, y ) ∈ ˇ F and let ε >
0. There are i ∈ I and x ∈ F i with k ϕ i x − y k < ε .Then ψψ i ( β, x ) = ˇ ϕ i ( β, x ) = ( β, ϕ i x ) , k ψψ i ( β, x ) − ( β, y ) k = k ˇ ϕ i ( β, x ) − ( β, y ) k = k ϕ i x − y k < ε . Thus ψ ( G ) is dense in ˇ F and ψ is surjective. Hence ψ is a C*-isomorphism. COROLLARY 1.2.10
We put Φ E ( F ) := ˇ F for every E -module F and simi-larly Φ E ( ϕ ) := ˇ ϕ for every E -linear C*-homomorphism ϕ .a) Φ E is a covariant functor from the category M E in the category C E .b) The categories C E and M E possess inductive limits and the functor Φ E is continuous with respect to the inductive limits. a) follows from Proposition 1.2.6.b) follows from Proposition 1.2.9. Remark.
The category C E does not possess inductive limits in general. Thishappens for instance if ϕ ij = 0 for all i, j ∈ I .22 .3 Some topologies T is only a set in this subsectionIf the group T is infinite then different topologies play a certain role in theconstruction of the projective representations of T . It will be shown that allthese topologies conduct to the same construction, but the use of them simplifiesthe manipulations.We introduce the following notation in order to unify the cases of C*-algebras and (resp. W*-algebras). DEFINITION 1.3.1 e (cid:13)| := (cid:13)| (resp. e (cid:13)| := W (cid:13)| ) , e ⊗ := ⊗ (resp. e ⊗ := ¯ ⊗ ) , gX := X (resp. gX := ¨ E X ) . If T is a Hausdorff topology on L E ( H ) then for every G ⊂ L E ( H ) , G T denotesthe set G endowed with the relative topology T and T ¯ G denotes the closure of G in L E ( H ) T . Moreover T P denotes the sum with respect to T . LEMMA 1.3.2
For x ∈ E , by the above identification of E with L E ( ˘ E ) , x e ⊗ K : H −→ H , ξ ( xξ t ) t ∈ T is well-defined and belongs to L E ( H ) .a) The map ϕ : E −→ L E ( H ) , x x e ⊗ K is an injective unital C*-homomorphism. ) Assume E is a W*-algebra. Then for every ( a, ξ, η ) ∈ ¨ E × H × H , thefamily ( ξ t a η ∗ t ) t ∈ T is summable in ¨ E E and for every x ∈ E , D ϕx , ^ ( a, ξ, η ) E = * x , E X t ∈ T ξ t a η ∗ t + . Thus ϕ is a W*-homomorphism ([C1] Theorem 5.6.3.5 d)) with ¨ ϕ ^ ( a, ξ, η ) = E X t ∈ T ξ t a η ∗ t , where ¨ ϕ denotes the pretranspose of ϕ .c) If we consider E as a canonical unital C**-subalgebra of L E ( H ) by usingthe embedding of a) then L E ( H ) is an E -C**-algebra. a) follows from [L] page 37 (resp. [C3] Proposition 1.4).b) We have D x ¯ ⊗ K , ^ ( a, ξ, η ) E = h h ( x ¯ ⊗ K ) ξ | η i , a i = * ¨ E X t ∈ T η ∗ t x ξ t , a + == X t ∈ T h η ∗ t x ξ t , a i = X t ∈ T h x , ξ t a η ∗ t i . Thus the family ( ξ t a η ∗ t ) t ∈ T is summable in ¨ E E and D ϕx , ^ ( a, ξ, η ) E = * x , E X t ∈ T ξ t a η ∗ t + . If ϕ ′ : L E ( H ) → E ′ denotes the transpose of ϕ then ϕ ′ ^ ( a, ξ, η ) = E X t ∈ T ξ t a η ∗ t ∈ ¨ E .
By continuity ϕ ′ .. z }| { L E ( H ) ! ⊂ ¨ E and ϕ is a unital W*-homomorphism.24) Let x ∈ E c and ξ, η ∈ L E ( H ). By [C1] Proposition 5.6.3.17 d), (cid:10) ( x e ⊗ K ) ξ (cid:12)(cid:12) η (cid:11) = gX t ∈ T η ∗ t (( x e ⊗ K ) ξ ) t = gX t ∈ T η ∗ t xξ t == gX t ∈ T xη ∗ t ξ t = x gX t ∈ T η ∗ t ξ t = x h ξ | η i . Thus for u ∈ L E ( H ), (cid:10) u ( x e ⊗ K ) ξ (cid:12)(cid:12) η (cid:11) = (cid:10) ( x e ⊗ K ) ξ (cid:12)(cid:12) u ∗ η (cid:11) = x h ξ | u ∗ η i = x h uξ | η i ,u ( x e ⊗ K ) = ( x e ⊗ K ) u , and so x e ⊗ K ∈ L E ( H ) c . DEFINITION 1.3.3
We put for all ξ, η ∈ H (resp. and a ∈ ¨ E + ) p ξ,η : L E ( H ) −→ IR + , X Xξ | η ik , (resp. p ξ,η,a : L E ( H ) −→ IR + , X Xξ | η i , a i | ) ,p ξ : L E ( H ) −→ IR + , X Xξ k = kh Xξ | Xξ ik / , (resp. p ξ,a : L E ( H ) −→ IR + , X Xξ | Xξ i , a i / ) ,q ξ : L E ( H ) −→ IR + , X p ξ ( X ∗ ) , (resp. q ξ,a : L E ( H ) −→ IR + , X p ξ,a ( X ∗ )) . and denote, respectively, by T , T , T the topologies on L E ( H ) generated bythe set of seminorms { p ξ,η | ξ, η ∈ H } , (cid:16) resp. n p ξ,η,a | ξ, η ∈ H, a ∈ ¨ E + o(cid:17) , { p ξ | ξ ∈ H } , (cid:16) resp. n p ξ,a | ξ ∈ H, a ∈ ¨ E + o(cid:17) , { p ξ | ξ ∈ H } ∪ { q ξ | ξ ∈ H } , (cid:16) resp. n p ξ,a | ξ ∈ H, a ∈ ¨ E + o ∪ n q ξ,a | ξ ∈ H, a ∈ ¨ E + o(cid:17) . Moreover k·k denotes the norm topology on L E ( H ) .
25f course T ⊂ T . In the C*-case, T is the topology of pointwise conver-gence. If E is finite-dimensional then the C*-case and the W*-case coincide. PROPOSITION 1.3.4
Let X ∈ L E ( H ) and ξ, η ∈ H (resp. and a ∈ ¨ E ).a) p ξ,η ( X ) = p η,ξ ( X ∗ ) (resp. p ξ,η, | a | ( X ) = p η,ξ, | a | ( X ∗ )) .b) p ξ,η ( X ) ≤ p ξ ( X ) k η k .c) If E is a W*-algebra and a = x | a | is the polar representation of a then p ξx,η, | a | ( X ) = (cid:12)(cid:12)(cid:12)D X , ^ ( a, ξ, η ) E(cid:12)(cid:12)(cid:12) ≤ p ξx, | a | ( X ) h h η | η i , | a | i / . d) If Y, Z ∈ L E ( H ) then p ξ,η ( Y XZ ) = p Zξ,Y ∗ η ( X ) (resp. p ξ,η, | a | ( Y XZ ) = p Zξ,Y ∗ η, | a | ( X )) ,p ξ ( Y XZ ) ≤ k Y k p Zξ ( X ) (resp. p ξ, | a | ( Y XZ ) ≤ k Y k p Zξ, | a | ( X )) . a) From h Xξ | η i = h ξ | X ∗ η i = h X ∗ η | ξ i ∗ it follows p ξ,η ( X ) = kh Xξ | η ik = kh X ∗ η | ξ ik = p η,ξ ( X ∗ ) , (resp. p ξ,η, | a | ( X ) = | h h X ∗ η | ξ i , | a | i | = p η,ξ, | a | ( X ∗ )) . b) p ξ,η ( X ) = kh Xξ | η ik ≤ p ξ ( X ) k η k .c) We have p ξx,η, | a | ( X ) = | h h X ( ξx ) | η i , | a | i | = | h h Xξ | η i x , | a | i | == | h h Xξ | η i , x | a | i | = | h h Xξ | η i , a i | = (cid:12)(cid:12)(cid:12)D X , ^ ( a, ξ, η ) E(cid:12)(cid:12)(cid:12) . By Schwarz’ inequality ([C1] Proposition 2.3.3.9), | h h X ( ξx ) | η i , | a | i | ≤ h h X ( ξx ) | X ( ξx ) i , | a | i h h η | η i , | a | i , p ξx,η, | a | ( X ) ≤ p ξx, | a | ( X ) h h η | η i , | a | i / . d) The first equation follows from p ξ,η ( Y XZ ) = kh Y XZξ | η ik = kh XZξ | Y ∗ η ik = p Zξ,Y ∗ η ( X )(resp. p ξ,η, | a | ( Y XZ ) = | h h
Y XZξ | η i , | a | i | == | h h XZξ | Y ∗ η i , | a | i | = p Zξ,Y ∗ η, | a | ( X ))and the second from p ξ ( Y XZ ) = k Y XZξ k ≤ k Y k k XZξ k = k Y k p Zξ ( X )(resp. p ξ, | a | ( Y XZ ) = h h
Y XZξ | Y XZξ i , | a | i / ≤≤ k Y k h h XZξ | XZξ i , | a | i / = k Y k p Zξ, | a | ( X )) . LEMMA 1.3.5
Let n ∈ IN and ( x i ) i ∈ IN n a family in E . Then X i ∈ IN n x i ! ∗ X i ∈ IN n x i ! ≤ n X i ∈ IN n x ∗ i x i . We prove the relation by induction with respect to n . By [C1] Corollary4.2.2.4 and by the hypothesis of the induction, X i ∈ IN n x i ! ∗ X i ∈ IN n x i ! = x ∗ n + X i ∈ IN n − x ∗ i x n + X i ∈ IN n − x i == x ∗ n x n + X i ∈ IN n − ( x ∗ n x i + x ∗ i x n ) + X i ∈ IN n − x i ∗ X i ∈ IN n − x i ≤≤ x ∗ n x n + X i ∈ IN n − ( x ∗ n x n + x ∗ i x i ) + ( n − X i ∈ IN n − x ∗ i x i = n X i ∈ IN n x ∗ i x i . EMMA 1.3.6
Let n ∈ IN , x ∈ E n,n , and for every j ∈ IN n put η j := ( δ ji E ) i ∈ IN n ∈ (cid:13)| i ∈ IN n ˘ E .
Then k x k ≤ √ n sup j ∈ IN n k xη j k . For ξ ∈ (cid:13)| i ∈ IN n ˘ E ! , by Lemma 1.3.5, h xξ | xξ i = X i ∈ IN n h ( xξ ) i | ( xξ ) i i = X i ∈ IN n X j ∈ IN n x ij ξ j ∗ X j ∈ IN n x ijξ j ≤≤ n X i ∈ IN n X j ∈ IN n ( x ij ξ j ) ∗ ( x ij ξ j ) = n X i ∈ IN n X j ∈ IN n ξ ∗ j x ∗ ij x ij ξ j == n X j ∈ IN n ξ ∗ j X i ∈ IN n x ∗ ij x ij ! ξ j . For i, j ∈ IN n , ( xη j ) i = X k ∈ IN n x ik η jk = x ij , h xη j | xη j i = X i ∈ IN n ( xη j ) ∗ i ( xη j ) i = X i ∈ IN n x ∗ ij x ij , so h xξ | xξ i ≤ n X j ∈ IN n ξ ∗ j h xη j | xη j i ξ j ≤ n X j ∈ IN n k xη j k ξ ∗ j ξ j ≤≤ n sup j ∈ IN n k xη j k X j ∈ IN n ξ ∗ j ξ j ≤ n sup j ∈ IN n k xη j k E , k x k ≤ n sup j ∈ IN n k xη j k . COROLLARY 1.3.7 ) The map L E ( H ) T −→ L E ( H ) T , X X ∗ is continuous. In particular Re L E ( H ) is a closed set of L E ( H ) T .b) T ⊂ T ⊂ T ⊂ norm topology.c) If E is a W*-algebra then the identity map L E ( H ) ... H −→ L E ( H ) T is continuous so L E ( H ) T = L E ( H ) ... H is compact.d) For Y, Z ∈ L E ( H ) and k ∈ { , } , the map L E ( H ) T k −→ L E ( H ) T k , X Y XZ is continuous.e) L E ( H ) T is complete in the C*-case.f ) If T is finite then T is the norm topology in the C*-case.g) K E ( H ) is dense in L E ( H ) T . a) follows from Proposition 1.3.4 a).b) T ⊂ T follows from Proposition 1.3.4 b),c). T ⊂ T ⊂ norm topologyis trivial.c) follows from Proposition 1.3.4 c) (and [C1] Theorem 5.6.3.5 a)).d) follows from Proposition 1.3.4 d).e) Let F be a Cauchy filter on L E ( H ) T . Put Y : H −→ H , ξ lim X, F ( Xξ ) ,Z : H −→ H , ξ lim X, F ( X ∗ ξ ) , H . For ξ, η ∈ H , h Y ξ | η i = lim X, F h Xξ | η i = lim X, F h ξ | X ∗ η i = h ξ | Zη i , so Y, Z ∈ L E ( H ) and Z = Y ∗ . Thus F converges to Y in L E ( H ) T and L E ( H ) T is complete.f) follows from b) and Lemma 1.3.6.g) Let X ∈ L E ( H ) and ξ ∈ H . For every S ∈ P f ( T ) put P S := X s ∈ S e s h · | e s i ∈ P r K E ( H )and let F T be the upper section filter or P f ( T ). Then P S X ∈ K E ( H ) for every S ∈ P f ( T ) and lim S, F T P S Xξ = Xξ in H (resp. in H ¨ H ) ([C1] Proposition 5.6.4.1 e) (resp. [C1] Proposition 5.6.4.6c))). Thus lim S, F T P S X = X with respect to the topology T . Since the same holds for X ∗ , it follows that X belongs to the closure of K E ( H ) in L E ( H ) T . Remark.
The inclusions in b) can be strict as it is known from the case E := IK. LEMMA 1.3.8
Let G be a W*-algebra and F a C*-subalgebra of G . Then thefollowing are equivalent.a) F generates G as a W*-algebra.b) F is dense in G G .c) F is dense in G ¨ G . a = ⇒ b follows from [C1] Corollary 6.3.8.7.30 = ⇒ c is trivial. c = ⇒ a follows from [C1] Corollary 4.4.4.12 a). PROPOSITION 1.3.9
Let G be a W*-algebra, F a C*-subalgebra of G gen-erating it as W*-algebra, I a set, and L := (cid:13)| i ∈ I ˘ F , M := W (cid:13)| i ∈ I ˘ G . a) M is the extension of L to a selfdual Hilbert right G -module ([C2] Propo-sition 1.3 f)) and L is dense in M M .b) If we denote for every X ∈ L F ( L ) by ¯ X ∈ L G ( M ) its unique extension ([C3] Proposition 1.4 a)) then the map L F ( L ) −→ L G ( M ) , X ¯ X is an injective C*-homomorphism and its image is dense in L G ( M ) ... M .c) The map L F ( L ) T −→ L G ( M ) T , X ¯ X is continuous. a) By Lemma 1.3.8 a ⇒ b , F is dense in G G so ˘ F is dense in ˘ G G and˘ G is the extension of ˘ F to a selfdual Hilbert right G -module ([C3] Corollary1.5 a ⇒ a ). By [C3] Proposition 1.8, M is the extension of L to a selfdualHilbert right G -module. By [C3] Corollary 1.5 a ⇒ a , L is dense in M M .b) By a) and [C3] Proposition 1.4 e), the map L F ( L ) −→ L G ( M ) , X ¯ X is an injective C*-homomorphism. By [C3] Proposition 1.9 b), its image isdense in L G ( M ) ... M .c) Denote by N the vector subspace of ... M generated by n ^ ( a, ξ, η ) (cid:12)(cid:12)(cid:12) ( a, ξ, η ) ∈ ¨ G × L × L o .
31y a) and [C3] Proposition 1.9 a), N is dense in ... M so by Corollary 1.3.7 c), L G ( M ) T = L G ( M ) N . For ( a, ξ, η ) ∈ ¨ G + × L × L and X ∈ L F ( L ), by Proposition 1.3.4 c), p ξ,η,a ( ¯ X ) = | (cid:10) (cid:10) ¯ Xξ (cid:12)(cid:12) η (cid:11) , a (cid:11) | == | h h Xξ | η i , a i | ≤ p ξx, | a | ( X ) h h η | η i , | a | i , where a = x | a | is the polar representation of a , so the map L F ( L ) T −→ L G ( M ) T , X ¯ X is continuous. LEMMA 1.3.10
Let n ∈ IN , ξ ∈ (cid:13)| i ∈ IN n ˘ E , and x := [ ξ i δ j, ] i,j ∈ IN n ∈ E n,n . Then k x k = k ξ k . For η ∈ (cid:13)| i ∈ IN n ˘ E and i ∈ IN n ,( xη ) i = X j ∈ IN n x ij η j = X j ∈ IN n ξ i δ j, η j = ξ i η , h xη | xη i = X i ∈ IN n h ( xη ) i | ( xη ) i i = X i ∈ IN n h ξ i η | ξ i η i = X i ∈ IN n η ∗ ξ ∗ i ξ i η == η ∗ X i ∈ IN n ξ ∗ i ξ i ! η = η ∗ h ξ | ξ i η ≤ k ξ k η ∗ η , k xη k ≤ k ξ k k η k ≤ k ξ k k η k , k x k ≤ k ξ k . On the other hand if we put ζ := ( δ i, E ) i ∈ IN n then for i ∈ IN n ,( xζ ) i = X j ∈ IN n x ij ζ j = X j ∈ IN n ξ i δ j, E = ξ i , h xζ | xζ i = X i ∈ IN n ( xζ ) ∗ i ( xζ ) i = X i ∈ IN n ξ ∗ i ξ i = h ξ | ξ i , k x k ≥ k xζ k = k ξ k , k x k = k ξ k . EMMA 1.3.11
Let
F, G be unital C**-algebras, ϕ : F → G a surjectiveC**-homomorphism, I a set, L := g (cid:13)| i ∈ I ˘ F ≈ ˘ F e ⊗ l ( I ) , M := g (cid:13)| i ∈ I ˘ G ≈ ˘ G e ⊗ l ( I ) , and for every ξ ∈ L put ˜ ξ := ( ϕξ i ) i ∈ I .a) If ξ, η ∈ L and x ∈ F then ˜ ξ ∈ M , (cid:13)(cid:13)(cid:13) ˜ ξ (cid:13)(cid:13)(cid:13) ≤ k ξ k , g ( ξx ) = ( ˜ ξ ) ϕx , D ˜ ξ (cid:12)(cid:12)(cid:12) ˜ η E = ϕ h ξ | η i . b) For every η ∈ M there is a ξ ∈ L with ˜ ξ = η, k ξ k = k η k .c) In the W*-case the map L ¨ L −→ M ¨ M , ξ ˜ ξ is continuous. a) For J ∈ P f ( I ), X i ∈ J h ϕξ i | ϕη i i = X i ∈ J ( ϕη i ) ∗ ( ϕξ i ) = ϕ X i ∈ J η ∗ i ξ i . It follows ˜ ξ ∈ M, (cid:13)(cid:13)(cid:13) ˜ ξ (cid:13)(cid:13)(cid:13) ≤ k ξ k , D ˜ ξ (cid:12)(cid:12)(cid:12) ˜ η E = ϕ h ξ | η i . Moreover for i ∈ I ,( f ξx ) i = ϕ ( ξx ) i = ϕ ( ξ i x ) = ( ϕξ i )( ϕx ) = ˜ ξ i ( ϕx ) , f ξx = ˜ ξ ( ϕx ) . b) Case 1 { i ∈ I | η i = 0 } is finiteFor simplicity we assume { i ∈ I | η i = 0 } = IN n for some n ∈ IN. We put θ : F n,n −→ G n,n , [ x ij ] i,j ∈ IN n [ ϕx ij ] i,j ∈ IN n . is obviously a surjective C*-homomorphism. So if we put y := [ η i δ j, ] i,j ∈ IN n ∈ G n,n , then there is an x ∈ F n,n with θx = y, k x k = k y k ([K] Theorem 10.1.7). If weput ξ : I −→ ˘ F , i (cid:26) x i if i ∈ IN n i ∈ I \ IN n and z := [ x ij δ j ] i,j ∈ IN n ∈ F n,n then θz = [ ϕ ( x ij δ j )] i,j ∈ IN n = [ y ij δ j ] i,j ∈ IN n = y and by [C1] Theorem 5.6.6.1 a), k z k ≤ k x k . We get for i ∈ IN n ,˜ ξ i = ϕξ i = ϕx i = y i = η i . By a) and Lemma 1.3.10, k ξ k = k z k ≤ k x k = k y k = k η k = (cid:13)(cid:13)(cid:13) ˜ ξ (cid:13)(cid:13)(cid:13) ≤ k ξ k , k ξ k = k η k . Case 2 η arbitrary in the W*-caseWe may assume k η k = 1. We put for every J ∈ P f ( I ), η J : I −→ G , i (cid:26) η i if i ∈ J i ∈ I \ J .
By Case 1, for every J ∈ P f ( I ) there is a ξ J ∈ L with ˜ ξ J = η J and k ξ J k = k η J k ≤
1. Let F be an ultrafilter on P f ( I ) finer than the upper section filter of P f ( I ). By [C1] Proposition 5.6.3.3 a ⇒ b , ξ := lim J, F ξ J exists in L L . For i ∈ I ,˜ ξ i = ϕξ i = ϕ lim J, F ( ξ J ) i = lim J, F ϕ ( ξ J ) i = η i so ˜ ξ = η . By a), 1 = k η k = (cid:13)(cid:13)(cid:13) ˜ ξ (cid:13)(cid:13)(cid:13) ≤ k ξ k ≤
1, so k ξ k = k η k .34ase 3 η arbitrary in the C*-caseWe put for every J ∈ P f ( I ) and every ζ ∈ M , ζ J : I −→ G , i (cid:26) ζ i if i ∈ J i ∈ I \ J .
Moreover we denote by F I the upper section filter of P f ( I ), set M := { ζ ∈ M | { i ∈ I | ζ i = 0 } is finite } , and denote by M the vector subspace of K G ( M ) generated by the set { ζ h · | ζ i | ζ , ζ ∈ M } . Let G be the vector subspace of K F ( L ) generated by the set { α h · | β i | α, β ∈ L } . G is an involutive subalgebra of K F ( L ). Let ( α q ) q ∈ Q , ( β q ) q ∈ Q be finite familiesin L such that X q ∈ Q α q h · | β q i = 0 . Let further α ′ , β ′ ∈ M . By Case 1, there are α, β ∈ L with ˜ α = α ′ , ˜ β = β ′ andwe get by a), * X q ∈ Q ˜ α q D β ′ (cid:12)(cid:12) ˜ β q E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ′ + = X q ∈ Q (cid:10) ˜ α q | α ′ (cid:11) D β ′ (cid:12)(cid:12) ˜ β q E == X q ∈ Q h ˜ α q | ˜ α i D ˜ β (cid:12)(cid:12)(cid:12) ˜ β q E = ϕ X q ∈ Q h α q | α i h β | β q i == ϕ * X q ∈ Q α q h · | β q i β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + = 0 . It follows ([C1] Proposition 5.6.4.1 e)) X q ∈ Q ˜ α q D · | ˜ β q E = 0 . ψ : G −→ K G ( M ) , X q ∈ Q α q h · | β q i 7−→ X q ∈ Q ˜ α q D · | ˜ β q E is well-defined and it is easy to see (by a)) that ψ is an involutive algebrahomomorphism.Step 1 k ψ k ≤
1; we extend ψ by continuity to a map ψ : K F ( L ) → K G ( M )Let u := X q ∈ Q α q h · | β q i ∈ G and let ζ ∈ M . By Case 1, there is an α ∈ L with ˜ α = ζ . By a),( ψu ) ζ = X q ∈ Q ˜ α q D ˜ α | ˜ β q E = X q ∈ Q ˜ α q ϕ h α | β q i = X q ∈ Q ^ z }| { α q h α | β q i = f uα , k ( ψu ) ζ k = k f uα k ≤ k uα k ≤ k u k . Since M is dense in M ([C1] Proposition 5.6.4.1 e)), it follows k ψu k ≤ k u k , k ψ k ≤ . Step 2 M is dense in K G ( M )Let α, β ∈ M . By [C1] Proposition 5.6.4.1 e), α = lim J, F I α J , β = lim J, F I β J so by [C1] Proposition 5.6.5.2 a), α h · | β i = lim J, F I α J h · | β J i , which proves the assertion.Step 3 ψ is a surjective C*-homomorphism36y Step 1, ψ is a C*-homomorphism. Since its image contains M (by Case1) it is surjective by Step 2. Step 4 The assertionLet j ∈ I . By Step 3 and [K] Theorem 10.1.7 (and [C1] Proposition 5.6.5.2a)), there is a u ∈ K F ( L ) with ψu = η h · | G ⊗ e j i , k u k = k η h · | G ⊗ e j ik = k η k . From ψ ( u ((1 F ⊗ e j ) h · | F ⊗ e j i )) = ( η h · | G ⊗ e j i )((1 G ⊗ e j ) h · | G ⊗ e j i ) == η h · | G ⊗ e j i , k η k = k η h · | G ⊗ e j ik ≤ k u ((1 F ⊗ e j ) h · | F ⊗ e j i ) k ≤≤ k u k k (1 F ⊗ e j ) h · | F ⊗ e j ik = k u k = k η k , k u ((1 F ⊗ e j ) h · | F ⊗ e j i ) k = k η k we see that we may assume u = u ((1 F ⊗ e j ) h · | F ⊗ e j i ) . Then u = ( u (1 F ⊗ e j )) h · | F ⊗ e j i . If we put ξ := u (1 F ⊗ e j ) ∈ L then u = ξ h · | F ⊗ e j i , k η k = k u k = k ξ k , η h · | G ⊗ e j i = ψu = ˜ ξ h · | G ⊗ e j i ) ,η = η h G ⊗ e j | G ⊗ e j i = ˜ ξ h G ⊗ e j | G ⊗ e j i = ˜ ξ . c) Let ( a, η ) ∈ ¨ G × M . By b), there is a ξ ∈ L with ˜ ξ = η . By a), for ξ ∈ L , D ˜ ξ , ^ ( a, η ) E = D D ˜ ξ (cid:12)(cid:12)(cid:12) η E , a E = D D ˜ ξ (cid:12)(cid:12)(cid:12) ˜ ξ E , a E == h ϕ h ξ | ξ i , a i = h h ξ | ξ i , ¨ ϕa i = D ξ , ^ ( ¨ ϕa, ξ ) E . We put θ : L −→ M , ξ ˜ ξ and denote by θ ′ : M ′ → L ′ its transpose. By the above, θ ′ ^ ( a, η ) ∈ ¨ L . Since θ ′ is continuous, θ ′ ( ¨ M ) ⊂ ¨ L and this proves the assertion.37 ROPOSITION 1.3.12
We use the notation of
Lemma 1.3.11 .a) If X ∈ L F ( L ) and ξ ∈ L with ˜ ξ = 0 then f Xξ = 0 ; we define ˜ X : M −→ M , η f Xξ , where ξ ∈ L with ˜ ξ = η (Lemma 1.3.11 b)) .b) For every X ∈ L F ( L ) , ˜ X belongs to L G ( M ) and the map L F ( L ) −→ L G ( M ) , X ˜ X is a surjective C**-homomorphism continuous with respect to the topolo-gies T k with k ∈ { , , } .c) For ξ, η ∈ L , ^ z }| { η h · | ξ i = ˜ η D · | ˜ ξ E and K G ( M ) = n ˜ X (cid:12)(cid:12)(cid:12) X ∈ K F ( L ) o . a) For i ∈ I , ϕξ i = ˜ ξ i = 0 so by Lemma 1.3.11 a), ^ X ( e i ξ i ) = ^ ( Xe i ) ξ i = ^ ( Xe i ) ϕξ i = 0 . By [C1] Proposition 5.6.4.1 e) (resp. [C1] Proposition 5.6.4.6 c) and [C1] Propo-sition 5.6.3.4 c)), Xξ = X X i ∈ I e i ξ i ! = X i ∈ I X ( e i ξ i ) resp. Xξ = X ¨ L X i ∈ I e i ξ i = ¨ L X i ∈ I X ( e i ξ i ) so by Lemma 1.3.11 a) (resp. c)), f Xξ = ^ z }| {X i ∈ I X ( e i ξ i ) = X i ∈ I ^ X ( e i ξ i ) = 038 resp. f Xξ = ^ z }| { ¨ L X i ∈ I X ( e i ξ i ) = ¨ M X i ∈ I ^ X ( e i ξ i ) = 0 . b) For X, Y ∈ L F ( L ) and ξ, η ∈ L , by Lemma 1.3.11 a), D ˜ X ˜ ξ (cid:12)(cid:12)(cid:12) ˜ η E = D f Xξ (cid:12)(cid:12)(cid:12) ˜ η E = ϕ h Xξ | η i == ϕ h ξ | X ∗ η i = D ˜ ξ (cid:12)(cid:12)(cid:12) g X ∗ η E = D ˜ ξ (cid:12)(cid:12)(cid:12) f X ∗ ˜ η E , ˜ X ˜ Y ˜ ξ = ˜ X f Y ξ = ^ X ( Y ξ ) = ^ ( XY ) ξ = g XY ˜ ξ . By Lemma 1.3.11 b), ˜ X ∈ L G ( M ), ( ˜ X ) ∗ = ˜ X ∗ , and ˜ X ˜ Y = g XY , i.e. the mapis a C*-homomorphism.For X ∈ L F ( L ) and ξ, η ∈ L (resp. and a ∈ ¨ M + ), by Lemma 1.3.11 a), p ˜ ξ, ˜ η ( ˜ X ) = (cid:13)(cid:13)(cid:13)D ˜ X ˜ ξ (cid:12)(cid:12)(cid:12) ˜ η E(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)D f Xξ (cid:12)(cid:12)(cid:12) ˜ η E(cid:13)(cid:13)(cid:13) = k ϕ h Xξ | η ik ≤ p ξ,η ( X )(resp. p ˜ ξ, ˜ η,a ( X ) = (cid:12)(cid:12)(cid:12)D D ˜ X ˜ ξ (cid:12)(cid:12)(cid:12) ˜ η E , a E(cid:12)(cid:12)(cid:12) = | h ϕ h Xξ | η i , a i | == | h h Xξ | η i , ¨ ϕa i | = p ξ,η, ¨ ϕa ( X )) , so by Lemma 1.3.11 b), the map is continuous with respect to the topology T .The proof for the other topologies is similar.c) For ζ ∈ L , by Lemma 1.3.11 a), ^ z }| { η h · | ξ i ˜ ζ = ^ z }| { ( η h · | ξ i ) ζ = ^ z }| { η h ζ | ξ i == ˜ η ϕ h ζ | ξ i = ˜ η D ˜ ζ (cid:12)(cid:12)(cid:12) ˜ ξ E = (cid:16) ˜ η D · | ˜ ξ E(cid:17) ˜ ζ so by Lemma 1.3.11 b), ^ z }| { η h · | ξ i = ˜ η D · | ˜ ξ E . The last assertion follows now from b).39
Main Part
Throughout this section we fix f ∈ F ( T, E ) We present here the projective representation of the groups and its mainproperties.
DEFINITION 2.1.1
We put for every t ∈ T and ξ ∈ H , u t : ˘ E −→ H , ζ ζ ⊗ e t ,V t ξ : T −→ ˘ E , s f ( t, t − s ) ξ ( t − s ) . If we want to emphasize the role of f then we put V ft instead of V t . For x ∈ E , ( x e ⊗ K ) V t ξ : T −→ ˘ E , s f ( t, t − s ) xξ ( t − s ) . PROPOSITION 2.1.2
Let s, t ∈ T , x ∈ E , ζ ∈ ˘ E , and ξ ∈ H .a) V t ξ ∈ H .b) V s V t = ( f ( s, t ) e ⊗ K ) V st .c) V t ( ζ ⊗ e s ) = ( f ( t, s ) ζ ) ⊗ e ts .d) V t ( x e ⊗ K ) = ( x e ⊗ K ) V t .e) V t ∈ U n L E ( H ) , V ∗ t = ( ˜ f ( t ) e ⊗ K ) V t − .f ) ( x e ⊗ K ) V t ( ζ ⊗ e s ) = ( f ( t, s ) xζ ) ⊗ e ts .g) If T is infinite and F denotes the filter on T of cofinite subsets, i.e. F := { S | S ∈ P ( T ) , T \ S ∈ P f ( T ) } , then lim t, F V t = 0 in L E ( H ) T .
40) For R ∈ P f ( T ), X r ∈ R h ( V t ξ ) r | ( V t ξ ) r i = X r ∈ R (cid:10) f ( t, t − r ) ξ t − r (cid:12)(cid:12) f ( t, t − r ) ξ t − r (cid:11) == X r ∈ R h ξ t − r | ξ t − r i = X r ∈ R h ξ r | ξ r i ≤ h ξ | ξ i so V t ξ ∈ H .b) For r ∈ T ,( V s V t ξ ) r = f ( s, s − r )( V t ξ ) s − r = f ( s, s − r ) f ( t, t − s − r ) ξ t − s − r == f ( s, t ) f ( st, t − s − r ) ξ t − s − r = f ( s, t )( V st ξ ) r = (( f ( s, t ) e ⊗ K ) V st ξ ) r so V s V t = ( f ( s, t ) e ⊗ K ) V st . c) For r ∈ T , ( V t ( ζ ⊗ e s )) r = f ( t, t − r )( ζ ⊗ e s ) t − r == δ s,t − r f ( t, t − r ) ζ = δ r,ts f ( t, s ) ζ = (( f ( t, s ) ζ ) ⊗ e ts ) r so V t ( ζ ⊗ e s ) = ( f ( t, s ) ζ ) ⊗ e ts . d) We have( V t ( x e ⊗ K ) ξ ) s = f ( t, t − s )(( x e ⊗ K ) ξ ) t − s = f ( t, t − s ) xξ t − s = (( x e ⊗ K ) V t ξ ) s so V t ( x e ⊗ K ) = ( x e ⊗ K ) V t . e) For η ∈ H , by Proposition 1.1.2 a),b), h V t ξ | η i = gX s ∈ T h ( V t ξ ) s | η s i = gX s ∈ T (cid:10) f ( t, t − s ) ξ t − s (cid:12)(cid:12) η s (cid:11) == gX r ∈ T h f ( t, r ) ξ r | η tr i = gX r ∈ T D ξ r | ˜ f ( t ) f ( t − , tr ) η tr E =41 gX r ∈ T D ξ r | ((( ˜ f ( t ) e ⊗ K ) V t − ) η ) r E = D ξ | (( ˜ f ( t ) e ⊗ K ) V t − ) η E so V t ∈ L E ( H ) with V ∗ t = ( ˜ f ( t ) e ⊗ K ) V t − . By b) and d), V ∗ t V t = ( ˜ f ( t ) e ⊗ K ) V t − V t = ( ˜ f ( t ) e ⊗ K )( f ( t − , t ) e ⊗ K ) V t − t = id H ,V t V ∗ t = V t ( ˜ f ( t ) e ⊗ K ) V t − = ( ˜ f ( t ) e ⊗ K ) V t V t − == ( ˜ f ( t ) e ⊗ K )( f ( t, t − ) e ⊗ K ) V tt − = id H . f) follows from c).g) Let us consider first the C*-case. Let ξ, η ∈ H , t ∈ T , and ε >
0. Thereis an S ∈ P f ( T ) such that (cid:13)(cid:13) ηe T \ S (cid:13)(cid:13) < ε . By e), | (cid:10) V t ξ | ηe T \ S (cid:11) | ≤ k V t ξ k (cid:13)(cid:13) ηe T \ S (cid:13)(cid:13) ≤ ε k ξ k so p ξ,η ( V t ) = | h V t ξ | η i | ≤ | h V t ξ | ηe S i | + | (cid:10) V t ξ | ηe T \ S (cid:11) | < | h V t ξ | ηe S i | + ε . From h V t ξ | ηe S i = X s ∈ S η ∗ s f ( t, t − s ) ξ t − s it follows lim t, F h V t ξ | ηe S i = 0 , lim t, F p ξ,η ( V t ) = 0 . The W*-case can be proved similarly.
Remark.
By e), T cannot be replaced by T in g). PROPOSITION 2.1.3
Let s, t ∈ T .a) u t ∈ L E ( ˘ E, H ) , u ∗ t = h · | E ⊗ e t i .b) u ∗ s u t = δ s,t E .c) u s u ∗ t = 1 E e ⊗ ( h · | e t i e s ) . ) T P r ∈ T u r u ∗ r = id H . a) For ζ ∈ ˘ E and ξ ∈ H , h u t ζ | ξ i = h ζ ⊗ e t | ξ i = gX s ∈ T ξ ∗ s ( ζ ⊗ e t ) s = ξ ∗ t ζ = h ζ | ξ t i so u t ∈ L E ( ˘ E, H ) , u ∗ t ξ = ξ t = h ξ | E ⊗ e t i . b) For ζ ∈ ˘ E , by a), u ∗ s u t ζ = u ∗ s ( ζ ⊗ e t ) = h ζ ⊗ e t | E ⊗ e s i = δ s,t ζ so u ∗ s u t = δ s,t E .c) For ζ ∈ ˘ E and r ∈ T , by a), u s u ∗ t ( ζ ⊗ e r ) = u s δ r,t ζ = δ r,t ( ζ ⊗ e s ) == ζ ⊗ h e r | e t i e s = (1 E e ⊗ ( h · | e t i e s ))( ζ ⊗ e r ) , so (by a) and [C1] Proposition 5.6.4.1 e) (resp. and [C1] Proposition 5.6.4.6 c),[C1] Proposition 5.6.3.4 c))) u s u ∗ t = 1 E e ⊗ ( h · | e t i e s ).d) For ξ ∈ H (resp. and a ∈ ¨ E + ) and S ∈ P f ( T ), by c), p ξ X t ∈ S u t u ∗ t − id H ! = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X t ∈ T \ S h ξ | ξ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / resp. p ξ,a X t ∈ S u t u ∗ t − id H ! == * * X t ∈ S ( u t u ∗ t − id H ) ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ∈ S ( u t u ∗ t − id H ) ξ + , a + / == ( X t ∈ T \ S h h ξ | ξ i , a i ) / ! and the assertion follows. 43 ROPOSITION 2.1.4
Let s, t ∈ T and x ∈ E .a) V s u t = u st f ( s, t ) .b) u ∗ s V t = f ( t, t − s ) u ∗ t − s .c) ( x e ⊗ K ) u t = u t x .d) xu ∗ t = u ∗ t ( x e ⊗ K ) . a) For ζ ∈ ˘ E , by Proposition 2.1.2 c), V s u t ζ = V s ( ζ ⊗ e t ) = ( f ( s, t ) ζ ) ⊗ e st = u st f ( s, t ) ζ so V s u t = u st f ( s, t ) . b) For ζ ∈ ˘ E and r ∈ T , by Proposition 2.1.2 c) and Proposition 2.1.3 a), u ∗ s V t ( ζ ⊗ e r ) = u ∗ s (( f ( t, r ) ζ ) ⊗ e tr ) = δ s,tr f ( t, r ) ζ == δ t − s,r f ( t, t − s ) ζ = f ( t, t − s ) u ∗ t − s ( ζ ⊗ e r )so u ∗ s V t = f ( t, t − s ) u ∗ t − s .c) For ζ ∈ ˘ E ,( x e ⊗ K ) u t ζ = ( x e ⊗ K )( ζ ⊗ e t ) = ( xζ ) ⊗ e t = u t xζ so ( x e ⊗ K ) u t = u t x .d) follows from c). DEFINITION 2.1.5
We put for all s, t ∈ T (Proposition 2.1.3 a)) ϕ s,t : L E ( H ) −→ L E ( ˘ E ) ≈ E , X u ∗ s Xu t and set X t := ϕ t, X for every X ∈ L E ( H ) . PROPOSITION 2.1.6
Let s, t ∈ T . ) ϕ s,t is linear with k ϕ s,t k = 1 .b) For X ∈ L E ( H ) and x, y ∈ ˘ E , h ( ϕ s,t X ) x | y i = h X ( x ⊗ e t ) | y ⊗ e s i . c) The map ϕ s,t : L E ( H ) T −→ E (resp. E ¨ E ) is continuous.d) ϕ t,t is involutive and completely positive.e) For r ∈ T and x ∈ E , ϕ s,t (( x e ⊗ K ) V r ) = δ s,rt f ( r, t ) x . f ) If ( x r ) r ∈ T ∈ E ( T ) and X := X r ∈ T ( x r e ⊗ K ) V r then ϕ s,t X = f ( st − , t ) x st − , X t = x t . g) For X ∈ L E ( H ) and x, y ∈ E , ϕ s,t (( x e ⊗ K ) X ( y e ⊗ K )) = x ( ϕ s,t X ) y , (( x e ⊗ K ) X ( y e ⊗ K )) t = xX t y . a) follows from Proposition 2.1.3 a),b).b) We have h ( ϕ s,t X ) x | y i = h u ∗ s Xu t x | y i = h Xu t x | u s y i = h X ( x ⊗ e t ) | y ⊗ e s i . c) The C*-case
45y b), for X ∈ L E ( H ), k ϕ s,t X k = kh ( ϕ s,t X )1 E | E ik == kh X (1 E ⊗ e t ) | E ⊗ e s ik = p E ⊗ e t , E ⊗ e s ( X ) . The W*-case
Let a ∈ ¨ E and let a = x | a | be its polar representation. By b), for X ∈L E ( H ), | h ϕ s,t X , a i | = | h h ( ϕ s,t X )1 E | E i , x | a | i | = | h h ( ϕ s,t X ) x | E i , | a | i | == | h h X ( x ⊗ e t ) | E ⊗ e s i , | a | i | = p x ⊗ e t , E ⊗ e s , | a | ( X ) . d) For X ∈ L E ( H ),( ϕ t,t X ) ∗ = ( u ∗ t Xu t ) ∗ = u ∗ t X ∗ u t = ϕ t,t ( X ∗ )so ϕ t,t is involutive. For n ∈ IN, X ∈ (( L E ( H )) n,n ) + , and ζ ∈ ˘ E n , X i ∈ IN n * X j ∈ IN n (( ϕ t,t X ij ) ζ j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ i + = X i,j ∈ IN n h u ∗ t X ij u t ζ j | ζ i i == X i,j ∈ IN n h X ij u t ζ j | u t ζ i i ≥ c ⇒ c ) so ϕ t,t is com-pletely positive ([C1] Theorem 5.6.6.1 f) and [C1] Theorem 5.6.1.11 c ⇒ c ).e) By Proposition 2.1.4 a),d) and Proposition 2.1.3 b), ϕ s,t (( x e ⊗ K ) V r ) = u ∗ s ( x e ⊗ K ) V r u t = xu ∗ s V r u t = xu ∗ s u rt f ( r, t ) = δ s,rt f ( r, t ) x . f) By e) (and Proposition 1.1.2 a)), ϕ s,t X = X r ∈ T ϕ s,t (( x r e ⊗ K ) V r ) = X r ∈ T δ s,rt f ( r, t ) x r = f ( st − , t ) x st − , t = ϕ t, X = f ( t, x t = x t . g) By Proposition 2.1.4 c),d), ϕ s,t (( x e ⊗ K ) X ( y e ⊗ K )) = u ∗ s ( x e ⊗ K ) X ( y e ⊗ K ) u t == xu ∗ s Xu t y = x ( ϕ s,t X ) y . DEFINITION 2.1.7
We put R ( f ) := ( X t ∈ T ( x t e ⊗ K ) V t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x t ) t ∈ T ∈ E ( T ) ) , S ( f ) := T R ( f ) , S k·k ( f ) := k·k R ( f ) . Moreover we put S C ( f ) := S ( f ) in the C*-case and S W ( f ) := S ( f ) in theW*-case. If F is a subset of E then we put S ( f, F ) := { X ∈ S ( f ) | t ∈ T = ⇒ X t ∈ F } and use similar notation for the other S . By Proposition 2.1.2 b),d),e), R ( f ) is an involutive unital E -subalgebra of L E ( H ) (with V as unit). In particular S k·k ( f ) is an E -C*-subalgebra of L E ( H ).If T is finite then R ( f ) = S ( f ). By Corollary 1.3.7 e), S C ( f ) T is complete. PROPOSITION 2.1.8
For X ∈ T R ( f ) and s, t ∈ T , ϕ s,t X = f ( st − , t ) X st − . Let F be a filter on R ( f ) converging to X in the T -topology. By Proposition2.1.6 c),f) (and Corollary 1.3.7 d)), ϕ s,t X = lim Y, F ϕ s,t Y = lim Y, F f ( st − , t ) Y st − = f ( st − , t ) lim Y, F Y st − == f ( st − , t ) lim Y, F ϕ st − , Y = f ( st − , t ) ϕ st − , X = f ( st − , t ) X st − . HEOREM 2.1.9
Let X ∈ T R ( f ) .a) If ( x t ) t ∈ T is a family in E such that X = T X t ∈ T ( x t e ⊗ K ) V t then X t = x t for every t ∈ T . In particular, if T is finite then the map E T −→ S ( f ) , x X t ∈ T ( x t ⊗ K ) V t is bijective and E -linear (Proposition 2.1.2 d)) .b) We have X = T X t ∈ T ( X t e ⊗ K ) V t ∈ S ( f ) . c) ( X ∗ ) t = ˜ f ( t )( X t − ) ∗ for every t ∈ T and X ∗ = T X t ∈ T (( X t ) ∗ e ⊗ K ) V ∗ t ∈ T R ( f ) . d) S ( f ) = T R ( f )= T R ( f ) .e) For ξ ∈ H and t ∈ T , ( Xξ ) t = gX s ∈ T f ( s, s − t ) X s ξ s − t . f ) If T is finite and if we identify L E ( H ) with E T,T then X is identified withthe matrix [ f ( st − , t ) X st − ] s,t ∈ T , and for every r ∈ T , V r is identified with the matrix [ f ( st − , t ) δ s,rt ] s,t ∈ T . ) If X, Y ∈ S ( f ) and t ∈ T then XY ∈ S ( f ) and ( XY ) t = gX s ∈ T f ( s, s − t ) X s Y s − t , ( X ∗ Y ) t = gX s ∈ T f ( s, t ) ∗ X ∗ s Y st , ( XY ∗ ) t = gX s ∈ T f ( t, s ) ∗ X ts Y ∗ s , ( X ∗ Y ) = gX s ∈ T X ∗ s Y s , ( XY ∗ ) = gX s ∈ T X s Y ∗ s . h) The map E −→ S ( f ) , x x e ⊗ K is an injective unital C**-homomorphism and so S ( f ) is an E -C**-subalge-bra of L E ( H ) and Re S ( f ) is closed in S ( f ) T . In the W*-case, S W ( f ) is the W*-subalgebra of L E ( H ) generated by R ( f ) and R ( f ) is dense in S W ( f ) T = S W ( f ) ... H , which is compact.i) If E is a W*-algebra then S C ( f ) may be identified canonically with aunital C*-subalgebra of S W ( f ) by using the map of Proposition 1.3.9 b).By this identification S C ( f ) generates S W ( f ) as W*-algebra.j) If F is a closed ideal of E (resp. of E ¨ E ) then S ( f, F ) is a closed ideal of S ( f ) ( resp. of S ( f ) .. z }| { S ( f ) ) .k) If F is a unital C**-subalgebra of E such that f ( s, t ) ∈ F for all s, t ∈ T then S ( f, F ) is a unital C**-subalgebra of S ( f ) and the map S ( f, F ) −→ S ( g ) , X T X t ∈ T ( X t e ⊗ K ) V gt is an injective C**-homomorphism, where g : T × T −→ U n F c , ( s, t ) f ( s, t ) . This map induces a C*-isomorphism S k·k ( f, F ) → S k·k ( g ) .l) ( X, Y ) ∈ ◦ z }| { S ( f ) + = ⇒ ( X , Y ) ∈ ◦ E + .
49) By Proposition 2.1.6 c),e), X t = ϕ t, X = gX s ∈ T ϕ t, (( x s e ⊗ K ) V s ) = gX s ∈ T δ t,s f ( s, x s = x t . b&c&d Step 1 X = T P t ∈ T ( X t e ⊗ K ) V t By Proposition 2.1.3 d), Corollary 1.3.7 d), Proposition 2.1.8, and Proposi-tion 2.1.4 b),d), X = T X s ∈ T u s u ∗ s ! X T X t ∈ T u t u ∗ t ! = T X s ∈ T T X t ∈ T u s u ∗ s Xu t u ∗ t == T X s ∈ T T X t ∈ T u s ( ϕ s,t X ) u ∗ t = T X s ∈ T T X t ∈ T u s f ( st − , t ) X st − u ∗ t == T X s ∈ T T X r ∈ T u s X r f ( r, r − s ) u ∗ r − s = T X s ∈ T T X r ∈ T u s X r u ∗ s V r == T X s ∈ T T X r ∈ T u s u ∗ s ( X r e ⊗ K ) V r = T X s ∈ T u s u ∗ s T X t ∈ T ( X t e ⊗ K ) V t ! = T X t ∈ T ( X t e ⊗ K ) V t . Step 2 b&c&dBy Step 1, Corollary 1.3.7 a), and Proposition 2.1.2 d),e) (and Proposition1.1.2 a)), X ∗ = T X s ∈ T ( X s e ⊗ K ) V s ! ∗ = T X s ∈ T ( X ∗ s e ⊗ K ) V ∗ s == T X s ∈ T ( X ∗ s e ⊗ K )( ˜ f ( s ) e ⊗ K ) V s − = T X r ∈ T (( ˜ f ( r ) X ∗ r − ) e ⊗ K ) V r ∈ T R ( f ) .
50y a), ( X ∗ ) t = ˜ f ( t )( X t − ) ∗ . By Step 1 and Proposition 2.1.2 e) (and Proposition 1.1.2 a)), X ∗ = T X t ∈ T (( X ∗ ) t e ⊗ K ) V t = T X t ∈ T (( X t − ) ∗ e ⊗ K )( ˜ f ( t ) e ⊗ K ) V t == T X t ∈ T (( X t − ) ∗ e ⊗ K ) V ∗ t − = T X t ∈ T (( X t ) ∗ e ⊗ K ) V ∗ t . Together with Step 1 this proves X = T X t ∈ T ( X t e ⊗ K ) V t ∈ S ( f ) , X ∗ = T X t ∈ T (( X t ) ∗ e ⊗ K ) V ∗ t ∈ S ( f ) . In particular S ( f ) = T R ( f )= T R ( f ).e) By b) and Corollary 1.3.7 b), in the C*-case,( Xξ ) t = * T X s ∈ T ( X s e ⊗ K ) V s ! ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ⊗ e t + = X s ∈ T (cid:10) ( X s e ⊗ K ) V s ξ (cid:12)(cid:12) E ⊗ e t (cid:11) == X s ∈ T X s f ( s, s − t ) ξ s − t = X s ∈ T f ( s, s − t ) X s ξ s − t . The proof is similar in the W*-case.f) For ξ ∈ H and s ∈ T , by e),( Xξ ) s = X t ∈ T f ( t, t − s ) X t ξ t − s = X r ∈ T f ( sr − , r ) X sr − ξ r . g) By b), Corollary 1.3.7 b),d), and Proposition 2.1.2 b),d), XY = T X s ∈ T ( X s e ⊗ K ) V s ! T X t ∈ T ( X t e ⊗ K ) V t ! == T X s ∈ T T X t ∈ T ( X s e ⊗ K ) V s ( Y t e ⊗ K ) V t = T X s ∈ T T X t ∈ T ( X s e ⊗ K )( Y t e ⊗ K ) V s V t =51 T X s ∈ T T X t ∈ T ( X s e ⊗ K )( Y t e ⊗ K )( f ( s, t ) e ⊗ K ) V st == T X s ∈ T T X r ∈ T (( f ( s, s − r ) X s Y s − r ) e ⊗ K ) V r . Since by d), T X r ∈ T (( f ( s, s − r ) X s Y s − r ) e ⊗ K ) V r ∈ S ( f )for every s ∈ T we get XY ∈ S ( f ), again by d). By Corollary 1.3.7 b) andProposition 2.1.6 c),e),( XY ) t = ϕ t, ( XY ) = gX s ∈ T gX r ∈ T ϕ t, (( f ( s, s − r ) X s Y s − r ) e ⊗ K ) V r == gX s ∈ T gX r ∈ T δ t,r f ( r, f ( s, s − r ) X s Y s − r = gX s ∈ T f ( s, s − t ) X s Y s − t . By the above, c), and Proposition 1.1.2 b),( X ∗ Y ) t = gX s ∈ T f ( s, s − t )( X ∗ ) s Y s − t = gX s ∈ T f ( s, s − t ) ˜ f ( s )( X s − ) ∗ Y s − t == gX s ∈ T f ( s − , t ) ∗ ( X s − ) ∗ Y s − t = gX s ∈ T f ( s, t ) ∗ X ∗ s Y st , ( XY ∗ ) t = gX s ∈ T f ( s, s − t ) X s ( Y ∗ ) s − t = gX s ∈ T f ( s, s − t ) X s ˜ f ( s − t )( Y t − s ) ∗ == gX s ∈ T f ( t, t − s ) ∗ X s ( Y t − s ) ∗ = gX s ∈ T f ( t, s ) ∗ X ts Y ∗ s . It follows by Proposition 1.1.2 a),( X ∗ Y ) = gX s ∈ T X ∗ s Y s , ( XY ∗ ) = gX s ∈ T X s Y ∗ s . h) By c) and g), S ( f ) is an involutive unital subalgebra of L E ( H ). Be-ing closed (resp. closed in L E ( H ) ... H (d) and Corollary 1.3.7 c))) it is a C**-subalgebra of L E ( H ) (resp. generated by R ( f ) [C1] Theorem 5.6.3.5 b) and [C1]52orollary 4.4.4.12 a) and by [C1] Corollary 6.3.8.7 R ( f ) is dense in S W ( f ) T ,which is compact by Corollary 1.3.7 c)). The assertion concerning E followsfrom Proposition 2.1.2 d) and Lemma 1.3.2 c). By Corollary 1.3.7 a), Re S ( f )is a closed set of S ( f ) T .i) The assertion follows from h), Proposition 1.3.9 b), and Lemma 1.3.8 c ) ⇒ a ).j) For X ∈ S ( f, F ), Y ∈ S ( f ), and t ∈ T , by g), ( XY ) t , ( Y X ) t ∈ S ( f, F )so S ( f, F ) is an ideal of S ( f ). The closure properties follow from Proposition2.1.6 c).k) By c) and g), S ( f, F ) is a unital involutive subalgebra of S ( f ) and byProposition 2.1.6 c), S ( f, F ) is a C**-subalgebra of S ( f ). The last assertionfollows from the fact that the image of the map contains R ( g ).l) There are U, V ∈ S ( f ) with( X, Y ) = (
U, V ) ∗ ( U, V ) = ( U ∗ , − V ∗ )( U, V ) = ( U ∗ U + V ∗ V, U ∗ V − V ∗ U ) . For t ∈ T ,0 ≤ ( U t , V t ) ∗ ( U t , V t ) = ( U ∗ t , − V ∗ t )( U t , V t ) = ( U ∗ t U t + V ∗ t V t , U ∗ t V t − V ∗ t U t ) . By g), X = ( U ∗ U + V ∗ V ) = gX t ∈ T ( U ∗ t U t + V ∗ t V t ) ,Y = ( U ∗ V − V ∗ U ) = gX t ∈ T ( U ∗ t V t − V ∗ t U t )so ( X , Y ) = gX t ∈ T ( U ∗ t U t + V ∗ t V t , U ∗ t V t − V ∗ t U t ) ∈ ◦ E + . Remark.
It may happen that by the identification of i), S C ( f ) = S W ( f )(Remark of Proposition 2.1.23). COROLLARY 2.1.10 ) If ( x t ) t ∈ T is a family in E such that ( k x t k ) t ∈ T is summable then (( x t e ⊗ K ) V t ) t ∈ T is norm summable in L E ( H ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ T ( x t e ⊗ K ) V t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X t ∈ T k x t k . b) The set A := ( X ∈ S ( f ) | X t ∈ T k X t k < ∞ ) is a dense involutive unital subalgebra of S k·k ( f ) with X t ∈ T k ( X ∗ ) t k = X t ∈ T k X t k , X t ∈ T k ( XY ) t k ≤ X t ∈ T k X t k ! X t ∈ T k Y t k ! for all X, Y ∈ A .c) A endowed with the norm A −→ IR + , X X t ∈ T k X t k is an involutive Banach algebra and S k·k ( f ) is its C*-hull. a) For S ∈ P f ( T ), by Proposition 2.1.2 e), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ S ( x t e ⊗ K ) V t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X t ∈ S (cid:13)(cid:13) x t e ⊗ K (cid:13)(cid:13) k V t k = X t ∈ S k x t k and the assertion follows.b) By Theorem 2.1.9 c), X ∗ ∈ S ( f ) and k ( X ∗ ) t k = k ( X t − ) ∗ k = k X t − k t ∈ T so X t ∈ T k ( X ∗ ) t k = X t ∈ T k X t − k = X t ∈ T k X t k . By Theorem 2.1.9 g), XY ∈ S ( f ) and k ( XY ) t k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)gX s ∈ T f ( s, s − t ) X s Y s − t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X s ∈ T k X s k k Y s − t k for every t ∈ T so X t ∈ T k ( XY ) t k ≤ X t ∈ T X s ∈ T k X s k k Y s − t k = X s ∈ T k X s k X t ∈ T k Y s − t k ! == X s ∈ T k X s k X t ∈ T k Y t k ! = X t ∈ T k X t k ! X t ∈ T k Y t k ! . c) is easy to see. Remark.
There may exist X ∈ S k·k ( f ) for which (( X t e ⊗ K ) V t ) t ∈ T is notnorm summable, as it is known from the theory of trigonometric series (seeProposition 3.5.1). In particular the inclusion A ⊂ S k·k ( f ) may be strict. COROLLARY 2.1.11
Let F be a unital C**-algebra and τ : E → F a posi-tive continuous (resp. W*-continuous) unital trace.a) τ ◦ ϕ , is a positive continuous (resp. W*-continuous) unital trace.b) If τ is faithful then τ ◦ ϕ , is faithful and V is finite.c) In the W*-case, S W ( f ) is finite iff E is finite. a) Let X, Y ∈ S ( f ). By Theorem 2.1.9 g) (and Proposition 1.1.2 a)), τ ϕ , ( XY ) = τ gX t ∈ T f ( t, t − ) X t Y t − ! = τ gX t ∈ T f ( t, t − ) X t − Y t ! =55 gX t ∈ T τ ( f ( t, t − ) X t − Y t ) = gX t ∈ T τ ( f ( t, t − ) Y t X t − ) = τ gX t ∈ T f ( t, t − ) Y t X t − ! == τ ϕ , ( Y X ) . Thus τ ◦ ϕ , is a trace which is obviously positive, continuous (resp. W*-continuous), and unital (Proposition 2.1.6 c),d)).b) By Theorem 2.1.9 g), ϕ , is faithful, so τ ◦ ϕ is also faithful. Let X ∈ S ( f )with X ∗ X = V . By a), τ ϕ , ( XX ∗ ) = τ ◦ ϕ , ( X ∗ X ) = τ ϕ , V = 1 F so τ ϕ , ( V − XX ∗ ) = 1 F − F = 0 , V = XX ∗ , and V is finite.c) By b), if E is finite then S W ( f ) is also finite. The reverse implicationfollows from the fact that E ¯ ⊗ K is a unital W*-subalgebra of S W ( f ) (Theorem2.1.9 h)). COROLLARY 2.1.12
Assume T finite and for every x ′ ∈ ( E ′ ) T put e x ′ : S ( f ) −→ IK , X X t ∈ T (cid:10) X t , x ′ t (cid:11) . a) e x ′ ∈ S ( f ) ′ and sup t ∈ T (cid:13)(cid:13) x ′ t (cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) e x ′ (cid:13)(cid:13)(cid:13) ≤ X t ∈ T (cid:13)(cid:13) x ′ t (cid:13)(cid:13) for every x ′ ∈ ( E ′ ) T and the map ϕ : ( E ′ ) T −→ S ( f ) ′ , x ′ e x ′ is an isomorphism of involutive vector spaces such that ϕ ( xx ′ ) = ( x ⊗ K )( ϕx ′ ) , ϕ ( x ′ x ) = ( ϕx ′ )( x ⊗ K ) ( [C1] Proposition 2.2.7.2 ) for every x ∈ E and x ′ ∈ ( E ′ ) T . ) If E is a W*-algebra then the map ψ : ( ¨ E ) T −→ .. z }| { S ( f ) , ( a t ) t ∈ T (˜ a t ) t ∈ T is an isomorphism of involutive vector spaces such that ψ ( xa ) = ( x ⊗ K )( ψa ) , ψ ( ax ) = ( ψa )( x ⊗ K ) for every x ∈ E and a ∈ ( ¨ E ) T . COROLLARY 2.1.13
Assume T finite and let M be a Hilbert right S ( f ) -module. M endowed with the right multiplication M × E −→ M, ( ξ, x ) ξ ( x ˜ ⊗ K ) and with the inner-product M × M −→ E, ( ξ, η ) ξ | η i is a Hilbert right E -module denoted by f M , L S ( f ) ( M ) is a unital C*-subalgebraof L E ( f M ) , and M is selfdual if f M is so. By Proposition 2.1.6 d),g) and Theorem 2.1.9 g),l), for
X, Y ∈ S ( f ) and x ∈ E , ϕ , ( X ( x ˜ ⊗ K )) = ( ϕ , X ) x , X ≥ ⇒ ϕ , X ≥ , ( X, Y ) ∈ ◦ z }| { S ( f ) + = ⇒ ( ϕ , X, ϕ , Y ) ∈ ◦ E + , inf { k ϕ , X k | X ∈ S ( f ) + , k X k = 1 } > COROLLARY 2.1.14
Let n ∈ IN and let ϕ : S ( f ) → E n,n be an E -C*-homomorphism. Then ( ϕV t ) i,j ∈ E c for all t ∈ T and all i, j ∈ IN n . For x ∈ E , by Proposition 2.1.2 d) and Theorem 2.1.9 h), x ( ϕV t ) = ϕ ( x e ⊗ K )( ϕV t ) = ϕ (( x e ⊗ K ) V t ) == ϕ ( V t ( x e ⊗ K )) = ( ϕV t ) ϕ ( x e ⊗ K ) = ( ϕV t ) x so ( ϕV t ) i,j ∈ E c . 57 OROLLARY 2.1.15
Let S be a group and g ∈ F ( S, S ( f )) . If we put h : ( T × S ) × ( T × S ) −→ U n S ( f ) c , (( t , s ) , ( t , s )) ( f ( t , t ) e ⊗ K ) g ( s , s ) then h ∈ F ( T × S, S ( f )) . The assertion follows from Theorem 2.1.9 h).
COROLLARY 2.1.16
Let X ∈ S ( f ) ( resp. X ∈ S k·k ( f )) .a) For every S ⊂ T , T X s ∈ S ( X s e ⊗ K ) V s ∈ S ( f ) (resp. k·k X s ∈ S ( X s e ⊗ K ) V s ∈ S k·k ( f )) and γ := sup ( (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ S ( X t e ⊗ K ) V t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ∈ P f ( T ) ) < ∞ . b) We put for every α ∈ l ∞ ( T ) αX : T −→ E , t α t X t . Then αX ∈ S ( f ) ( resp. αX ∈ S k·k ( f )) for every α ∈ l ∞ ( T ) and the map l ∞ ( T ) −→ S ( f ) (resp. S k·k ( f )) , α αX is norm-continuous.c) Assume E is a W*-algebra and let l ∞ ( T, E ) be the C*-direct product ofthe family ( E ) t ∈ T , which is a W*-algebra ([C1] Proposition 4.4.4.21 a)) .We put for every α ∈ l ∞ ( T, E ) , αX : T −→ E , t α t X t . Then αX ∈ S W ( f ) for every α ∈ l ∞ ( T, E ) and the map l ∞ ( T, E ) −→ S W ( f ) , α αX is continuous and W*-continuous.
58) In the C*-case the family (( X s ⊗ K ) V s ) s ∈ S is summable since S C ( f ) T is complete. By Banach-Steinhaus Theorem, γ is finite. In the W*-case thesummability follows now from Corollary 1.3.7 b),c) and Theorem 2.1.9 b).b) Let G be the vector subspace { α ∈ l ∞ ( T ) | α ( T ) is finite } of l ∞ ( T ). Bya), the map G −→ S ( f ) (resp. S k·k ( f )) , α αX is well-defined, linear, and continuous. The assertion follows by continuity.c) Let x ∈ E , S ⊂ T , and α := xe S . For ξ, η ∈ H and a ∈ ¨ E , by a) andLemma 1.3.2 b) (and Theorem 2.1.9 b)), * αX , ^ z }| { ( a, ξ, η ) + = h h αXξ | η i , a i = * ¨ E X t ∈ T η ∗ t x (( e S X ) ξ ) t , a + == X t ∈ T h x , (( e S X ) ξ ) t aη ∗ t i = * x , E X t ∈ T (( e S X ) ξ ) t aη ∗ t + . Let G be the involutive subalgebra { α ∈ l ∞ ( T, E ) | α ( T ) is finite } of l ∞ ( T, E )and let ¯ G be its norm-closure in l ∞ ( T, E ), which is a C*-subalgebra of l ∞ ( T, E ).By [C1] Proposition 4.4.4.21 a), G is dense in l ∞ ( T, E ) ¨ F , where F := l ∞ ( T, E ).Let α ∈ l ∞ ( T, E ) and let F be a filter on G converging to α in l ∞ ( T, E ) ¨ F ([C1] Corollary 6.3.8.7). By the above (and by Theorem 2.1.9 h)),lim β, F βX = αX in S W ( f ) .. z }| { S W ( f ) and so αX ∈ S W ( f ). The assertion follows. COROLLARY 2.1.17
Let S be a subgroup of T . Put f S := f | ( S × S ) , K S := l ( S ) , G := { X ∈ S ( f ) | t ∈ T \ S = ⇒ X t = 0 } . a) f S ∈ F ( S, E ) .b) G is an E -C**-subalgebra of S ( f ) . ) For every X ∈ G , the family (( X s e ⊗ K S ) V f S s ) s ∈ S is summable in L E ( K S ) T and the map ϕ : G −→ S ( f S ) , X T X s ∈ S ( X s e ⊗ K S ) V f S s is an injective E -C**-homomorphism.d) If X ∈ G ∩ S k·k ( f ) then ϕX ∈ S k·k ( f S ) and the map G ∩ S k·k ( f ) −→ S k·k ( f S ) , X ϕX is an E -C*-isomorphism.e) If S is finite then the map G −→ S ( f S ) , X X t ∈ S ( X t ⊗ K S ) V f S t is an E -C*-isomorphism. a) is obvious.b) By Theorem 2.1.9 c),g), G is an involutive unital subalgebra of S ( f ) andby Proposition 2.1.6 a) (resp. Proposition 2.1.6 c) and Corollary 1.3.7 c)) andTheorem 2.1.9 h), it is an E -C**-subalgebra of S ( f ).c) follows from Theorem 2.1.9 b) and Corollary 2.1.16 a).d) follows from c).e) is contained in d). DEFINITION 2.1.18
We denote by S T the set of finite subgroups of T andcall T locally finite if S T is upward directed and [ S ∈ S T S = T .
60 is locally finite iff the subgroups of T generated by finite subsets of T arefinite. COROLLARY 2.1.19
Assume T locally finite. We put f S := f | ( S × S ) for every S ∈ S T and identify S ( f S ) with { X ∈ S ( f ) | t ∈ T \ S ⇒ X t = 0 } (Corollary 2.1.17 e)) .a) For every X ∈ S k·k ( f ) and ε > there is an S ∈ S T such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ R ( X t ⊗ K ) V t − X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε for every R ∈ S T with S ⊂ R .b) S k·k ( f ) is the norm closure of ∪ s ∈ S T S ( f S ) and so it is canonically iso-morphic to the inductive limit of the inductive system { S ( f S ) | S ∈ S T } and for every S ∈ S T the inclusion map S ( f S ) → S k·k ( f ) is the associatedcanonical morphism. a) There is a Y ∈ R ( f ) with k X − Y k < ε . Let S ∈ S T with Y ∈ S ( f S ).By Corollary 2.1.17 b), for R ∈ S T with S ⊂ R , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ R (( X t − Y t ) e ⊗ K ) V t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k X − Y k < ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ R ( X t e ⊗ K ) V t − X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ R (( X t − Y t ) e ⊗ K ) V t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + k Y − X k < ε ε ε . b) follows from a). Remark.
The C*-algebras of the form S k·k ( f ) with T locally finite can beseen as a kind of AF- E -C*-algebras. PROPOSITION 2.1.20
The following are equivalent for all t ∈ T with t =1 and α ∈ U n E . ) ( V + ( α e ⊗ K ) V t ) ∈ P r S ( f ) .b) α = ˜ f ( t ) . By Proposition 2.1.2 b),d),e),( V t ) ∗ = ( ˜ f ( t ) e ⊗ K ) V t , ( V t ) = ( ˜ f ( t ) ∗ e ⊗ K ) V so 12 ( V + ( α e ⊗ K ) V t ) ∗ = 12 ( V + (( α ∗ ˜ f ( t )) e ⊗ K ) V t ) , (cid:18)
12 ( V + ( α e ⊗ K ) V t ) (cid:19) = 14 ((1 E + α ˜ f ( t ) ∗ ) e ⊗ K ) V + 12 ( α e ⊗ K ) V t . Thus a) is equivalent to α ∗ ˜ f ( t ) = α and α ˜ f ( t ) ∗ = 1 E , which is equivalent tob). COROLLARY 2.1.21
Let t ∈ T such that t = 1 and ˜ f ( t ) = 1 E . Then
12 ( V ± V t ) ∈ P r S ( f ) , ( V + V t )( V − V t ) = 0 . The assertion follows from Proposition 2.1.20.
COROLLARY 2.1.22
Let α, β ∈ U n E , s, t ∈ T with s = t = 1 , st = ts , γ := 12 ( α ∗ βf ( s, st ) ∗ + β ∗ αf ( t, st ) ∗ ) , γ ′ := 12 ( αβ ∗ f ( st, t ) ∗ + βα ∗ f ( st, s ) ∗ ) , and X := 12 (( α e ⊗ K ) V s + ( β e ⊗ K ) V t ) . a) f ( s, st ) f ( t, st ) = f ( st, t ) f ( st, s ) = ˜ f ( st ) ∗ .b) f ( st, t ) f ( s, st ) = f ( st, s ) f ( t, st ) .c) X ∗ X = ( V + ( γ e ⊗ K ) V st ) , XX ∗ = ( V + ( γ ′ e ⊗ K ) V st ) .d) The following are equivalent. ) X ∗ X ∈ P r S ( f ) . d ) XX ∗ ∈ P r S ( f ) . d ) α ∗ βf ( t, st ) = β ∗ αf ( s, st ) . d ) α ∗ βf ( st, t ) = β ∗ αf ( st, s ) . a) and b) follow from the equation of Schur functions (Definition1.1.1) andProposition 1.1.2 a).c) By Proposition 2.1.2 b),e) and Proposition 1.1.2 b), X ∗ = 12 ((( α ∗ ˜ f ( s )) e ⊗ K ) V s + (( β ∗ ˜ f ( t )) e ⊗ K ) V t ) ,X ∗ X = 12 V + 14 (( α ∗ β ˜ f ( s ) f ( s, t ) + β ∗ α ˜ f ( t ) f ( t, s )) e ⊗ K ) V st == 12 V + 14 (( α ∗ βf ( s, st ) ∗ + β ∗ αf ( t, st ) ∗ ) e ⊗ K ) V st = 12 ( V + ( γ e ⊗ K ) V st ) ,XX ∗ = 12 V + 14 (( αβ ∗ ˜ f ( t ) f ( s, t ) + βα ∗ ˜ f ( s ) f ( t, s )) e ⊗ K ) V st == 12 V + 14 (( αβ ∗ f ( st, t ) ∗ + βα ∗ f ( st, s ) ∗ ) e ⊗ K ) V st = 12 ( V + ( γ ′ e ⊗ K ) V st ) .d ⇔ d is known. d ⇔ d . By a), γ − ˜ f ( st ) = 14 ( α ∗ βα ∗ βf ( s, st ) ∗ + β ∗ αβ ∗ αf ( t, st ) ∗ + 2 f ( s, st ) ∗ f ( t, st ) ∗ ) −− f ( s, st ) ∗ f ( t, st ) ∗ = 14 ( α ∗ βf ( s, st ) ∗ − β ∗ αf ( t, st ) ∗ ) . By Proposition 2.1.20 d ) is equivalent to γ = ˜ f ( st ) so, by the above, since α ∗ βf ( s, st ) ∗ − β ∗ αf ( t, st ) ∗ is normal, it is equivalent to α ∗ βf ( s, st ) ∗ = β ∗ αf ( t, st ) ∗ or to β ∗ αf ( s, st ) = α ∗ βf ( t, st ) .d ⇔ d follows from b). 63 ROPOSITION 2.1.23
Let X ∈ S ( f ) .a) fP t ∈ T X ∗ t X t = ( X ∗ X ) , fP t ∈ T ( X t X ∗ t ) = ( XX ∗ ) .b) ( X t ) t ∈ T , ( X ∗ t ) t ∈ T ∈ g (cid:13)| t ∈ T ˘ E , k ( X t ) t ∈ T k ≤ k X k , k ( X ∗ t ) t ∈ T k ≤ k X k . c) If T is finite and f is constant then there is an X ∈ S ( f ) with k X k ≥ √ Card T k ( X t ) t ∈ T k , k X k ≥ √ Card T k ( X ∗ t ) t ∈ T k . d) If T is infinite and locally finite and f is constant then the map S ( f ) −→ g (cid:13)| t ∈ T ˘ E, X ( X t ) t ∈ T is not surjective. a) follows from Theorem 2.1.9 g).b) By a), ( X t ) t ∈ T , ( X ∗ t ) t ∈ T ∈ g (cid:13)| t ∈ T ˘ E and by Proposition 2.1.6 a), k ( X t ) t ∈ T k = k ϕ , ( X ∗ X ) k ≤ k X ∗ X k = k X k , k ( X ∗ t ) t ∈ T k = k ϕ , ( XX ∗ ) k ≤ k XX ∗ k = k X k . c) Let n := Card T and for every t ∈ T put X t := 1 E , ξ t := 1 E . Then k ( X t ) t ∈ T k = k ( X ∗ t ) t ∈ T k = n , k ( ξ t ) t ∈ T k = n . For t ∈ T , by Theorem 2.1.9 e),( Xξ ) t = X s ∈ T f ( s, s − t ) X s ξ s − t = n E h Xξ | Xξ i = n E , n k X k = k X k k ξ k ≥ k Xξ k = n , k X k ≥ n k ( X t ) t ∈ T k , k X k ≥ √ n k ( X t ) t ∈ T k . d) follows from c), Theorem 2.1.9 a), and the Principle of Inverse Operator. Remark. If E is a W*-algebra then it may exist a family ( x t ) t ∈ T in E suchthat the family (( x t e ⊗ K ) V t ) t ∈ T is summable in L E ( H ) T in the W*-case butnot in the C*-case as the following example shows. Take T := ZZ , f constant, E := l ∞ ( ZZ ), and x t := ( δ t,s ) s ∈ T ∈ E for every t ∈ T . By Proposition 2.1.23 b),(( x t ⊗ K ) V t ) t ∈ T is not summable in L E ( H ) T in the C*-case. In the W*-casefor ξ ∈ H and s, t ∈ T , h (( x t ¯ ⊗ K ) V t ξ ) s | (( x t ¯ ⊗ K ) V t ξ ) s i = e t | ξ s − t | , h ( x t ¯ ⊗ K ) V t ξ | ( x t ¯ ⊗ K ) V t ξ i = e t k ξ k . Thus X := T X t ∈ T ( x t ¯ ⊗ K ) V t ∈ S W ( f ) . Using the identification of Theorem 2.1.9 i), we get X ∈ S W ( f ) \ S C ( f ). COROLLARY 2.1.24
Let X ∈ S ( f ) .a) X ∈ (cid:8) x e ⊗ K (cid:12)(cid:12) x ∈ E (cid:9) c iff X t ∈ E c for all t ∈ T .b) X ∈ { V t | t ∈ T } c iff X s − ts = f ( s, s − ts ) ∗ f ( t, s ) X t = f ( s − , ts ) f ( t, s ) ˜ f ( s ) X t for all s, t ∈ T .c) X ∈ S ( f ) c iff for all s, t ∈ TX t ∈ E c , X s − ts = f ( s, s − ts ) ∗ f ( t, s ) X t = f ( s − , ts ) f ( t, s ) ˜ f ( s ) X t . In particular if f ( s, t ) = f ( t, s ) for all s, t ∈ T then X ∈ S ( f ) c iff X t ∈ E c for all t ∈ T . ) ϕ , ( S ( f ) c ) = E c .e) If the conjugacy class of t ∈ T (i.e. the set (cid:8) s − ts (cid:12)(cid:12) s ∈ T (cid:9) ) is infiniteand X ∈ { V t | t ∈ T } c then X t = 0 .f ) If the conjugacy class of every t ∈ T \ { } is infinite then { V t | t ∈ T } c = (cid:8) x e ⊗ K (cid:12)(cid:12) x ∈ E (cid:9) , S ( f ) c = (cid:8) x e ⊗ K (cid:12)(cid:12) x ∈ E c (cid:9) . Thus in this case S ( f ) is a kind of E -factor.g) The following are equivalent: g ) S ( f ) is commutative. g ) T and E are commutative and f ( s, t ) = f ( t, s ) for all s, t ∈ T . For s, t ∈ T , x ∈ E , and Y := ( x e ⊗ K ) V s , by Theorem 2.1.9 g),( XY ) t = gX r ∈ T f ( r, r − t ) X r Y r − t = gX r ∈ T f ( r, r − t ) X r δ s,r − t x = f ( ts − , s ) X ts − x , ( Y X ) t = gX r ∈ T f ( r, r − t ) Y r X r − t = gX r ∈ T f ( r, r − t ) δ r,s xX r − t = f ( s, s − t ) xX s − t . a) follows from the above by putting s := 1 (Proposition 1.1.2 a)).b) follows from the above by putting x := 1 E and t := rs (Proposition1.1.2).c) follows from a),b), and Corollary 1.3.7 d). The last assertion follows usingProposition 1.1.5 a).d) follows from c) (and Proposition 1.1.2 a)).e) follows from b) and Proposition 2.1.23 b).f) follows from c), e), and Proposition 2.1.2 d). g ⇒ g . By a), E is commutative. By Proposition 2.1.2 b), f ( s, t ) V st = V s V t = V t V s = f ( t, s ) V t V s = f ( t, s ) V ts st = ts and f ( s, t ) = f ( t, s ). g ⇒ g follows from c). COROLLARY 2.1.25 If IK = IR then the following are equivalent:a) S ( f ) c = S ( f ) = Re S ( f ) .b) T is commutative, E c = E = Re E , and f ( s, t ) = f ( t, s ) , ˜ f ( t ) = 1 E , t = 1 for all s, t ∈ T . a ⇒ b . By Corollary 2.1.24 g ⇒ g , T is commutative, E = E c , and f ( s, t ) = f ( t, s ) for all s, t ∈ T . Since E is isomorphic with a C*-subalgebra of S ( f ) (Theorem 2.1.9 h)), E = Re E . By Proposition 2.1.2 e), V t = V ∗ t = ( ˜ f ( t ) e ⊗ K ) V t − so by Theorem 2.1.9 a), t = t − , ˜ f ( t ) = 1 E , so t = 1. b ⇒ a . By Corollary 2.1.24 g ⇒ g , S ( f ) c = S ( f ). For X ∈ S ( f ) and t ∈ T , by Theorem 2.1.9 c),( X ∗ ) t = ˜ f ( t )( X t − ) ∗ = ( X t ) ∗ = X t so X ∗ = X (Theorem 2.1.9 a)). PROPOSITION 2.1.26
Let ( E i ) i ∈ I be a family of unital C**-algebras suchthat E is the C*-direct product of this family. For every i ∈ I , we identify E i with the corresponding closed ideal of E (resp. of E ¨ E ) and put f i : T × T −→ U n E ci , ( s, t ) f ( s, t ) i . a) For every i ∈ I , f i ∈ F ( T, E i ) . We put (by Theorem 2.1.9 b)) ϕ i : S ( f ) −→ S ( f i ) , X T X t ∈ T (( X t ) i e ⊗ K ) V f i t .ϕ i is a surjective C**-homomorphism. ) In the C*-case, if T is finite then R ( f ) = S k·k ( f ) = S C ( f ) is isomorphicto the C*-direct product of the family ( R ( f i ) = S k·k ( f i ) = S C ( f i )) i ∈ I . c) In the C*-case, if I is finite then S C ( f ) (resp. S k·k ( f ) ) is isomorphic to Q i ∈ I S C ( f i ) (resp. Q i ∈ I S k·k ( f i ) ).d) In the W*-case, S W ( f ) is isomorphic to the C*-direct product of the family ( S W ( f i )) i ∈ I .Remark. The C*-isomorphisms of b) and c) cease to be surjective in generalif T and I are both infinite. Take T := ( ZZ ) IN , I := IN, E i := IK for every i ∈ I , and E := l ∞ (i.e. E is the C*-direct product of the family ( E i ) i ∈ I ). Forevery n ∈ IN put t n := ( δ m,n ) m ∈ IN ∈ T . Assume there is an X ∈ S C ( f ) (resp. X ∈ S k·k ( f )) with ψX = ( V f i t i ) i ∈ I (resp. ϕX = ( V f i t i ) i ∈ I ), where ψ and ϕ arethe maps of b) and c), respectively. Then ( X t n ) i = δ i,n for all i, n ∈ IN and thisimplies ( X t ) t ∈ T
6∈ (cid:13)| t ∈ T ˘ E , which contradicts Proposition 2.1.23 b). PROPOSITION 2.1.27
Let S be a finite group, K ′ := l ( S ) , K ′′ := l ( S × T ) , and g ∈ F ( S, S ( f )) such that g ( s , s ) ∈ U n E c (where U n E c is identifiedwith ( U n E c ) e ⊗ K ⊂ U n S ( f ) c ) for all s , s ∈ S and put h : ( S × T ) × ( S × T ) −→ U n E c , (( s , t ) , ( s , t )) g ( s , s ) f ( t , t ) . a) h ∈ F ( S × T, E ) ; for every X ∈ S ( g ) put ϕX := X s ∈ S T X t ∈ T (( X s ) t e ⊗ K ′′ ) V h ( s,t ) ∈ S ( h ) . b) ϕ : S ( g ) −→ S ( h ) is an E -C*-isomorphism. a) is obvious.b) For X, Y ∈ S ( g ) and ( s, t ) ∈ S × T , by Theorem 2.1.9 c),g) and Propo-sition 2.1.6 g), ( ϕX ∗ ) ( s,t ) = (( X ∗ ) s ) t = ˜ g ( s )(( X s − ) ∗ ) t =68 ˜ g ( s ) ˜ f ( t )(( X s − ) t − ) ∗ = ˜ h ( s, t )( X ( s,t ) − ) ∗ = (( ϕX ) ∗ ) ( s,t ) , ( ϕ ( XY )) ( s,t ) = (( XY ) s ) t = X r ∈ S g ( r, r − s )( X r Y r − s ) t == X r ∈ S g ( r, r − s ) gX q ∈ T f ( q, q − t )( X r ) q ( Y r − s ) q − t == ^ X ( r,q ) ∈ S × T h (( r, q ) , ( r, q ) − ( s, t )) X ( r,q ) Y ( r,q ) − ( s,t ) = (( ϕX )( ϕY )) ( s,t ) , so ϕ is a C*-homomorphism. If ϕX = 0 then X ( s,t ) = 0 for all ( s, t ) ∈ S × T ,so X = 0 and ϕ is injective. Let Z ∈ S ( h ). For every s ∈ S put X s := T X t ∈ T ( Z ( s,t ) ˜ ⊗ K ) V ft ∈ S ( f ) ,X := X s ∈ S ( X s ⊗ K ′ ) V gs ∈ S ( g ) . Then ϕX = Z and ϕ is surjective. PROPOSITION 2.1.28 If T is infinite and X ∈ S ( f ) \ { } then X ( H ) isnot precompact. Let t ∈ T with X t = 0. There is an x ′ ∈ E ′ + (resp. x ′ ∈ ¨ E + ) with h X ∗ t X t , x ′ i >
0. We put t := 1 and construct a sequence ( t n ) n ∈ IN recursivelyin T such that for all m, n ∈ IN , m < n , (cid:12)(cid:12)(cid:12)D f ( t, t m ) ∗ f ( tt m t − n , t n ) X ∗ t X tt m t − n , x ′ E(cid:12)(cid:12)(cid:12) < (cid:10) X ∗ t X t , x ′ (cid:11) . Let n ∈ IN \ { } and assume the sequence was constructed up to n −
1. Since(Proposition 2.1.23 a)) X s ∈ T (cid:10) X ∗ tt m s − X tt m s − , x ′ (cid:11) < ∞ for all m ∈ IN n − there is a t n ∈ T with D X ∗ tt m t − n X tt m t − n , x ′ E < (cid:10) X ∗ t X t , x ′ (cid:11) m ∈ IN n − . By Schwarz’ inequality ([C1] Proposition 2.3.4.6 c)) for m ∈ IN n − , (cid:12)(cid:12)(cid:12)D f ( t, t m ) ∗ f ( tt m t − n , t n ) X ∗ t X tt m t − n , x ′ E(cid:12)(cid:12)(cid:12) ≤≤ (cid:10) X ∗ t X t , x ′ (cid:11) D X ∗ tt m t − n X tt m t − n , x ′ E < (cid:10) X ∗ t X t , x ′ (cid:11) . This finishes the recursive construction.For r, s ∈ T , by Theorem 2.1.9 e),( X (1 E ⊗ e r )) s = gX q ∈ T f ( q, q − s ) X q δ r,q − s = f ( sr − , r ) X sr − , h X (1 E ⊗ e r ) | X t ⊗ e s i = f ( sr − , r ) X ∗ t X sr − . For m, n ∈ IN , m < n , it follows h X (1 E ⊗ e t m ) | X t ⊗ e tt m i = f ( t, t m ) X ∗ t X t , (cid:10) h X (1 E ⊗ e t m ) | X t ⊗ e tt m i , x ′ f ( t, t m ) ∗ (cid:11) = (cid:10) X ∗ t X t , x ′ (cid:11) , h X (1 E ⊗ e t n ) | X t ⊗ e tt m i = f ( tt m t − n , t n ) X ∗ t X tt m t − n , | (cid:10) h X (1 E ⊗ e t n ) | X t ⊗ e tt m i , x ′ f ( t, t m ) ∗ (cid:11) | = (cid:12)(cid:12)(cid:12)D f ( t, t m ) ∗ f ( tt m t − n , t n ) X ∗ t X tt m t − n , x ′ E(cid:12)(cid:12)(cid:12) < (cid:10) X ∗ t X t , x ′ (cid:11) , (cid:13)(cid:13) x ′ (cid:13)(cid:13) k X (1 E ⊗ e t m ) − X (1 E ⊗ e t n ) k k X t k ≥≥ (cid:12)(cid:12)(cid:10) h X (1 E ⊗ e t m ) − X (1 E ⊗ e t n ) | X t ⊗ e tt m i , x ′ f ( t, t m ) ∗ (cid:11)(cid:12)(cid:12) ≥≥ (cid:12)(cid:12)(cid:10) h X (1 E ⊗ e t m ) | X t ⊗ e tt m i , x ′ f ( t, t m ) ∗ (cid:11)(cid:12)(cid:12) −− (cid:12)(cid:12)(cid:10) h X (1 E ⊗ e t n ) | X t ⊗ e tt m i , x ′ f ( t, t m ) ∗ (cid:11)(cid:12)(cid:12) >> (cid:10) X ∗ t X t , x ′ (cid:11) − (cid:10) X ∗ t X t , x ′ (cid:11) = 12 (cid:10) X ∗ T X T , x ′ (cid:11) . Thus the sequence ( X (1 E ⊗ e t n )) n ∈ IN has no Cauchy subsequence and therefore X ( H ) is not precompact. 70 ROPOSITION 2.1.29
Assume T finite and let Ω be a compact space, ω ∈ Ω , g : T × T −→ U n C (Ω , E ) , ( s, t ) f ( s, t )1 Ω ,A := { X ∈ S ( g ) | t ∈ T, t = 1 = ⇒ X t ( ω ) = 0 } ,B := { Y ∈ C (Ω , S ( f )) | t ∈ T, t = 1 = ⇒ Y ( ω ) t = 0 } . Then g ∈ F ( T, C (Ω , E )) and we define for every X ∈ A and Y ∈ B , ϕX : Ω −→ S ( f ) , ω X t ∈ T ( X t ( ω ) ⊗ K ) V ft ,ψY := X t ∈ T ( Y ( · ) t ⊗ K ) V gt . Then A (resp. B ) is a unital C*-subalgebra of S ( g ) (resp. of C (Ω , S ( f )) ) ϕ : A −→ B , ψ : B −→ A are C*-isomorphisms, and ϕ = ψ − . It is easy to see that A (resp. B ) is a unital C*-subalgebra of S ( g ) (resp.of C (Ω , S ( f ))) and that ϕ and ψ are well-defined. For X, X ′ ∈ A , t ∈ T , and ω ∈ Ω, by Theorem 2.1.9 c),g) and Proposition 2.1.2 e),((( ϕX )( ϕX ′ ))( ω )) t = X s ∈ T f ( s, s − t )(( ϕX )( ω )) s (( ϕX ′ )( ω )) s − t == X s ∈ T f ( s, s − t ) X s ( ω ) X ′ s − t ( ω ) = X s ∈ T ( f ( s, s − t ) X s X ′ s − t )( ω ) == ( XX ′ ) t ( ω ) = ( ϕ ( XX ′ )( ω )) t , ( ϕX ∗ )( ω ) = X s ∈ T ((( X ∗ ) s ( ω )) ⊗ K ) V fs = X s ∈ T (( ˜ f ( s )(( X s − ) ∗ ( ω ))) ⊗ K ) V fs == X s ∈ T (( X s − )( ω ) ∗ ⊗ K )( V fs − ) ∗ = X s ∈ T ( X s ( ω ) ∗ ⊗ K )( V fs ) ∗ = ( ϕX ) ∗ ( ω )so ϕ is a C*-homomorphism and we have( ψϕX ) t = ( ϕX ) t = X t . Moreover for Y ∈ B , ( ϕψY ) t ( ω ) = (( ψY )( ω )) t = Y t ( ω )which proves the assertion. 71 .2 Variation of the parameters In this subsection we examine the changes produced by the replacement ofthe groups and of the Schur functions.
DEFINITION 2.2.1
We put for every λ ∈ Λ( T, E ) (Definition 1.1.3) U λ : H −→ H , ξ ( λ ( t ) ξ t ) t ∈ T . It is easy to see that U λ is well-defined, U λ ∈ U n L E ( H ), and the mapΛ( T, E ) −→ U n L E ( H ) , λ U λ is an injective group homomorphism with U ∗ λ = U λ ∗ (Proposition 1.1.4 c)).Moreover k U λ − U µ k ≤ k λ − µ k ∞ for all λ, µ ∈ Λ( T, E ). PROPOSITION 2.2.2
Let f, g ∈ F ( T, E ) and λ ∈ Λ( T, E ) .a) The following are equivalent: a ) g = f δλ . a ) There is a (unique) E -C*-isomorphism ϕ : S ( f ) −→ S ( g ) continuous with respect to the T -topologies such that for all t ∈ T and x ∈ E , ϕV ft = ( λ ( t ) ∗ e ⊗ K ) V gt (we call such an isomorphism an S -isomorphism and denote it by ≈ S )b) If the above equivalent assertions are fulfilled then for X ∈ S ( f ) and t ∈ T , ϕX = U ∗ λ XU λ , ( ϕX ) t = λ ( t ) ∗ X t . ) There is a natural bijection { S ( f ) | f ∈ F ( T, E ) } / ≈ S −→ F ( T, E ) / { δλ | λ ∈ Λ( T, E ) } . By Proposition 1.1.4 c), δλ ∈ F ( T, E ) for every λ ∈ Λ( T, E ). a ⇒ a & b . For s, t ∈ T and ζ ∈ ˘ E , by Proposition 2.1.2 c), U ∗ λ V ft U λ ( ζ ⊗ e s ) = U ∗ λ V ft (( λ ( s ) ζ ) ⊗ e s ) = U ∗ λ (( f ( t, s ) λ ( s ) ζ ) ⊗ e ts ) == ( λ ( ts ) ∗ f ( t, s ) λ ( s ) ζ ) ⊗ e ts = ( λ ( t ) ∗ g ( t, s ) ζ ) ⊗ e ts = ( λ ( t ) ∗ e ⊗ K ) V gt ( ζ ⊗ e s )so (by Proposition 2.1.2 e)) U ∗ λ V ft U λ = ( λ ( t ) ∗ e ⊗ K ) V gt . Thus the map ϕ : S ( f ) −→ S ( g ) , X U ∗ λ XU λ is well-defined. It is obvious that it has the properties described in a ). Theuniqueness follows from Theorem 2.1.9 b).We have ϕ (( X t e ⊗ K ) V ft ) = ( X t e ⊗ K )( λ ( t ) ∗ e ⊗ K ) V gt = (( λ ( t ) ∗ X t ) e ⊗ K ) V gt so ( ϕX ) t = λ ( t ) ∗ X t . a ⇒ a . Put h := f δλ . By the above, for t ∈ T ,( λ ( t ) ∗ e ⊗ K ) V gt = ϕV ft = ( λ ( t ) ∗ e ⊗ K ) V ht so V gt = V ht and this implies g = h .c) follows from a). Remark.
Not every E -C*-isomorphism S ( f ) → S ( g ) is an S isomorphism(see Remark of Proposition 3.2.3). COROLLARY 2.2.3
Let Λ ( T, E ) := { λ ∈ Λ( T, E ) | λ is a group homomorphism } nd for every λ ∈ Λ ( T, E ) put ϕ λ : S ( f ) −→ S ( f ) , X U ∗ λ XU λ . Then the map λ ϕ λ is an injective group homomorphism. By Proposition 1.1.4 c), Λ ( T, E ) is the kernel of the mapΛ(
T, E ) −→ F ( T, E ) , λ δλ so by Proposition 2.2.2, ϕ λ is well-defined. Thus only the injectivity of the maphas to be proved. For t ∈ T and ζ ∈ ˘ E , by Proposition 2.1.2 c), U ∗ λ V t U λ ( ζ ⊗ e ) = U ∗ λ V t ( ζ ⊗ e ) = U ∗ λ ( ζ ⊗ e t ) == ( λ ( t ) ∗ ζ ) ⊗ e t = ( λ ( t ) ∗ e ⊗ K ) V t ( ζ ⊗ e ) . So if ϕ λ is the identity map then λ ( t ) = 1 E for every t ∈ T . PROPOSITION 2.2.4
Let F be a unital C**-algebra, ϕ : E → F a surjectiveC**-homomorphism, g := ϕ ◦ f ∈ F ( T, F ) , and L := g (cid:13)| t ∈ T ˘ F . We put for all ξ ∈ H, η ∈ L , and X ∈ L E ( H ) , ˜ ξ := ( ϕξ i ) i ∈ I ∈ L , ˜ Xη := f Xζ ∈ L , where ζ ∈ H with ˜ ζ = η (Lemma 1.3.11 a),b) and Proposition 1.3.12 a)) . Then ˜ X = T X t ∈ T (( ϕX t ) e ⊗ K ) V gt ∈ S ( g ) for every X ∈ S ( f ) and the map ˜ ϕ : S ( f ) −→ S ( g ) , X ˜ X is a surjective C**-homomorphism, continuous with respect to the topologies T k , k ∈ { , , } such that Ker ˜ ϕ = { X ∈ S ( f ) | t ∈ T = ⇒ X t ∈ Ker ϕ } . s, t ∈ T and ξ ∈ H , ^ z }| { ( X t e ⊗ K ) V ft ˜ ξ s = ( ^ z }| { ( X t e ⊗ K ) V ft ξ ) s = ϕ (( X t e ⊗ K ) V ft ξ ) s == ϕ ( f ( t, t − s ) X t ξ t − s ) = g ( t, t − s )( ϕX t ) ˜ ξ t − s = ((( ϕX t ) e ⊗ K ) V gt ˜ ξ ) s so by Lemma 1.3.11 b), ^ z }| { ( X t e ⊗ K ) V ft = (( ϕX t ) e ⊗ K ) V gt . By Theorem 2.1.9 b), X = T X t ∈ T ( X t e ⊗ K ) V ft so by the above and by Proposition 1.3.12 b),˜ X = T X t ∈ T (( ϕX t ) e ⊗ K ) V gt ∈ S ( g ) . By Proposition 1.3.12 b), ˜ ϕ is a surjective C**-homomorphism, continuous withrespect to the topologies T k ( k ∈ { , , } ). The last assertion is easy to see. COROLLARY 2.2.5
Let F be a unital C*-algebra, ϕ : E → F a unitalC*-homomorphism such that ϕ ( U n E c ) ⊂ F c , g := ϕ ◦ f ∈ F ( T, F ) , and L := (cid:13)| t ∈ T ˘ F . Then the map ˜ ϕ : S k·k ( f ) −→ S k·k ( g ) , X k·k X t ∈ T (( ϕX t ) ⊗ L ) V gt is C*-homomorphism. Put G := E/Ker ϕ and denote by ϕ : E → G the quotient map and by ϕ : G → F the corresponding injective C*-homomorphism. By Proposition2.2.4, the corresponding map˜ ϕ : S k·k ( f ) −→ S k·k ( ϕ ◦ f )75s a C*-homomorphism and by Theorem 2.1.9 k), the corresponding map˜ ϕ : S k·k ( ϕ ◦ f ) −→ S k·k ( g )is also a C*-homomorphism. The assertion follows from ˜ ϕ = ˜ ϕ ◦ ˜ ϕ . PROPOSITION 2.2.6
Let T ′ be a group, K ′ := l ( T ′ ) , H ′ := ˘ E e ⊗ K ′ , ψ : T → T ′ a surjective group homomorphism such that sup t ′ ∈ T ′ Card − ψ ( t ′ ) ∈ IN , and f ′ ∈ F ( T ′ , E ) such that f ′ ◦ ( ψ × ψ ) = f . If we put X ′ t ′ := X t ∈ − ψ ( t ′ ) X t for every X ∈ S ( f ) and t ′ ∈ T ′ then the family (( X ′ t ′ e ⊗ K ′ ) V f ′ t ′ ) t ′ ∈ T ′ is summablein L E ( H ′ ) T for every X ∈ S ( f ) and the map ˜ ψ : S ( f ) −→ S ( f ′ ) , X X ′ := T X t ′ ∈ T ′ ( X ′ t ′ e ⊗ K ′ ) V f ′ t ′ is a surjective E -C**-homomorphism.We may drop the hypothesis that ψ is surjective if we replace S by S k·k . Let X ∈ S ( f ). By Corollary 2.1.16 a), since ψ is surjective andsup t ′ ∈ T ′ Card − ψ ( t ′ ) ∈ IN , it follows that the family (( X ′ t ′ e ⊗ K ′ ) V f ′ t ′ ) t ′ ∈ T ′ is summable in L E ( H ′ ) T andtherefore X ′ ∈ S ( f ′ ).Let X, Y ∈ S ( f ). By Theorem 2.1.9 c),g), for t ′ ∈ T ′ ,( X ′∗ ) t ′ = e f ′ ( t ′ )( X t ′− ) ∗ = e f ′ ( t ′ ) X t ∈ − ψ ( t ′− ) X t ∗ = e f ′ ( t ′ ) X s ∈ − ψ ( t ′ ) ( X s − ) ∗ =76 X s ∈ − ψ ( t ′ ) ˜ f ( s )( X s − ) ∗ = X s ∈ − ψ ( t ′ ) ( X ∗ ) s = ( X ∗ ) ′ t ′ , ( X ′ Y ′ ) t ′ = gX s ′ ∈ T ′ f ′ ( s ′ , s ′− t ′ ) X ′ s ′ Y ′ s ′− t ′ == gX s ′ ∈ T ′ f ′ ( s ′ , s ′− t ′ ) X s ∈ − ψ ( s ′ ) X s X r ∈ − ψ ( s ′− t ′ ) Y r == gX s ′ ∈ T ′ f ′ ( s ′ , s ′− t ′ ) X s ∈ − ψ ( s ′ ) X t ∈ − ψ ( t ′ ) X s Y s − t == gX s ′ ∈ T ′ X s ∈ − ψ ( s ′ ) X t ∈ − ψ ( t ′ ) f ( s, s − t ) X s Y s − t == X t ∈ − ψ ( t ′ ) gX s ∈ T f ( s, s − t ) X s Y s − t = X t ∈ − ψ ( t ′ ) ( XY ) t = ( XY ) ′ t ′ . Thus ψ is a C*-homomorphism. The other assertions are easy to see.The last assertion follows from Corollary 2.1.17 d). COROLLARY 2.2.7
If we use the notation of
Proposition 2.2.6 and
Corol-lary 2.2.5 and define e ϕ ′ and e ψ ′ in an obvious way then e ϕ ′ ◦ ˜ ψ = e ψ ′ ◦ ˜ ϕ . For X ∈ S ( f ) and t ′ ∈ T ′ ,( e ϕ ′ ˜ ψX ) t ′ = ϕ (( ˜ ψX ) t ′ ) = ϕ X t ∈ − ψ ( t ′ ) X t = X t ∈ − ψ ( t ′ ) ϕX t , ( e ψ ′ ˜ ϕX ) t ′ = X t ∈ − ψ ( t ′ ) ( ˜ ϕX ) t = X t ∈ − ψ ( t ′ ) ϕX t , so e ϕ ′ ◦ ˜ ψ = e ψ ′ ◦ ˜ ϕ . ROPOSITION 2.2.8
Let F be a unital C*-subalgebra of E such that f ( s, t ) ∈ F for all s, t ∈ T . We denote by ψ : F → E the inclusion map and put f F : T × T −→ U n F c , ( s, t ) f ( s, t ) ,H F := (cid:13)| t ∈ T ˘ F ≈ ˘ F ⊗ K , ˜ ψ : H F −→ H , ξ ( ψξ t ) t ∈ T . Moreover we denote for all s, t ∈ T by u Ft , V Ft , and ϕ Fs,t the correspondingoperators associated with F ( f F ∈ F ( T, F )) . Let X ∈ S C ( f ) such that X ( ˜ ψξ ) ∈ ˜ ψ ( H F ) for every ξ ∈ H F and put X F : H F −→ H F , ξ ξ ′ , where ξ ′ ∈ H F with ˜ ψξ ′ = X ( ˜ ψξ ) , and X Ft := ( u F ) ∗ X F u Ft ∈ F (by the canoni-cal identification of F with L F ( ˘ F ) ) for every t ∈ T .a) ξ, η ∈ H F ⇒ D ˜ ψξ (cid:12)(cid:12)(cid:12) ˜ ψη E = ψ h ξ | η i .b) ˜ ψ is linear and continuous with (cid:13)(cid:13)(cid:13) ˜ ψ (cid:13)(cid:13)(cid:13) = 1 .c) X F is linear and continuous with (cid:13)(cid:13) X F (cid:13)(cid:13) = k X k .d) For s, t ∈ T , ψϕ Fs,t X F = ϕ s,t X , ψX Ft = X t , ϕ Fs,t X F = f F ( st − , t ) X Fst − . e) X F ∈ S ( f F ) .f ) ξ ∈ H F ⇒ X ( ˜ ψξ ) = k·k P t ∈ T ( X t ⊗ K ) V t ( ˜ ψξ ) . a&b&c are easy to see.d) By a) and Proposition 2.1.6 b), ϕ Fs,t X F = (cid:10) X F (1 F ⊗ e t ) (cid:12)(cid:12) F ⊗ e s (cid:11) ,ψϕ Fs,t X F = ψ (cid:10) X F (1 F ⊗ e t ) (cid:12)(cid:12) F ⊗ e s (cid:11) =78 D ˜ ψ ( X F (1 F ⊗ e t )) (cid:12)(cid:12)(cid:12) ˜ ψ (1 F ⊗ e s ) E = h X (1 E ⊗ e t ) | E ⊗ e s i = ϕ s,t X .
In particular ψX Ft = ψϕ F ,t X F = ϕ ,t X = X t and by Proposition 2.1.8, ψϕ Fs,t X F = ϕ s,t X = f ( st − , t ) X st − = ψ ( f F ( st − , t ) X Fst − ) ,ϕ Fs,t X F = f F ( st − , t ) X Fst − . e) By c) and Proposition 2.1.3 d), for ξ ∈ H F , k·k X t ∈ T u Ft ( u Ft ) ∗ ξ = ξ ,X F ξ = X F k·k X t ∈ T u Ft ( u Ft ) ∗ ξ = k·k X t ∈ T X F u Ft ( u Ft ) ∗ ξ ,X F ξ = k·k X s ∈ T u Fs ( u Fs ) ∗ X F ξ = k·k X s ∈ T k·k X t ∈ T u Fs (( u Fs ) ∗ X F u Ft )( u Ft ) ∗ ξ . By d) and Proposition 2.1.4 b),d), X F ξ = k·k X s ∈ T k·k X t ∈ T u Fs f F ( st − , t ) X Fst − ( u Ft ) ∗ ξ = k·k X s ∈ T k·k X t ∈ T u Fs X Fst − ( u Fs ) ∗ V Fst − ξ == k·k X s ∈ T k·k X r ∈ T u Fs X Fr ( u Fs ) ∗ V Fr ξ = k·k X s ∈ T k·k X r ∈ T u Fs ( u Fs ) ∗ ( X Fr ⊗ F ) V Fr ξ == k·k X s ∈ T u Fs ( u Fs ) ∗ k·k X t ∈ T ( X Ft ⊗ K ) V Ft ξ = k·k X t ∈ T ( X Ft ⊗ K ) V Ft ξ by Proposition 2.1.3 d), again. Thus X F = T X t ∈ T ( X Ft ⊗ K ) V Ft ∈ S C ( f F ) .
79) For s, t ∈ T , by d),( ˜ ψ (( X Ft ⊗ K ) V Ft ξ )) s = ψ (( X Ft ⊗ K ) V Ft ξ ) s = ψ ( f F ( t, t − s ) X Ft ξ t − s ) == f ( t, t − s ) X t ( ˜ ψξ ) t − s = (( X t ⊗ K ) V t ˜ ψξ ) s , ˜ ψ (( X Ft ⊗ K ) V Ft ξ ) = ( X t ⊗ K ) V t ˜ ψξ so by b) and e), X ( ˜ ψξ ) = ˜ ψ ( X F ξ ) = ˜ ψ k·k X t ∈ T ( X Ft ⊗ K ) V Ft ξ == k·k X t ∈ T ˜ ψ (( X Ft ⊗ K ) V Ft ξ ) = k·k X t ∈ T ( X t ⊗ K ) V t ( ˜ ψξ ) . PROPOSITION 2.2.9
Let F be a W*-algebra such that E is a unital C*-subalgebra of F generating it as W*-algebra, ϕ : E → F the inclusion map, and ˜ ξ := ( ϕξ t ) t ∈ T ∈ L for every ξ ∈ H , where L := W (cid:13)| t ∈ T ˘ F ≈ ˘ F ¯ ⊗ K . a) ϕ ( U n E c ) ⊂ U n F c and g := ϕ ◦ f ∈ F ( T, F ) .b) If ψ : L E ( H ) −→ L F ( L ) , X ¯ X is the injective C*-homomorphism defined in Proposition 1.3.9 b), then ψ ( S C ( f )) ⊂ S W ( g ) , ψ ( S C ( f )) generates S W ( g ) as W*-algebra, and forevery X ∈ S C ( f ) and t ∈ T we have ( ¯ X ) t = ϕX t .c) The following are equivalent for every Y ∈ S W ( g ) : c ) Y ∈ ψ ( S C ( f )) . c ) ξ ∈ H ⇒ Y ˜ ξ ∈ H .If these conditions are fulfilled then c ) ( Y t ) t ∈ T ∈ H . c ) ( Y ∗ t ) t ∈ T ∈ H . ) ξ ∈ H ⇒ Y ˜ ξ = k·k P t ∈ T ( Y t ¯ ⊗ K ) V gt ˜ ξ ∈ H . a) follows from the density of ϕ ( E ) in F ¨ F (Lemma 1.3.8 a ⇒ c ).b) For x ∈ E , t ∈ T , and ξ ∈ H ,((( ϕx ) ¯ ⊗ K ) V gt ˜ ξ ) s = g ( t, t − s )( ϕx ) ˜ ξ t − s == ϕ ( f ( t, t − s ) xξ t − s ) = ϕ (( x ⊗ K ) V t ξ s )so (( ϕx ) ¯ ⊗ K ) V gt = ( x ⊗ K ) V ft . Let now X ∈ S ( f ). By Theorem 2.1.9 b), X = T X t ∈ T ( X t ⊗ K ) V ft so by the above and by Proposition 1.3.9 c) (and Theorem 2.1.9 d)),¯ X = T X t ∈ T ( X t ⊗ K ) V ft = T X t ∈ T (( ϕX t ) ¯ ⊗ K ) V gt ∈ S W ( g )so ψ ( S C ( f )) ⊂ S W ( f ). By Theorem 2.1.9 a), ( ¯ X ) t = ϕX t for every t ∈ T .Since ϕ ( E ) is dense in F ¨ F (Lemma 1.3.8 a ) ⇒ c )) it follows that R ( g ) ⊂ T ϕ ( R ( f ))so ψ ( S ( f )) is dense in S ( g ) .. z}|{ S ( g ) and therefore generates S ( g ) as W*-algebra(Lemma 1.3.8 c ⇒ a ). c ⇒ c follows from the definition of ψ . c ⇒ c follows from Proposition 2.2.8 e). c ⇒ c & c follows from Proposition 2.1.23 b). c ⇒ c follows from Proposition 2.2.8 f).81 EMMA 2.2.10
Let
E, F be W*-algebras, G := E ¯ ⊗ F , and L := W (cid:13)| t ∈ T ˘ G ≈ ˘ G ¯ ⊗ K . a) If z ∈ G then z ¯ ⊗ K belongs to the closure of { w ¯ ⊗ K | w ∈ E ⊙ F, k w k ≤ } in L G ( L ) ... L .b) For every y ∈ F , the map E E −→ G ¨ G , x x ⊗ y is continuous. a) By [C1] Corollary 6.3.8.7, there is a filter F on { w ∈ E ⊙ F | k w k ≤ } converging to z in G G . By Lemma 1.3.2 b), for ( a, ξ, η ) ∈ ¨ G × L × L , D z ¯ ⊗ K , ^ ( a, ξ, η ) E = * z , G X t ∈ T ξ t a η ∗ t + == lim w, F * w , G X t ∈ T ξ t a η ∗ t + = lim w, F D w ¯ ⊗ K , ^ ( a, ξ, η ) E which proves the assertion.b) Let ( a i , b i ) i ∈ I be a finite family in ¨ E × ¨ F . For x ∈ E , * x ⊗ y , X i ∈ I a i ⊗ b i + = X i ∈ I h x , a i i h y , b i i = * x , X i ∈ I h y , b i i a i + . Since (cid:8) x ⊗ y | x ∈ E (cid:9) is a bounded set of G , the above identity proves thecontinuity. 82 ROPOSITION 2.2.11
Let F be a unital C**-algebra, S a group, and g ∈F ( S, F ) . We denote by ⊗ σ the spatial tensor product and put G := E ⊗ σ F ( resp. G := E ¯ ⊗ F ) ,L := g (cid:13)| s ∈ S ˘ F ≈ ˘ F e ⊗ l ( S ) , M := ^ (cid:13)| ( t,s ) ∈ T × S ˘ G ≈ ˘ G e ⊗ l ( T × S ) ,h : ( T × S ) × ( T × S ) −→ U n G c , (( t , s ) , ( t , s )) f ( t , t ) ⊗ g ( s , s ) . a) h ∈ F ( T × S, G ) , M ≈ H e ⊗ L, L E ( H ) ⊗ σ L F ( L ) ⊂ L G ( M ) in the C*-case , L E ( H ) ¯ ⊗L F ( L ) ≈ L G ( M ) in the W*-case . b) For t ∈ T , s ∈ S , x ∈ E , y ∈ F , (( x e ⊗ l ( T ) ) V ft ) ⊗ (( y e ⊗ l ( S ) ) V gs ) = (( x ⊗ y ) e ⊗ l ( T × S ) ) V h ( t,s ) . c) In the C*-case, S k·k ( f ) ⊗ σ S k·k ( g ) ≈ S k·k ( h ) and S C ( f ) ⊗ σ S C ( g ) ≈ S C ( h ) .d) In the W*-case, if z ∈ G and ( t, s ) ∈ T × S then ( z ¯ ⊗ l ( T × S ) ) V h ( t,s ) be-longs to the closure of n ( w ¯ ⊗ l ( T × S ) ) V h ( t,s ) (cid:12)(cid:12)(cid:12) w ∈ ( E ⊙ F ) o in L G ( M ) ... M e) In the W*-case, S W ( f ) ¯ ⊗S W ( g ) ≈ S W ( h ) . a) h ∈ F ( T × S, G ) is obvious.Let us treat the C*-case first. For ξ, ξ ′ ∈ H and η, η ′ ∈ L , (cid:10) ξ ′ ⊗ η ′ (cid:12)(cid:12) ξ ⊗ η (cid:11) = (cid:10) ξ ′ (cid:12)(cid:12) ξ (cid:11) ⊗ (cid:10) η ′ (cid:12)(cid:12) η (cid:11) = X t ∈ T ξ ∗ t ξ ′ t ! ⊗ X s ∈ S η ∗ s η ′ s ! == X ( t,s ) ∈ T × S (( ξ ∗ t ξ ′ t ) ⊗ ( η ∗ s η ′ s )) = X ( t,s ) ∈ T × S ( ξ ∗ t ⊗ η ∗ s )( ξ ′ t ⊗ η ′ s ) == X ( t,s ) ∈ T × S ( ξ t ⊗ η s ) ∗ ( ξ ′ t ⊗ η ′ s ) , so the linear map H ⊙ L −→ M, ξ ⊗ η ( ξ t ⊗ η s ) ( t,s ) ∈ T × S ϕ : H ⊗ L → M preserving the scalar products.Let z ∈ G , ( t, s ) ∈ T × S , and ε >
0. There is a finite family ( x i , y i ) i ∈ I in E × F such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I x i ⊗ y i − z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I ( x i ⊗ e t ) ⊗ ( y i ⊗ e s ) − z ⊗ e ( t,s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε so z ⊗ e ( t,s ) ∈ ϕ ( H ⊗ L ) = ϕ ( H ⊗ L ). It follows that ϕ is surjective and so H ⊗ L ≈ M .The proof for the inclusion L E ( H ) ⊗ σ L F ( L ) ⊂ L G ( M ) can be found in [L]page 37.Let us now discus the W*-case. ˘ E ¯ ⊗ ˘ F ≈ ˘ G follows from [C2] Proposition 1.3e), M ≈ H ¯ ⊗ L follows from [C3] Corollary 2.2, and L E ( H ) ¯ ⊗L F ( L ) ≈ L G ( M )follows from [C2] Theorem 2.4 d) or [C3] Theorem 2.4.b) For t , t ∈ T , s , s ∈ S , ξ ∈ ˘ E , and η ∈ ˘ F , by Proposition 2.1.2 f) and[C3] Corollary 2.11,((( x e ⊗ l ( T ) ) V ft ) e ⊗ (( y e ⊗ l ( S ) ) V gs ))(( ξ ⊗ e t ) ⊗ ( η ⊗ e s )) == ((( x e ⊗ l ( T ) ) V ft )( ξ ⊗ e t )) e ⊗ ((( y e ⊗ l ( S ) ) V gs )( η ⊗ e s )) , (((( x ⊗ y ) e ⊗ l ( T × S ) )) V h ( t ,s ) )(( ξ ⊗ η ) ⊗ e ( t ,s ) ) == ( h (( t , s ) , ( t , s ))( x ⊗ y )( ξ ⊗ η )) ⊗ e ( t t ,s s ) == (( f ( t , t ) xξ ) ⊗ ( g ( s , s ) yη )) ⊗ e t t ⊗ e s s == ((( x e ⊗ l ( T ) ) V ft )( ξ ⊗ e t )) e ⊗ ((( y e ⊗ l ( S ) ) V gs )( η ⊗ e s )) . We put u := (( x e ⊗ l ( T ) ) V ft ) e ⊗ (( y e ⊗ l ( S ) ) V gs ) − (( x ⊗ y ) e ⊗ l ( T × S ) ) V ht,s ∈ L G ( M ) . By the above, u ( ζ ⊗ e r ) = 0 for all ζ ∈ ˘ E ⊙ ˘ F and r ∈ T × S .84et us consider the C*-case first. Since ˘ E ⊙ ˘ F is dense in ˘ G , we get u ( z ⊗ e r ) = 0 for all z ∈ ˘ G and r ∈ T × S . For ζ ∈ M , by [C1] Proposition5.6.4.1 e), uζ = u X r ∈ T × S ( ζ r ⊗ e r ) ! = X r ∈ T × S u ( ζ r ⊗ e r ) = 0 , which proves the assertion in this case.Let us consider now the W*-case. Let z ∈ G and r ∈ T × S and let F be afilter on ( E ⊙ F ) converging to z in G ¨ G ([C1] Corollary 6.3.8.7). For η ∈ M , a ∈ ¨ G , and r ∈ T × S , D z ⊗ e r , ] ( a, η ) E = h h z ⊗ e r | η i , a i = h η ∗ r z , a i = h z , aη ∗ r i == lim w, F h w , aη ∗ r i = lim w, F D w ⊗ e r , ] ( a, η ) E , lim w, F w ⊗ e r = z ⊗ e r in M ¨ M . Since u : M ¨ M → M ¨ M is continuous ([C1] Proposition 5.6.3.4 c)), weget by the above u ( z ⊗ e r ) = 0. For ζ ∈ M it follows by [C1] Proposition 5.6.4.6c), uζ = u ¨ M X r ∈ T × S ( ζ r ⊗ e r ) = ¨ M X r ∈ T × S u ( ζ r ⊗ e r ) = 0which proves the assertion in the W*-case.c) By b), R ( f ) ⊙ R ( g ) ⊂ R ( h ) so by a), S k·k ( f ) ⊙ S k·k ( g ) ⊂ S k·k ( h ) , S C ( f ) ⊙ S C ( g ) ⊂ S C ( h ) , S k·k ( f ) ⊗ σ S k·k ( g ) ⊂ S k·k ( h ) , S C ( f ) ⊗ σ S C ( g ) ⊂ S C ( h ) . Let z ∈ G , ( t, s ) ∈ T × S , and ε >
0. There is a finite family ( x i , y i ) i ∈ I in E × F such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I ( x i ⊗ y i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I ( x i ⊗ y i ) − z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . By b), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I ((( x i ⊗ l ( T ) ) V ft ) ⊗ (( y i ⊗ l ( S ) ) V gs )) − ( z ⊗ l ( T × S ) ) V h ( t,s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε R ( h ) ⊂ k·k R ( f ) ⊙ R ( g ) ⊂ T R ( f ) ⊙ R ( g ) , S k·k ( h ) ⊂ S k·k ( f ) ⊗ σ S k·k ( g ) , S C ( h ) ⊂ S C ( f ) ⊗ σ S C ( g ) . d) By a) and Lemma 2.2.10 a), there is a filter F on n w ¯ ⊗ l ( T × S ) (cid:12)(cid:12) w ∈ ( E ⊙ F ) o converging to z ¯ ⊗ l ( T × S ) in L G ( M ) ... M . For ξ, η ∈ M and a ∈ ¨ G , D ( z ¯ ⊗ l ( T × S ) ) V h ( t,s ) , ^ ( a, ξ, η ) E = D z ¯ ⊗ l ( T × S ) , V h ( t,s ) ^ ( a, ξ, η ) E == lim w, F D w ¯ ⊗ l ( T × S ) V h ( t,s ) , ^ ( a, ξ, η ) E = lim w, F D ( w ¯ ⊗ l ( T × S ) ) V h ( t,s ) , ^ ( a, ξ, η ) E , which proves the assertion.e) By Theorem 2.1.9 h), ... H R ( f ) = S W ( f ) ⊂ L E ( H ) , ... L R ( g ) = S W ( g ) ⊂ L F ( L ) . By b), R ( f ) ⊙ R ( g ) ⊂ R ( h ), so by Lemma 2.2.10 b), S W ( f ) ⊙ R ( g ) ⊂ S W ( h ) , S W ( f ) ⊗ S W ( g ) ⊂ S W ( h ) . S W ( f ) ⊗ S W ( g ) ⊂ S W ( h )By [C3] Proposition 2.5, S W ( f ) ¯ ⊗S W ( g ) ≈ ... M S W ( f ) ⊗ S W ( g ) ⊂ S W ( h ) . For x ∈ E , y ∈ F , and ( t, s ) ∈ T × S , by b),(( x ⊗ y ) ¯ ⊗ l ( T × S ) ) V h ( t,s ) = (( x ¯ ⊗ l ( T ) ) V ft ) ¯ ⊗ (( y ¯ ⊗ l ( S ) ) V gs ) ∈ S W ( f ) ¯ ⊗S W ( g ) . Let z ∈ G . By d), there is a filter F on n ( w ¯ ⊗ l ( T × S ) ) V h ( t,s ) (cid:12)(cid:12)(cid:12) w ∈ ( E ⊙ F ) o z ¯ ⊗ l ( T × S ) ) V h ( t,s ) in L G ( M ) ... M , so by the above( z ¯ ⊗ l ( T × S ) ) V h ( t,s ) ∈ S W ( f ) ¯ ⊗S W ( g ) . We get R ( h ) ⊂ S W ( f ) ¯ ⊗S W ( g ) , S W ( h ) ⊂ S W ( f ) ¯ ⊗S W ( g ) , S W ( h ) = S W ( f ) ¯ ⊗S W ( g ) . COROLLARY 2.2.12
Let n ∈ IN and g : T × T −→ U n ( E n,n ) c , ( s, t ) [ δ i,j f ( s, t )] i,j ∈ IN n . a) ( S ( f )) n,n ≈ S ( g ) , ( S k·k ( f )) n,n ≈ S k·k ( g ) .b) Let us denote by ρ : S ( g ) → ( S ( f )) n,n the isomorphism of a). For X ∈S ( g ) , t ∈ T , and i, j ∈ IN n , (( ρX ) i,j ) t = ( X t ) i,j . a) Take F := IK n,n and S := { } in Proposition 2.2.11. Then G ≈ E n,n and g : T × T −→ U n G c , ( s, t ) f ( s, t ) ⊗ F . By Proposition 2.2.11 c),e), S ( g ) ≈ S ( f ) ⊗ IK n,n ≈ ( S ( f )) n,n S k·k ( g ) ≈ S k·k ( f ) ⊗ IK n,n ≈ ( S k·k ( f )) n,n . b) By Theorem 2.1.9 b), X = T X s ∈ T ( X s e ⊗ K ) V gs so ( ρX ) i,j = T X s ∈ t (( X s ) i,j e ⊗ K ) V fs , (( ρX ) i,j ) t = ( X t ) i,j by Theorem 2.1.9 a). 87 OROLLARY 2.2.13
Let n ∈ IN . If IK = IC (resp. if n = 4 m for some m ∈ IN ) then there is an f ∈ F ( ZZ n × ZZ n , E ) (resp. f ∈ F ( ( ZZ ) m , E ) ) suchthat R ( f ) = S ( f ) ≈ E n,n . By [C1] Proposition 7.1.4.9 b),d) (resp. [C1] Theorem 7.2.2.7 i),k)) there isa g ∈ F ( ZZ n × ZZ n , IC) (resp. g ∈ F ( ( ZZ ) m , IK)) such that S ( g ) ≈ IC n,n (resp. S ( g ) ≈ IK n,n ) . If we put f : ( ZZ n × ZZ n ) × ( ZZ n × ZZ n ) −→ U n E c , ( s, t ) g ( s, t ) ⊗ E (resp. f : ( ZZ ) m × ( ZZ ) m −→ U n E c , ( s, t ) g ( s, t ) ⊗ E )then by Proposition 2.2.11 a),e), f ∈ F ( ZZ n × ZZ n , E ) (resp. f ∈ F ( ( ZZ ) m , E ))and S ( f ) ≈ S ( g ) ⊗ E ≈ IK n,n ⊗ E ≈ E n,n . COROLLARY 2.2.14
Let F be a unital C**-algebra, G := E e ⊗ F , and h : T × T −→ U n G c , ( s, t ) f ( s, t ) ⊗ F . Then h ∈ F ( T, G ) and S k·k ( h ) ≈ S k·k ( f ) ⊗ F , S ( h ) ≈ S ( f ) e ⊗ F .
COROLLARY 2.2.15 If E is a W*-algebra then the following are equivalent:a) E is semifinite.b) S W ( f ) is semifinite. a ⇒ b. Assume first that there are a finite W*-algebra F and a Hilbertspace L such that E ≈ F ¯ ⊗L ( L ). Put g : T × T −→ U n F c , ( s, t ) f ( s, t ) .
88y Corollary 2.2.14, S W ( f ) ≈ S W ( g ) ¯ ⊗L ( L ) . By Corollary 2.1.11 c), S W ( g ) is finite and so S W ( f ) is semifinite.The general case follows from the fact that E is the C*-direct product ofW*-algebras of the above form ([T] Proposition V.1.40). b ⇒ a. E is isomorphic to a W*-subalgebra of S W ( f ) (Theorem 2.1.9 h))and the assertion follows from [T] Theorem V.2.15. PROPOSITION 2.2.16
Let
S, T be finite groups and g ∈ F ( S, S ( f )) andput L := l ( S ) , M := l ( S × T ) , and h : ( S × T ) × ( S × T ) −→ U n S ( f ) c , (( s , t ) , ( s , t )) f ( t , t ) g ( s , s ) . Then h ∈ F ( S × T, S ( f )) and the map ϕ : S ( g ) −→ S ( h ) , X X ( s,t ) ∈ S × T (( X s ) t ⊗ M ) V h ( s,t ) is an S ( f ) -C*-isomorphism. For
X, Y ∈ S ( g ), Z ∈ S ( f ), and ( s, t ) ∈ S × T , by Theorem 2.1.9 c),g),( ϕ ( X ∗ )) ( s,t ) = (( X ∗ ) s ) t = (˜ g ( s )( X s − ) ∗ ) t = ((˜ g ( s ) ∗ X s − ) ∗ ) t == ˜ f ( t )((˜ g ( s ) ∗ X s − ) t − ) ∗ = ˜ f ( t )˜ g ( s )(( X s − ) t − ) ∗ == ˜ h ( s, t )(( ϕX ) ( s − ,t − ) ) ∗ = ˜ h ( s, t )(( ϕX ) ( s,t ) − ) ∗ = (( ϕX ) ∗ ) ( s,t ) , (( ϕX )( ϕY )) ( s,t ) = X ( r,u ) ∈ S × T h (( r, u ) , ( r, u ) − ( s, t ))( ϕX ) ( r,u ) ( ϕY ) ( r,u ) − ( s,t ) == X ( r,u ) ∈ S × T g ( r, r − s ) f ( u, u − t )( X r ) u ( Y r − s ) u − t = X r ∈ S g ( r, r − s )( X r Y r − s ) t == X r ∈ S g ( r, r − s ) X r Y r − s ! t = (( XY ) s ) t = ( ϕ ( XY )) ( s,t ) , ( ϕ ( ZX )) ( s,t ) = (( ZX ) s ) t = (( ZX ) s ) t = ( ZX s ) t = Z ( X s ) t = Z ( ϕX ) ( s,t ) ϕ ( X ∗ ) = ( ϕX ) ∗ , ϕ ( XY ) = ( ϕX )( ϕY ) , ϕ ( ZX ) = Zϕ ( X )and ϕ is an S ( f )-C*-homomorphism.If X ∈ S ( g ) with ϕX = 0 then for ( s, t ) ∈ S × T ,( X s ) t = ( ϕX ) ( s,t ) = 0 , X s = 0 , X = 0so ϕ is injective.Let x ∈ E and ( s, t ) ∈ S × T . Put Z := ( x ⊗ K ) V ft ∈ S ( f ) , X := ( Z ⊗ L ) V gs ∈ S ( g ) . Then for ( r, u ) ∈ S × T ,( ϕX ) ( r,u ) = ( X r ) u = δ r,s Z u = δ r,s δ u,t x so ϕX = ( x ⊗ M ) V h ( s,t ) and ϕ is surjective. PROPOSITION 2.2.17
Let S be a finite subgroup of T and g := f | ( S × S ) .We identify S ( g ) with the E -C**-subalgebra { Z ∈ S ( f ) | t ∈ T \ S ⇒ Z t = 0 } of S ( f ) (Corollary 2.1.17 e)) . Let X ∈ S ( f ) ∩ S ( g ) c , P + := X ∗ X , and P − := XX ∗ and assume P ± ∈ P r S ( f ) .a) P ± ∈ S ( g ) c .b) The map ϕ ± : S ( g ) −→ P ± S ( f ) P ± , Y P ± Y P ± is a unital C**-homomorphism.c) For every Z ∈ ϕ + ( S ( g )) , XZX ∗ ∈ ϕ − ( S ( g )) and the map ψ : ϕ + ( S ( g )) −→ ϕ − ( S ( g )) , Z XZX ∗ is a C*-isomorphism with inverse ϕ − ( S ( g )) −→ ϕ + ( S ( g )) , Z X ∗ ZX such that ϕ − = ψ ◦ ϕ + . ) If p ∈ P r S ( g ) then ( X ( ϕ + p )) ∗ ( X ( ϕ + p )) = ϕ + p , ( X (( ϕ + p ))( X ( ϕ + p )) ∗ = ϕ − p . e) If ϕ + is injective then ϕ − is also injective, the map E −→ P ± S ( f ) P ± , x P ± ( x e ⊗ K ) P ± is an injective unital C**-homomorphism, P ± S ( f ) P ± is an E -C**-algebra, ϕ ± ( S ( g )) is an E -C**-subalgebra of it, and ϕ ± and ψ are E -C**-homo-morphisms.f ) The above results still hold for an arbitrary subgroup S of T if we replace S by S k·k . a) follows from the hypothesis on X .b) follows from a).c) Let Y ∈ S ( g ) with Z = P + Y P + . By the hypotheses of the Proposition , XZX ∗ = XP + Y P + X ∗ = XX ∗ XY X ∗ XX ∗ == XX ∗ Y XX ∗ XX ∗ = P − Y P − ∈ ϕ − ( S ( g ))and ψ is a C*-homomorphism. The other assertions follow from X ∗ ( XZX ∗ ) X = P + ZP + = P + Y P + . d) By b) and c),( X ( ϕ + p )) ∗ ( X ( ϕ + p )) = ( ϕ + p ) X ∗ X ( ϕ + p ) = ( ϕ + p ) P + ( ϕ + p ) = ϕ + p , ( X ( ϕ + p ))( X ( ϕ + p )) ∗ = X ( ϕ + p )( ϕ + p ) ∗ X ∗ = X ( ϕ + p ) X ∗ = ψϕ + p = ϕ − p . e) follows from b), c), and Lemma 1.3.2.f) follows from Corollary 2.1.17 d). Remark.
Even if ϕ ± is injective P ± S ( f ) P ± is not an E -C*-subalgebra of S ( f ). 91 HEOREM 2.2.18
Let S be a finite subgroup of T , L := l ( S ) , g := f | ( S × S ) , ω : ZZ × ZZ → T an injective group homomorphism such that S ∩ ω ( ZZ × ZZ ) = { } , a := ω (1 , , b := ω (0 , , c := ω (1 , , α := f ( a, a ) , α := f ( b, b ) ,β , β ∈ U n E c such that α β + α β = 0 , γ := 12 ( α ∗ β ∗ β − α ∗ β β ∗ ) = α ∗ β ∗ β = − α ∗ β β ∗ ,X := 12 (( β e ⊗ K ) V fa + ( β e ⊗ K ) V fb ) , P + := X ∗ X , P − := XX ∗ . We assume f ( s, c ) = f ( c, s ) and cs = sc for every s ∈ S , and f ( a, b ) = − f ( b, a ) = 1 E . Moreover we consider S ( g ) as an E -C**-subalgebra of S ( f )(Corollary 2.1.17 e)) .a) We have f ( a, c ) = − f ( c, a ) = α , f ( b, c ) = − f ( c, b ) = − α , f ( c, c ) = − α α ,γ = − α ∗ α ∗ , V fc ∈ S ( g ) c . b) We have P ± = 12 ( V f ± ( γ e ⊗ K ) V fc ) ∈ S ( g ) c ∩ P r S ( f ) , P + + P − = V f , P + P − = 0 ,X = 0 , XP + = X, P − X = X, P + X = XP − = 0 , X + X ∗ ∈ U n S ( f ) ,Y ∈ S ( g ) = ⇒ XY X = 0 . c) The map E −→ P ± S ( f ) P ± , x ( x e ⊗ K ) P ± is a unital injective C**-homomorphism; we shall consider P ± S ( f ) P ± asan E -C**-algebra using this map.d) The maps ϕ + : S ( g ) −→ P + S ( f ) P + , Y P + Y P + ,ϕ − : S ( g ) −→ P − S ( f ) P − , Y XY X ∗ re orthogonal injective E -C**-homomorphisms and ϕ + + ϕ − is an injec-tive E -C*-homomorphism. If Y , Y ∈ U n S ( g ) (resp. Y , Y ∈ P r S ( g ) )then ϕ + Y + ϕ − Y ∈ U n S ( f ) (resp. ϕ + Y + ϕ − Y ∈ P r S ( f ) ). Moreoverthe map ψ : S ( f ) −→ S ( f ) , Z ( X + X ∗ ) Z ( X + X ∗ ) is an E -C**-isomorphism such that ψ − = ψ , ψ ( P + S ( f ) P + ) = P − S ( f ) P − , ψ ◦ ϕ + = ϕ − . If IK = IC then X + X ∗ is homotopic to V f in U n S ( f ) and ψ is homotopicto the identity map of S ( f ) . Using this homotopy we find that ϕ + Y ishomotopic in the above sense to ϕ − Y for every Y ∈ S ( g ) and ϕ + Y + ϕ − Y , ϕ − Y + ϕ + Y , ϕ + ( Y Y ) + P − , and ϕ + ( Y Y + P − are homotopicin the above sense for all Y , Y ∈ S ( g ) .e) Let s ∈ S such that sa = as . Then sb = bs , f ( sc, c ) f ( s, c ) = − α α ,f ( sa, c ) f ( c, sa ) ∗ = − E , f ( a, s ) f ( s, a ) ∗ = f ( b, s ) f ( s, b ) ∗ . f ) If sa = as for every s ∈ S then the map S × ( ZZ × ZZ ) −→ T, ( s, r ) s ( ωr ) is an injective group homomorphism.g) If T is generated by S ∪ ω ( ZZ × ZZ ) and sa = as for every s ∈ S then ϕ + and ψ − are E -C*-isomorphisms with inverse P ± S ( f ) P ± −→ S ( g ) , Z X s ∈ S ( Z s e ⊗ L ) V gs , where ψ − : S ( g ) −→ P − S ( f ) P − , Y P − Y P − . h) If sa = as and f ( a, s ) = f ( s, a ) for every s ∈ S then X ∈ S ( g ) c , ϕ − Y = P − Y for every Y ∈ S ( g ) , and there is a unique S ( g ) -C**-homomorphism φ : S ( g ) , → S ( f ) such that φ (cid:20) α β ) ⊗ L (cid:21) = X . is injective and φ (cid:20) V g
00 0 (cid:21) = P + , φ (cid:20) V g (cid:21) = P − . i) If sa = as and f ( a, s ) = f ( s, a ) for all s ∈ S and if T is generated by S ∪ ω ( ZZ × ZZ ) then φ is an S ( g ) -C*-isomorphism and φ − V f = (cid:20) E ⊗ L
00 1 E ⊗ L (cid:21) , φ − V fc = (cid:20) γ ∗ ⊗ L − γ ∗ ⊗ L (cid:21) ,φ − V fa = (cid:20) − β ∗ ⊗ L ( β γ ∗ ) ⊗ L (cid:21) ,φ − V fb = (cid:20) − β ∗ ⊗ L ( β γ ∗ ) ⊗ L (cid:21) ,φ − P + = (cid:20) V g
00 0 (cid:21) , φ − P − = (cid:20) V g (cid:21) , and for every s ∈ S φ − V fs = (cid:20) V gs V gs (cid:21) . j) The above results still hold for an arbitrary subgroup S of T if we replace S with S k·k . a) By the equation of the Schur functions, f ( a, a ) = f ( a, c ) f ( a, b ) , f ( a, b ) f ( c, a ) = f ( a, c ) f ( b, a ) , f ( a, b ) f ( c, b ) = f ( b, b ) ,f ( b, a ) f ( c, b ) = f ( b, c ) f ( a, b ) , f ( a, b ) f ( c, c ) = f ( a, a ) f ( b, c )and so α = f ( a, c ) , f ( c, a ) = − f ( a, c ) = − α , f ( c, b ) = α , − α = − f ( c, b ) = f ( b, c ) , f ( c, c ) = α f ( b, c ) = − α α . For s ∈ S , by Proposition 2.1.2 b), V fc V fs = ( f ( c, s ) e ⊗ K ) V fcs = ( f ( s, c ) e ⊗ K ) V fsc = V fs V fc and so V fc ∈ S ( g ) c (by Proposition 2.1.2 d)).94) By Proposition 2.1.2 b),d),e) (and Corollary 2.1.22 c)), X ∗ = 12 ((( α ∗ β ∗ ) e ⊗ K ) V fa + (( α ∗ β ∗ ) e ⊗ K ) V fb ) ,P + = 14 (2 V f + (( α ∗ β ∗ β ) e ⊗ K ) V fc − (( α ∗ β ∗ β ) e ⊗ K ) V fc ) = 12 ( V f + ( γ e ⊗ K ) V fc ) ,P − = 14 (2 V f + (( β α ∗ β ∗ ) e ⊗ K ) V fc − (( β α ∗ β ∗ ) e ⊗ K ) V fc ) = 12 ( V f − ( γ e ⊗ K ) V fc ) . By a), P ∗± = 12 ( V f ± ( γ ∗ e ⊗ K )(( − α ∗ α ∗ ) e ⊗ K ) V fc ) = P ± ,P ± = 14 ( V f ± γ e ⊗ K ) V fc + ( γ e ⊗ K )(( − α α ) e ⊗ K ) V f ) == 12 ( V f ± ( γ e ⊗ K ) V fc ) = P ± , so, by a) again, P ± ∈ S ( g ) c ∩ P r S ( f ). By Proposition 2.1.2 b),d), X = 14 ((( β α + β α ) e ⊗ K ) V f + (( β β ) e ⊗ K )( V fa V fb + V fb V fa )) = 0 , ( X + X ∗ ) = X + XX ∗ + X ∗ X + X ∗ = P + + P − = V f . For the last relation we remark that by the above,
XY X = X ( P + + P − ) Y X = XP + Y X = XY P + X = 0 . c) follows from b) and Lemma 1.3.2.d) By b) and c), the map ϕ ± is an E -C**-homomorphism. Let Y ∈ S ( g )with ϕ ± Y = 0. By b), Y = ∓ Y ( γ e ⊗ K ) V fc so by Proposition 2.1.2 b),d) andTheorem 2.1.9 b), X s ∈ S ( Y s e ⊗ K ) V fs = ∓ Y ( γ e ⊗ K ) V fc = ∓ X s ∈ S (( Y s γf ( s, c )) e ⊗ K ) V fsc , which implies Y s = 0 for every s ∈ S (Theorem 2.1.9 a)). Thus ϕ ± is injective.It follows that ϕ + + ϕ − is also injective.Assume first Y , Y ∈ U n S ( g ). By b),( ϕ + Y + ϕ − Y ) ∗ ( ϕ + Y + ϕ − Y ) = ( ϕ + Y ∗ + ϕ − Y ∗ )( ϕ + Y + ϕ − Y ) =95 ϕ + ( Y ∗ Y ) + ϕ − ( Y ∗ Y ) = P + + P − = V f . Similarly ( ϕ + Y + ϕ − Y )( ϕ + Y + ϕ − Y ) ∗ = V f . The case Y , Y ∈ P r S ( g ) iseasy to see.By b), ψ is an E -C**-isomorphism with ψ − = ψ , ψP + = ( X + X ∗ ) X ∗ X ( X + X ∗ ) = XX ∗ XX ∗ = P − . Moreover for Y ∈ S ( g ), ψϕ + Y = ( X + X ∗ ) P + Y P + ( X + X ∗ ) = XY X ∗ = ϕ − Y .
Assume now IK = IC. By b), X + X ∗ ∈ U n S ( f ). Being selfadjoint itsspectrum is contained in {− , +1 } and so it is homotopic to V f in U n S ( f ).e) We have sb = sac = asc = acs = bs . By a), f ( s, c ) f ( sc, c ) = f ( s, f ( c, c ) = − α α ,f ( s, a ) f ( sa, c ) = f ( s, b ) f ( a, c ) = α f ( s, b ) ,f ( c, as ) f ( a, s ) = f ( c, a ) f ( b, s ) = − α f ( b, s ) ,f ( c, bs ) f ( b, s ) = f ( c, b ) f ( a, s ) = α f ( a, s ) ,f ( s, c ) f ( sc, b ) = f ( s, a ) f ( c, b ) = α f ( s, a ) ,f ( c, s ) f ( cs, b ) = f ( c, sb ) f ( s, b )so f ( sa, c ) f ( c, as ) ∗ = − f ( s, b ) f ( s, a ) ∗ f ( b, s ) ∗ f ( a, s ) == − f ( c, s ) f ( cs, b ) f ( c, sb ) ∗ α f ( s, c ) ∗ f ( sc, b ) ∗ α ∗ f ( c, bs ) = − E . From f ( s, c ) f ( sc, a ) = f ( s, b ) f ( c, a ) , f ( c, a ) f ( b, s ) = f ( c, as ) f ( a, s ) ,f ( c, s ) f ( cs, a ) = f ( c, sa ) f ( s, a )we get f ( a, s ) f ( s, a ) ∗ = f ( b, s ) f ( s, b ) ∗ .
96) Since S and ω ( ZZ × ZZ ) commute, the map is a group homomorphism.If s ( ωr ) = 1 for ( s, r ) ∈ S × ( ZZ × ZZ ) then ωr = s − ∈ S ∩ ω ( ZZ × ZZ ), whichimplies s = 1 and r = (0 , t ∈ T there are uniquely s ∈ S and d ∈ { , a, b, c } with t = sd . Let Z ∈ P ± S ( f ) P ± . By b) and Theorem 2.1.9b) (and Corollary 1.3.7 d)), Z = ± ( γ e ⊗ K ) ZV fc = ± ( γ e ⊗ K ) V fc Z By Proposition 2.1.2 b), ZV fc = X s ∈ S (( Z s f ( s, c )) e ⊗ K ) V fsc + X s ∈ S (( Z sa f ( sa, c )) e ⊗ K ) V fsb ++ X s ∈ S (( Z sb f ( sb, c )) e ⊗ K ) V fsa + X s ∈ S (( Z sc f ( sc, c )) e ⊗ K ) V fs ,V fc Z = X s ∈ S (( f ( c, s ) Z s ) e ⊗ K ) V fsc + X s ∈ S (( f ( c, sa ) Z sa ) e ⊗ K ) V fsb ++ X s ∈ S (( f ( c, sb ) Z sb ) e ⊗ K ) V fsa + X s ∈ S (( f ( c, sc ) Z sc ) e ⊗ K ) V fs and so by Theorem 2.1.9 a), Z s = ± γf ( sc, c ) Z sc = ± γf ( c, sc ) Z sc ,Z sc = ± γf ( s, c ) Z s = ± γf ( c, s ) Z s ,Z sa = ± γf ( sb, c ) Z sb = ± γf ( c, sb ) Z sb ,Z sb = ± γf ( sa, c ) Z sa = ± γf ( c, sa ) Z sa . By e), Z sa = Z sb = 0 for every s ∈ S . We get (by a), d), and Proposition 2.1.2b)) ϕ ± (2 X s ∈ S ( Z s e ⊗ L ) V gs ) = X s ∈ S ( Z s e ⊗ K ) V fs ± ( γ e ⊗ K ) V fc X s ∈ S ( Z s e ⊗ K ) V fs == X s ∈ S ( Z s e ⊗ K ) V fs ± X s ∈ S (( γf ( c, s ) Z s ) e ⊗ K ) V fsc == X s ∈ S ( Z s e ⊗ K ) V fs + X s ∈ S ( Z sc e ⊗ K ) V fsc = Z . ϕ ± is an E -C*-isomorphism with the mentioned inverse.h) is a long calculation using e).i) follows from h).j) follows from Corollary 2.1.17 d). Remark.
An example in which the above hypotheses are fulfilled is given inTheorem 4.1.7. S Throughout this subsection we assume T finiteIn this subsection we present the construction in the frame of categorytheory. Some of the results still hold for T locally finite. DEFINITION 2.3.1
The above construction of S ( f ) can be done for an arbi-trary E -module F , in which case we shall denote the result by S ( F ) . Moreoverwe shall write V Ft instead of V ft in this case. If F is an E -module then S ( F ) is canonically an E -module. If in addition F is adapted then S ( F ) is adapted and isomorphic to S ( ˇ F , F ). If F is an E -C*-algebra then S ( F ) is also an E -C*-algebra. PROPOSITION 2.3.2 If F, G are E -modules and ϕ : F → G is an E -linearC*-homomorphism then the map S ( ϕ ) : S ( F ) −→ S ( G ) , X X t ∈ S (( ϕX t ) ⊗ K ) V Gt is an E -linear C*-homomorphism, injective or surjective if ϕ is so. The assertion follows from Theorem 2.1.9 a),c),g).98
OROLLARY 2.3.3
Let F , F , F be E -modules and let ϕ : F → F , ψ : F → F be E -linear C*-homomorphisms.a) S ( ψ ) ◦ S ( ϕ ) = S ( ψ ◦ ϕ ) .b) If the sequence −→ F ϕ −→ F ψ −→ F is exact then the sequence −→ S ( F ) S ( ϕ ) −→ S ( F ) S ( ψ ) −→ S ( F ) is also exact.c) The covariant functor S : M E → M E is exact. a) is obvious.b) Let Y ∈ Ker S ( ψ ). For every t ∈ T , Y t ∈ Ker ψ = Im ϕ . If we identify F with Im ϕ then Y t ∈ F . It follows Y ∈ Im S ( ϕ ), Ker S ( ψ ) = Im S ( ϕ ).c) follows from b) and Proposition 2.3.2. COROLLARY 2.3.4
Let F be an adapted E -module and put ι : F −→ ˇ F , x (0 , x ) ,π : ˇ F −→ E , ( α, x ) α ,λ : E −→ ˇ F , α ( α, . Then the sequence −→ S ( F ) S ( ι ) −→ S ( ˇ F ) S ( π ) −→S ( λ ) ←− S ( E ) −→ is split exact. PROPOSITION 2.3.5
The covariant functor S : M E → M E (resp. S : C E → C E ) (Proposition 2.3.2, Corollary 2.3.3 a)) is continuous with respect tothe inductive limits (Proposition 1.2.9 a),b)) . { ( F i ) i ∈ I , ( ϕ ij ) i,j ∈ I } be an inductive system in the category M E (resp. C E ) and let { F, ( ϕ i ) i ∈ I } be its limit in the category M E (resp. C E ). Then { ( S ( F i )) i ∈ I , ( S ( ϕ ij ) i,j ∈ I ) } is an inductive system in the category M E (resp. C E ). Let { G, ( ψ i ) i ∈ I } be its limit in this category and let ψ : G → S ( F ) bethe E -linear C*-homomorphism such that ψ ◦ ψ i = S ( ϕ i ) for every i ∈ I . Inthe C E case, for α ∈ E and i ∈ I , ψ ( α ⊗ K ) = ψ ◦ ψ i ( α ⊗ K ) = ( S ( ϕ i ))( α ⊗ K ) = α ⊗ K so that ψ is an E -C*-homomorphism.Let i ∈ I and let X ∈ Ker S ( ϕ i ). Then ϕ i X t = 0 for every t ∈ T . Since T is finite, for every ε > j ∈ I , j ≥ i , with k ϕ ji X t k < εCard T for every t ∈ T . Then k ( S ( ϕ ji )) X k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X t ∈ T (( ϕ ji X t ) ⊗ K ) V F j t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . It follows k ψ i X k = inf j ∈ I,j ≥ i k ( S ( ϕ ji )) X k = 0 ,ψ i X = 0 , X ∈ Ker ψ i , Ker S ( ϕ i ) ⊂ Ker ψ i . By Lemma 1.2.8, ψ is injective. Since [ i ∈ I Im S ( ϕ i ) ⊂ Im ψ ,Im ψ is dense in S ( F ). Thus ψ is surjective and so an E -C*-isomorphism. PROPOSITION 2.3.6
Let θ : F → G be a surjective morphism in the cate-gory C E . We use the notation of Theorem 2.2.18 and mark with an exponentif this notation is used with respect to F or to G . For every Y ∈ U n S ( g G ) ,there is a Z ∈ S ( g F ) such that Z ∗ Z = P F + , S ( θ ) Z = ϕ G + Y . S ( θ ) is surjective and so there is a Z ∈ S ( g F ) with k Z k = 1 and S ( θ ) Z = Y . Put Z := P F + Z + X F (1 − Z ∗ Z ) . By Theorem 2.2.18 b), Z ∗ Z = P F + Z ∗ Z + (1 − Z ∗ Z ) ( X F ) ∗ X F (1 − Z ∗ Z ) == P F + Z ∗ Z + P F + (1 − Z ∗ Z ) = P F + . Since S ( θ )(1 − Z ∗ Z ) = 1 − Y ∗ Y = 0we get S ( θ )(1 − Z ∗ Z ) = 0 , S ( θ ) Z = P G + Y = ϕ G + Y .
PROPOSITION 2.3.7
Let F be an adapted E -module and Ω a locally com-pact space. We define for X ∈ S ( C (Ω , F )) (see Corollary 1.2.5 d)) and Y ∈C (Ω , S ( F )) , ϕX : Ω −→ S ( F ) , ω X t ∈ T ( X t ( ω ) ⊗ K ) V Ft ,ψY := X t ∈ T ( Y ( · ) t ⊗ K ) V C (Ω ,F ) t . Then ϕ : S ( C (Ω , F )) −→ C (Ω , S ( F )) ,ψ : C (Ω , S ( F )) −→ S ( C (Ω , F )) are E -linear C*-isomorphisms and ϕ = ψ − .Let ω ∈ Ω and assume F is an E -C*-algebra. Then the above maps ϕ and ψ induce the following E -C*-isomorphisms S ( { X ∈ C (Ω , F ) | X ( ω ) ∈ E } ) −→←− { Y ∈ C (Ω , S ( F )) | Y ( ω ) ∈ S ( E ) } . Let
X, X ′ ∈ S ( C (Ω , F )) and Y, Y ′ ∈ C (Ω , S ( F )). By Proposition 2.1.23 b)and Corollary 2.1.10 a), ϕX ∈ C (Ω , S ( F )) , ψY ∈ S ( C (Ω , F ))101nd it is easy to see that ϕ and ψ are E -linear. By Theorem 2.1.9 c),g), for t ∈ T and ω ∈ Ω, (( ϕX ) ∗ ( ω )) t = ˜ f ( t )((( ϕX )( ω ) t − )) ∗ == ˜ f ( t ) X t − ( ω ) ∗ = ( X ∗ ( ω )) t = (( ϕX ∗ )( ω )) t , ((( ϕX )( ϕX ′ ))( ω )) t = X s ∈ T f ( s, s − t )(( ϕX )( ω )) s (( ϕX ′ )( ω )) s − t == X s ∈ T f ( s, s − t ) X s ( ω ) X ′ s − t ( ω ) = X s ∈ T f ( s, s − t ) X s X ′ s − t ! ( ω ) == ( XX ′ ) t ( ω ) = (( ϕ ( XX ′ ))( ω )) t , so ( ϕX ) ∗ = ϕX ∗ , ( ϕX )( ϕX ′ ) = ϕ ( XX ′ )and ϕ is a C*-homomorphism. Similarly( ψY ∗ ) t ( ω ) = ( Y ∗ ( ω )) t = ˜ f ( t )( Y ( ω ) t − ) ∗ = ˜ f ( t )(( ψY ) t − ( ω )) ∗ = (( ψY ) ∗ ) t ( ω ) , (( ψY )( ψY ′ )) t ( ω ) = X s ∈ T f ( s, s − t )( ψY ) s ( ψY ′ ) s − t ! ( ω ) == X s ∈ T f ( s, s − t )( ψY ) s ( ω )( ψY ′ ) s − t ( ω ) = X s ∈ T f ( s, s − t ) Y ( ω ) s Y ′ ( ω ) s − t == ( Y ( ω ) Y ′ ( ω )) t = ( ψ ( Y Y ′ ) t )( ω )so ψY ∗ = ( ψY ) ∗ , ( ψY )( ψY ′ ) = ψ ( Y Y ′ )and ψ is a C*-homomorphism. Moreover( ψϕX ) t ( ω ) = (( ϕX )( ω )) t = X t ( ω ) , (( ϕψY )( ω )) t = ( ψY ) t ( ω ) = ( Y ( ω )) t , so ψϕX = X and ϕψY = Y which proves the assertion.The last assertion is easy to see. PROPOSITION 2.3.8
Let F be an adapted E -module, −→ F ι −→ ˇ F π −→ E −→ , −→ S ( F ) ι −→ ˇ z }| { S ( F ) π −→ E −→ the associated exact sequences (Proposition 1.2.4 h)) , and j : E −→ S ( E ) , α ( α ⊗ K ) V E ,ϕ : ˇ z }| { S ( F ) −→ S ( ˇ F ) , ( α, X ) ( ι ) X + ( α ⊗ K ) V ˇ F . Then ϕ is an injective E -C*-homomorphism and S ( π ) ◦ ϕ = j ◦ π . PROPOSITION 2.3.9 If E is commutative and F is an E -module then themap ϕ : S ( E ) ⊗ F −→ S ( F ) , X ⊗ x X t ∈ T (( X t x ) ⊗ K ) V Ft is a surjective C*-homomorphism. If in addition E = IK then ϕ is a C*-isomorphism with inverse ψ : S ( F ) −→ S ( E ) ⊗ F , Y X t ∈ T ( V Et ⊗ Y t ) . It is obvious that ϕ is surjective. For X, Y ∈ S ( E ) and x, y ∈ F , by Theorem2.1.9 c),g) and Proposition 2.1.2 b),d),e), ϕ (( X ⊗ x ) ∗ ) = ϕ ( X ∗ ⊗ x ∗ ) = X t ∈ T ((( X ∗ ) t x ∗ ) ⊗ K ) V Ft == X t ∈ T (( ˜ f ( t )( X t − ) ∗ x ∗ ) ⊗ K ) V Ft = X t ∈ T ((( X t − ) ∗ x ∗ ) ⊗ K )( V Ft − ) ∗ == X t ∈ T (( x ∗ ( X t ) ∗ ) ⊗ K )( V Ft ) ∗ = ( ϕ ( X ⊗ x )) ∗ ,ϕ ( X ⊗ x ) ϕ ( Y ⊗ y ) = X s,t ∈ T (( X s xY t y ) ⊗ K ) V Fs V Ft == X s,t ∈ T (( f ( s, t ) X s xY t y ) ⊗ K ) V Fst = X r ∈ T X s ∈ T (( f ( s, s − r ) X s Y s − r xy ) ⊗ K ) V Fr == X r ∈ T ((( XY ) r xy ) ⊗ K ) V Fr = ϕ (( X ⊗ x )( Y ⊗ y ))so ϕ is a C*-homomorphism. 103ssume now E = IK and let X ∈ S ( E ) and x ∈ F . Then ψϕ ( X ⊗ x ) = ψ X t ∈ T (( X t x ) ⊗ K ) V Ft = X t ∈ T V Et ⊗ ( X t x ) == X t ∈ T X t V Et ! ⊗ x = X ⊗ x which proves the last assertion (by using the first assertion). We draw the reader’s attention to the fact that in additive groups the neutralelement is denoted by 0 and not by 1. T := ZZ PROPOSITION 3.1.1 a) The map ψ : F ( ZZ , E ) −→ U n E c , f f (1 , is a group isomorphism.b) ψ ( { δλ | λ ∈ Λ( ZZ , E ) } ) = (cid:8) x (cid:12)(cid:12) x ∈ U n E c (cid:9) .c) If there is an x ∈ E c with x = f (1 , (in which case x ∈ U n E c ) thenthe map ϕ : S ( f ) −→ E × E , X ( X + xX , X − xX ) is an E -C*-isomorphism.d) If IK = IC and if A is a connected and simply connected compact space ora totally disconnected compact space then for every x ∈ U n C ( A ) there isa y ∈ C ( A, IC) with x = e y .e) Assume IK = IR . ) There are uniquely p, q ∈ Pr E c with p + q = 1 E , pf (1 ,
1) = p, qf (1 ,
1) = − q .e ) The map ϕ : S ( f ) −→ ( pE ) × ( pE ) × ◦ z}|{ qE , X ˜ X , where ◦ z}|{ qE denotes the complexification of the C*-algebra qE and ˜ X := ( p ( X + X ) , p ( X − X ) , ( qX , qX )) for every X ∈ S ( f ) , is an E -C*-isomorphism. In particular if f (1 ,
1) = − E then S ( f ) is isomorphic to the complexification of E .f ) Assume IK = IC , let σ ( E c ) be the spectrum of E c , and let c f be thefunction of C ( σ ( E c ) , IC) corresponding to f by the Gelfand transform.Then n e iθ (cid:12)(cid:12)(cid:12) θ ∈ IR , e iθ ∈ c f ( σ ( E c )) o is the spectrum of V . a) follows from Proposition 1.1.2 a) (and Proposition 1.1.4 a) ).b) follows from Definition 1.1.3.c) For X, Y ∈ S ( f ), by Theorem 2.1.9 c),g) (and Proposition 1.1.2 a)),( X ∗ ) = ( X ) ∗ , ( X ∗ ) = ( x ∗ ) ( X ) ∗ , ( XY ) = X Y + x X Y , ( XY ) = X Y + X Y , so ϕ ( X ∗ ) = (( X ) ∗ + x ( x ∗ ) ( X ) ∗ , ( X ) ∗ − x ( x ∗ ) ( X ) ∗ ) == (( X ) ∗ + x ∗ ( X ) ∗ , ( X ) ∗ − x ∗ ( X ) ∗ ) = ( ϕX ) ∗ , ( ϕX )( ϕY ) = (( X + xX )( Y + xY ) , ( X − xX )( Y − xY )) == ( X Y + xX Y + xX Y + x X Y , X Y − xX Y − xX Y + x X Y ) =105 (( XY ) + x ( XY ) , ( XY ) − x ( XY ) = ϕ ( XY )i.e. ϕ is an E -C*-homomorphism. ϕ is obviously injective.Let ( y, z ) ∈ E × E . If we take X ∈ S ( f ) with X := 12 ( y + z ) , X := 12 x ∗ ( y − z )then ϕX = ( y, z ), i.e. ϕ is surjective.d) is known. e ) follows by using the spectrum of E c . e ) Put ψ : S ( f ) −→ ◦ z}|{ qE , X ( qX , qX ) . For
X, Y ∈ S ( f ), by Theorem 2.1.9 c),g), ψ ( X ∗ ) = ( q ( X ∗ ) , q ( X ∗ ) ) = ( q ( X ) ∗ , qf (1 , ∗ ( X ) ∗ ) == (( qX ) ∗ , − ( qX ) ∗ ) = ( ψX ) ∗ , ( ψX )( ψY ) = ( qX , qX )( qY , qY ) == ( q ( X Y − X Y ) , ( q ( X Y + X Y ))) = ψ ( XY )so ψ is an E -C*-homomorphism. Thus by c), ϕ is an E -C*-homomorphism.The bijectivity of ϕ is easy to see.f) By Proposition 2.1.2 e), V is unitary so its spectrum is contained in (cid:8) e iθ (cid:12)(cid:12) θ ∈ IR (cid:9) . For θ ∈ IR and X ∈ S ( f ),( e iθ V − V ) X = X ( e iθ − V ) == (( e iθ X ) ⊗ K ) V + (( e iθ X ) ⊗ K ) V − ( X ⊗ K ) V − (( f X ) ⊗ K ) V == (( e iθ X − f X ) ⊗ K ) V + (( e iθ X − X ) ⊗ K ) V . Thus X is the inverse of e iθ V − V iff X = e iθ X and e iθ X − f X = 1 E , i.e.( e iθ − f ) X = 1 E . Therefore e iθ V − V is invertible iff e iθ − c f does notvanish on σ ( E c ). 106 OROLLARY 3.1.2
Assume
IK := IR and let S be a group, F a unital C*-algebra, g ∈ F ( S, F ) , and h : ( S × ZZ ) × ( S × ZZ ) −→ U n F c , (( s , t ) , ( s , t )) (cid:26) − g ( s , s ) if ( t , t ) = (1 , g ( s , s ) if ( t , t ) = (1 , . a) h ∈ F ( S × ZZ , F ) .b) S ( h ) ≈ ◦ z}|{ S ( g ) , S k·k ( h ) ≈ ◦ z }| { S k·k ( g ) . Put E := IR in the above Proposition and define f ∈ F ( ZZ , IR) by f (1 ,
1) = − e ), S ( f ) ≈ IC. Thus by Propo-sition 2.2.11 c),e), S ( h ) ≈ S ( g ) ⊗ S ( f ) ≈ ◦ z}|{ S ( g ) , S k·k ( h ) ≈ S k·k ( g ) ⊗ S k·k ( f ) ≈ ◦ z }| { S k·k ( g ) . DEFINITION 3.1.3
We put
IT := { z ∈ IC | | z | = 1 } . EXAMPLE 3.1.4
Let E := C ( IT , IC) and f ∈ F ( ZZ , E ) with f (1 ,
1) : IT −→ U n IC , z z . If we put ˜ X : IT −→ IC , z X ( z ) + zX ( z ) for every X ∈ S ( f ) then the map ϕ : S ( f ) −→ E , X ˜ X is an isomorphism of C*-algebras (but not an E -C*-isomorphism). X, Y ∈ S ( f ), by Theorem 2.1.9 c),g),( X ∗ ) = ( X ) ∗ , ( X ∗ ) = f (1 , X ) ∗ , ( XY ) = X Y + f (1 , X Y , ( XY ) = X Y + X Y so for z ∈ IT, f X ∗ ( z ) = X ∗ ( z ) + z ¯ z X ∗ ( z ) = X ( z ) + zX ( z ) = ˜ X ∗ ( z ) , ( ˜ X ( z ))( ˜ Y ( z )) = ( X ( z ) + zX ( z ))( Y ( z ) + zY ( z )) == X ( z ) Y ( z ) + zX ( z ) Y ( z ) + zX ( z ) Y ( z ) + z X ( z ) Y ( z ) == ( XY ) ( z ) + z ( XY ) ( z ) = g XY ( z ) , f X ∗ = ˜ X ∗ , ˜ X ˜ Y = g XY , i.e. ϕ is a C*-homomorphism. If ϕX = 0 then for z ∈ IT, X ( z ) + zX ( z ) = 0so, successively, X ( z ) − zX ( z ) = 0 , X ( z ) = X ( z ) = 0 , X = X = 0 , X = 0and ϕ is injective.Put G := ( X k ∈ ZZ c k z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( c k ) k ∈ ZZ ∈ IC ( ZZ ) ) ⊂ E .
Let x := X k ∈ ZZ c k z k ∈ G and take X ∈ S ( f ) with X := X k ∈ ZZ c k z k , X := X k ∈ ZZ c k +1 z k . Then ˜ X = X k ∈ ZZ c k z k + z X k ∈ ZZ c k +1 z k = x so G ⊂ ϕ ( S ( f )). Since G is dense in E , ϕ ( S ( f )) = E and ϕ is surjective.108 EFINITION 3.1.5
For every x ∈ C ( IT , IC) which does not take the value we put w ( x ) := winding number of x := 12 πi Z x d zz = 12 πi [ log x ( e iθ )] θ =2 πθ =0 ∈ ZZ . If A is a connected compact space and γ is a cycle in A (i.e. a continuousmap of IT in A ), which is homologous to 0 (or more generally, if a multiple of γ is homologous to 0), then for every x ∈ C ( A, U n
IC) we have w ( x ◦ γ ) = 0. If A is a compact space and x ∈ C ( A, U n
IC) such that w ( x ◦ γ ) = 0 for every cycle γ in A then there is a y ∈ C ( A, IC) with x = e y . EXAMPLE 3.1.6
Let E := C ( IT , IC) , f ∈ F ( ZZ , E ) , and n := w ( f (1 , .a) If n is even then there is an x ∈ U n E with winding number equal to n such that the map S ( f ) −→ E × E, X ( X + xX , X − xX ) is an E -C*-isomorphism.b) If n is odd then S ( f ) is isomorphic to E .c) The group F ( ZZ , E ) / Λ( ZZ , E ) is isomorphic to ZZ and Card ( { S ( g ) | g ∈ F ( ZZ , E ) } / ≈ S ) = 2 . d) There is a complex unital C*-algebra E and a family ( f β ) β ∈ P (IN) in F ( ZZ , E ) such that for distinct β, γ ∈ P (IN) , S ( f β )
6≈ S ( f γ ) . Put α : IT −→ U n IC , z z . Since w ( f (1 , α − n ) = 0, there is a y ∈ U n E with w ( y ) = 0 and f (1 , α − n = y . a) If we put x := yα n then w ( x ) = n and f (1 ,
1) = x and the assertionfollows from Proposition 3.1.1 c). 109) We put x := yα n − . Then f (1 ,
1) = αx . Take g ∈ F ( ZZ , E ) with g (1 ,
1) = α and λ ∈ Λ( ZZ , E ) with ( δλ )(1 ,
1) = x (Proposition 3.1.1 a),b)).Then f = gδλ . By Example 3.1.4, S ( g ) is isomorphic to E and by Proposition2.2.2 a ⇒ a , S ( f ) is also isomorphic to E .c) follows from Proposition 3.1.1 b) and Proposition 2.2.2 a),c).d) Denote by E the C*-direct product of the sequence ( C ( IT , IC n,n )) n ∈ IN andfor every β ∈ { , } IN define f β ∈ F ( ZZ , E ) by f β (1 ,
1) : IN −→ U n E c , n α β ( n ) IC n,n . By a) and b), for distinct β, γ ∈ { , } IN , S ( f β )
6≈ S ( f γ ) (Proposition 2.1.26a)). EXAMPLE 3.1.7
Let
I, J be finite disjoint sets and for all i ∈ I ∪ J and j ∈ J put A i := B j := IT . We define the compact spaces A and B in thefollowing way. For A we take first the disjoint union of the spaces A i for all i ∈ I ∪ J and identify then the points ∈ A i for all i ∈ I ∪ J . For B we takefirst the disjoint union of all the spaces A i for all i ∈ I ∪ J and of the spaces B j for all j ∈ J and identify first the points ∈ A i for all i ∈ I ∪ J and identifythen also the points − ∈ A i for all i ∈ I and ∈ B j for all j ∈ J .Let E := C ( A, IC) and f ∈ F ( ZZ , E ) with f (1 ,
1) : A −→ U n IC , z (cid:26) z if z ∈ A i with i ∈ I if z ∈ A i with i ∈ J .
For every X ∈ S ( f ) define ˜ X ∈ C ( B, IC) by ˜ X : B −→ IC , z X ( z ) + zX ( z ) if z ∈ A i with i ∈ IX ( z ) + X ( z ) if z ∈ A i with i ∈ JX ( z ) − X ( z ) if z ∈ B j with j ∈ J .
Then the map ϕ : S ( f ) −→ C ( B, IC) , X ˜ X is an isomorphism of C*-algebras. Let
X, Y ∈ S ( f ). By Theorem 2.1.9 c),g),( X ∗ ) = ( X ) ∗ , ( X ∗ ) = f (1 , X ) ∗ , XY ) = X Y + f (1 , X Y , ( XY ) = X Y + X Y . For z ∈ A i with i ∈ I , f X ∗ ( z ) = ( X ∗ ) ( z ) + z ( X ∗ ) ( z ) = X ( z ) + z ¯ z X ( z ) == X ( z ) + zX ( z ) = ( ˜ X ) ∗ ( z ) , ˜ X ( z ) ˜ Y ( z ) = ( X ( z ) + zX ( z ))( Y ( z ) + zY ( z )) == X ( z ) Y ( z ) + zX ( z ) Y ( z ) + zX ( z ) Y ( z ) + z X ( z ) Y ( z ) == ( XY ) ( z ) + z ( XY ) ( z ) = g XY ( z ) . For z ∈ A j or z ∈ B j with j ∈ J , f X ∗ ( z ) = ( X ∗ ) ( z ) ± ( X ∗ ) ( z ) = X ( z ) ± X ( z ) = ( ˜ X ) ∗ ( z ) , ˜ X ( z ) ˜ Y ( z ) = ( X ( z ) ± X ( z ))( Y ( z ) ± Y ( z )) == X ( z ) Y ( z ) ± X ( z ) Y ( z ) ± X ( z ) Y ( z ) + X ( z ) Y ( z ) == ( XY ) ( z ) ± ( XY ) ( z ) = g XY ( z ) . Thus ϕ is a C*-homomorphism. Assume ˜ X = 0. For z ∈ A i with i ∈ I , X ( z ) + zX ( z ) = 0so, successively, X ( z ) − zX ( z ) = 0 , X ( z ) = X ( z ) = 0 , X ( z ) = 0 . For z ∈ A j with j ∈ J , (cid:26) X ( z ) + X ( z )= 0 X ( z ) − X ( z )= 0 , so X ( z ) = X ( z ) = 0 , X ( z ) = 0 . Thus ϕ is injective.Let x ∈ C ( B, IC) such that for every i ∈ I there is a family ( c i,k ) k ∈ ZZ ∈ IC ( ZZ ) with x ( z ) = X k ∈ ZZ c i,k z k z ∈ A i . Define X , X ∈ E in the following way. If z ∈ A i with i ∈ I weput X ( z ) := X k ∈ ZZ c i, k z k , X ( z ) := X k ∈ ZZ c i, k +1 z k . If z ∈ A j with j ∈ J then we put z ′ := z ∈ B j , X ( z ) := 12 ( x ( z ) + x ( z ′ )) , X ( z ) := 12 ( x ( z ) − x ( z ′ )) . It is easy to see that X and X are well defined. Then˜ X ( z ) = X k ∈ ZZ c i, k z k + z X k ∈ ZZ c i, k +1 z k = x ( z )for all z ∈ A i with i ∈ I and ˜ X ( z ) = x ( z ) for all z ∈ A j ∪ B j with j ∈ J . Sincethe elements x of the above form are dense in C ( B, IC), ϕ is surjective. EXAMPLE 3.1.8
Let E := C ( IT , IC) and f, g ∈ F ( ZZ , E ) with (cid:26) f (1 ,
1) : IT −→ U n IC , ( z , z ) z g (1 ,
1) : IT −→ U n IC , ( z , z ) z . Then the maps (cid:26) S ( f ) −→ E, X X ( z , z ) + z X ( z , z ) S ( g ) −→ E, X X ( z , z ) + z X ( z , z ) are isomorphisms of C*-algebras.Remark. S ( f ) and S ( g ) are isomorphic but not E -C*-isomorphic. EXAMPLE 3.1.9
Let E := C ( IT , IC) and f ∈ F ( ZZ , E ) with f (1 ,
1) : IT −→ U n IC , ( z , z ) z z . If we put ˜ X : IT −→ IC , ( z , z ) X ( z , z ) + z z X ( z , z ) for every X ∈ S ( f ) then the map ϕ : S ( f ) −→ E , X ˜ X s an injective unital C*-homomorphism with ϕ ( S ( f )) = G := (cid:8) x ∈ E | ( z , z ) ∈ IT = ⇒ x ( z , z ) = x ( − z , − z ) (cid:9) . In particular S ( f ) is isomorphic to E . Let
X, Y ∈ S ( f ). By Theorem 2.1.9 c),g),( X ∗ ) = ( X ) ∗ , ( X ∗ ) = f (1 , X ) ∗ , ( XY ) = X Y + f (1 , X Y , ( XY ) = X Y + X Y so for ( z , z ) ∈ IT , f X ∗ ( z , z ) = X ∗ ( z , z ) + z z ¯ z ¯ z X ∗ ( z , z ) == X ( z , z ) + z z X ( z , z ) = ˜ X ( z , z ) , ( ˜ X ( z , z ))( ˜ Y ( z , z )) == ( X ( z , z ) + z z X ( z , z ))( Y ( z , z ) + z z Y ( z , z )) == X ( z , z ) Y ( z , z ) + z z X ( z , z ) Y ( z , z )++ z z X ( z , z ) Y ( z , z ) + z z X ( z , z ) Y ( z , z ) == ( XY ) ( z , z ) + z z ( XY ) ( z , z ) = g XY ( z , z ) , i.e. ϕ is a unital C*-homomorphism. If ˜ X = 0 then for ( z , z ) ∈ IT , X ( z , z ) + z z X ( z , z ) = 0so, successively, X ( z , z ) − z z X ( z , z ) = 0 , X ( z , z ) = X ( z , z ) = 0 ,X = X = 0 , X = 0and ϕ is injective.The inclusion S ( f ) ⊂ G is obvious. Let ( a j,k ) j,k ∈ ZZ , ( b j,k ) j,k ∈ ZZ ∈ IC ( ZZ × ZZ ) and x = X j,k ∈ ZZ a j,k z j z k + X j,k ∈ ZZ b j,k z j +11 z k +12 ∈ G . X := X j,k ∈ ZZ a j,k z j z k , X := X j,k ∈ ZZ b j,k z j z k . Then ˜ X = x . Since the elements of the above form are dense in G , ϕ ( S ( f )) = G .If we consider the equivalence relation ∼ on IT defined by( z , z ) ∼ ( w , w ) : ⇐⇒ z = − w , z = − w then the quotient space IT / ∼ is homeomorphic to IT . Thus S ( f ) is isomorphicto E . EXAMPLE 3.1.10
Let E := C ( IT , IC) .a) For x ∈ U n E and z ∈ IT , w ( x ( · , z )) and w ( x ( z, · )) do not depend on z ,where w denotes the winding number (Definition 3.1.5) .b) If x ∈ U n E and if w ( x ( · , w ( x (1 , · )) = 0 then there is a y ∈ U n E with x = y .c) Let f ∈ F ( ZZ , E ) and put α : IT −→ IT , z ( z, , β : IT −→ IT , z (1 , z ) ,m := w ( f (1 , ◦ α ) , n := w ( f (1 , ◦ β ) .c ) If m + n is odd then S ( f ) is isomorphic to E . c ) If m and n are even then S ( f ) is isomorphic to E × E . c ) If m and n are odd then S ( f ) is isomorphic to E .d) The group F ( ZZ , E ) / Λ( ZZ , E ) is isomorphic to ZZ × ZZ and Card ( { S ( f ) | f ∈ F ( ZZ , E ) } / ≈ S ) = 4 . g ∈ F ( ZZ , E ) with g (1 ,
1) : IT −→ U n IC , ( z , z ) z m z n . Then w ( g (1 , ◦ α ) = m , w ( g (1 , ◦ β ) = n . By b), there is an x ∈ U n E with f (1 ,
1) = x g (1 , a ⇒ a , S ( f ) ≈ S ( g ). c ) Assume m even and put y : IT −→ U n IC , ( z , z ) z m z n − . If h ∈ F ( ZZ , E ) with h (1 ,
1) : IT −→ U n IC , ( z , z ) z then g (1 ,
1) = y h (1 , a ⇒ a , S ( g ) ≈ S ( h ) and by Example 3.1.8 a ⇒ a , S ( h ) ≈ E . Thus S ( f ) ≈ E . c ) If we put y : IT −→ U n IC , ( z , z ) z m z n then g (1 ,
1) = y and the assertion follows from Proposition 3.1.1 c). c ) We put y : IT −→ U n IC , ( z , z ) z m − z n − and take h ∈ F ( ZZ , E ) with h (1 ,
1) : IT −→ U n IC , ( z , z ) z z then g (1 ,
1) = y h (1 ,
1) so by Proposition 3.1.1 b) and Proposition 2.2.2 a ⇒ a , S ( g ) ≈ S ( h ). By Example 3.1.9 S ( h ) ≈ E , so S ( f ) ≈ E .d) follows from b), Proposition 3.1.1 b), and Proposition 2.2.2 a),c).115 emark. In a similar way it is possible to show that for every n ∈ IN, F ( ZZ , IT n ) / Λ( ZZ , IT n ) is isomorphic to ( ZZ ) n and Card ( { S ( f ) | f ∈ F ( ZZ , IT n ) } / ≈ S ) = 2 n . EXAMPLE 3.1.11
Let
I, J, K be finite pairwise disjoint sets and for every i ∈ I ∪ J ∪ K and k ∈ K put A i := B k := IT . We define the compact spaces A and B in the following way. For A we take first the disjoint union of thespaces A i with i ∈ I ∪ J ∪ K and then identify the points (1 , ∈ A i for all i ∈ I ∪ J ∪ K . For B we take first the disjoint union of the spaces A i with i ∈ I ∪ J ∪ K and of the spaces B k with k ∈ K . Then we identify the points (1 , ∈ A i for all i ∈ I ∪ J ∪ K and then we identify for every j ∈ J the points ( z , z ) ∈ A j with the points ( − z , − z ) ∈ A j and finally we identify the points ( − , ∈ A i for all i ∈ I ∪ J with the points (1 , ∈ B k for all k ∈ K .Let E := C ( A, IC) and f ∈ F ( ZZ , A ) such that f (1 ,
1) : A −→ U n IC , ( z , z ) z if ( z , z ) ∈ A i with i ∈ Iz z if ( z , z ) ∈ A i with i ∈ J if ( z , z ) ∈ A i with i ∈ K .
We define for every X ∈ S ( f ) a map ˜ X : B → IC by ( z , z ) X ( z , z ) + z X ( z , z ) if ( z , z ) ∈ A i with i ∈ IX ( z , z ) + z z X ( z , z ) if ( z , z ) ∈ A i with i ∈ JX ( z , z ) + X ( z , z ) if ( z , z ) ∈ A i with i ∈ KX ( z , z ) − X ( z , z ) if ( z , z ) ∈ B k with k ∈ K .
Then the map S ( f ) −→ C ( B, IC) , X ˜ X is an isomorphism of C*-algebras. The proof is similar to the proof of Example 3.1.7.
EXAMPLE 3.1.12 If n ∈ IN , E := C ( IT n , IC) , and f ∈ F ( ZZ , C ( IT n , IC)) then S ( f ) is isomorphic either to C ( IT n , IC) or to C ( IT n , IC) × C ( IT n , IC) . XAMPLE 3.1.13
Assume E := C ( A, IC) , where A denotes Moebius’s band(resp. Klein’s bottle), i.e. the topological space obtained from [0 , π ] × [ − π, π ] by identifying the points (0 , α ) and (2 π, − α ) for all α ∈ [ − π, π ] (resp. and thepoints ( θ, − π ) and ( θ, π ) for all θ ∈ [0 , π ] ). We put B := IT × [ − π, π ] (resp. B := IT ) and ˜ x : [0 , π ] × [ − π, π ] −→ IC , ( θ, α ) (cid:26) x (2 θ, α ) if θ ∈ [0 , π ] x (2( θ − π ) , − α ) if θ ∈ [ π, π ] for every x ∈ E .a) ˜ x is well-defined and belongs to C ( B, IC) for every x ∈ E .b) If f , ( θ, α ) = e iθ for all ( θ, α ) ∈ [0 , π ] × [ − π, π ] then the map ϕ : S ( f ) −→ C ( B, IC) , X f X + e iθ f X is a C*-isomorphism.c) Let x ∈ U n E . If w ( x ( · , (where w denotes the winding number)then there is a y ∈ E with e y = x .d) Let x ∈ U n E and put n := w ( x ( · , . Then there is a y ∈ E with e y = e − inθ x .e) The group F ( ZZ , A ) / Λ( ZZ , A ) is isomorphic to ZZ .f ) If w ( f , ( · , is even (resp. odd) then S ( f ) is isomorphic to E × E (resp. to C ( B, IC) ). a) For α ∈ [ − π, π ],˜ x ( π, α ) = x (2 π, α ) = x (0 , − α ) = ˜ x ( π, α )so ˜ x is well-defined. Moreover˜ x (0 , α ) = x (0 , α ) = x (2 π, − α ) = ˜ x (2 π, α )and in the case of Klein’s bottle (cid:26) ˜ x ( θ, − π ) = x (2 θ, − π ) = x (2 θ, π ) = ˜ x ( θ, π ) if θ ∈ [0 , π ]˜ x ( θ, − π ) = x (2( θ − π ) , π ) = x (2( θ − π ) , − π ) = ˜ x ( θ, π ) if θ ∈ [ π, π ]117.e. ˜ x ∈ C ( B, IC).b) For
X, Y ∈ S ( f ) and ( θ, α ) ∈ [0 , π ] × [ − π, π ], by Theorem 2.1.9 c),g),( ϕX ∗ )( θ, α ) = ^ ( X ∗ ) ( θ, α ) + e iθ ^ ( X ∗ ) ( θ, α ) == ^ ( X ) ∗ ( θ, α ) + e iθ ^ z }| { ( e − iθ ( X ) ∗ )( θ, α ) == (cid:26) X (2 θ, α ) + e iθ ( e − iθ X (2 θ, α )) if θ ∈ [0 , π ] X (2( θ − π ) , − α ) + e iθ ( e − i ( θ − π ) X (2( θ − π ) , − α )) if θ ∈ [ π, π ] == ( X (2 θ, α ) + e iθ X (2 θ, α ) if θ ∈ [0 , π ] X (2( θ − π ) , − α ) + e iθ X (2( θ − π ) , − α ) if θ ∈ [ π, π ] = ϕX ( θ, α ) , ( ϕX )( ϕY ) = ( f X + e iθ f X )( f Y + e iθ f Y ) = f X f Y + e iθ f X f Y + e iθ f X f Y + e iθ f X f Y ,ϕ ( XY ) = ^ ( XY ) + e iθ ^ ( XY ) == f X f Y + e iθ f X f Y + e iθ ( f X f Y + f X f Y ) = ( ϕX )( ϕY ) , i.e. ϕ is a C*-homomorphism. If ϕX = 0 then for α ∈ [ − π, π ], (cid:26) X (2 θ, α ) + e iθ X (2 θ, α ) = 0 if θ ∈ [0 , π ] X (2( θ − π ) , − α ) + e iθ X (2( θ − π ) , − α ) = 0 if θ ∈ [ π, π ]so for θ ∈ [0 , π ], replacing θ by θ + π and α by − α in the second relation, X (2 θ, α ) − e iθ X (2 θ, α ) = 0 . It follows successively X (2 θ, α ) = X (2 θ, α ) = 0 ,X = X = 0 , X = 0 . Thus ϕ is injective.Let y ∈ C ( B, IC). Put ( X : [0 , π ] × [ − π, π ] −→ IC , ( θ, α ) ( y ( θ , α ) + y ( θ + π, − α )) X : [0 , π ] × [ − π, π ] −→ IC , ( θ, α ) e − i θ ( y ( θ , α ) − y ( θ + π, − α )) . α ∈ [ − π, π ], (cid:26) X (0 , α ) = ( y (0 , α ) + y ( π, − α )) X (2 π, − α ) = ( y ( π, − α ) + y (2 π, α )) (cid:26) X (0 , α ) = ( y (0 , α ) − y ( π, − α )) X (2 π, − α ) = − ( y ( π, − α ) − y (2 π, α ))so X , X ∈ E . Moreover for ( θ, α ) ∈ [0 , π ] × [ − π, π ], f X ( θ, α ) + e iθ f X ( θ, α ) == (cid:26) X (2 θ, α ) + e iθ X (2 θ, α ) if θ ∈ [0 , π ] X (2( θ − π ) , − α ) + e iθ X (2( θ − π ) , − α ) if θ ∈ [ π, π ] == (cid:26) ( y ( θ, α ) + y ( θ + π, − α ) + y ( θ, α ) − y ( θ + π, − α )) if θ ∈ [0 , π ] ( y ( θ − π, − α ) + y ( θ, α ) − y ( θ − π, − α ) + y ( θ, α )) if θ ∈ [ π, π ] == y ( θ, α )i.e. ϕ is surjective.c) If A is Moebius’s band then the assertion is obvious so assume A is Klein’sbottle. The winding numbers of (cid:26) [0 , π ] −→ IC , α x (0 , α )[0 , π ] −→ IC , α x (2 π, α )are equal by homotopy, but their sum is equal to 0. Thus these winding numbersare equal to 0. The paths θ and α on A generate the homotopy group of A .Thus the winding number of x on any path of A is 0 and the assertion follows.d) The winding number of[0 , π ] −→ IC , θ e − inθ x ( θ, a ⇒ a , and Propo-sition 3.1.1 c). 119 .2 T := ZZ × ZZ PROPOSITION 3.2.1
Let E be a unital C*-algebra and let a, b, c be the threeelements of ( ZZ × ZZ ) \ { (0 , } . Put A := (cid:8) ( α, β, γ, ε ) ∈ ( U n E c ) (cid:12)(cid:12) ε = 1 E (cid:9) and for every ̺ ∈ A and σ ∈ ( U n E c ) denote by f ̺ and g σ the functionsdefined by the following tables: f ̺ a b ca βγ γ βb εγ εαγ αc εβ εα αβ g σ a b ca α αβγ ∗ αγβ ∗ b αβγ ∗ β βγα ∗ c αγβ ∗ βγα ∗ γ a) f ̺ ∈ F ( ZZ × ZZ , E ) for every ̺ ∈ A and the map A −→ F ( ZZ × ZZ , E ) , ̺ f ̺ is bijective.b) g σ ∈ { δλ | λ ∈ Λ( ZZ × ZZ , E ) } for every σ ∈ ( U n E c ) and the map ( U n E c ) −→ { δλ | λ ∈ Λ( ZZ × ZZ , E ) } , σ g σ is bijective.c) The following are equivalent for all ̺ := ( α, β, γ, ǫ ) ∈ A and ̺ ′ :=( α ′ , β ′ , γ ′ , ǫ ′ ) ∈ A : c ) S ( f ̺ ) ≈ S S ( f ̺ ′ ) . c ) ε = ε ′ and there are x, y, z ∈ U n E c with x = ββ ′∗ γγ ′∗ , y = αα ′∗ γγ ′∗ , z = αα ′∗ ββ ′∗ .c ) ε = ε ′ and there are x, y ∈ U n E c with x = ββ ′∗ γγ ′∗ , y = αα ′∗ γγ ′∗ . d) The following are equivalent for all ̺ := ( α, β, γ, ε ∈ A ) and X ∈ S ( f ̺ ) : ) X ∈ n V f ̺ t (cid:12)(cid:12)(cid:12) t ∈ ZZ × ZZ o c . d ) t ∈ ZZ × ZZ = ⇒ εX t = X t .e) The following are equivalent for all ̺ := ( α, β, γ, ε ∈ A ) and X ∈ S ( f ̺ ) : e ) X ∈ S ( f ̺ ) c . e ) t ∈ ZZ × ZZ = ⇒ εX t = X t ∈ E c .f ) For ̺ := ( α, β, γ, ε ) ∈ A and X, Y ∈ S ( f ̺ ) , ( X ∗ ) = X ∗ , ( X ∗ ) a = β ∗ γ ∗ X ∗ a , ( X ∗ ) b = εα ∗ γ ∗ X ∗ b , ( X ∗ ) c = α ∗ β ∗ X ∗ c , ( XY ) = X Y + βγX a Y a + εαγX b Y b + αβX c Y c , ( XY ) a = X Y a + X a Y + αX b Y c + εαX c Y b , ( XY ) b = X Y b + βX a Y c + X b Y + εβX c Y a , ( XY ) c = X Y c + γX a Y b + εγX b Y a + X c Y . g) Assume IK = IC , let σ ( E c ) be the spectrum of E c , and for every δ ∈ E c let ˆ δ be its Gelfand transform. Then σ ( V a ) = n e iθ (cid:12)(cid:12)(cid:12) θ ∈ IR , e iθ ∈ c βγ ( σ ( E c )) o ,σ ( V b ) = n e iθ (cid:12)(cid:12)(cid:12) θ ∈ IR , e iθ ∈ c αγ ( σ ( E c )) o ,σ ( V c ) = n e iθ (cid:12)(cid:12)(cid:12) θ ∈ IR , e iθ ∈ c αβ ( σ ( E c )) o . a) is a long calculation.b) is easy to verify. c ⇒ c By Proposition 2.2.2 a ⇒ a there is a λ ∈ Λ( ZZ × ZZ , E ) with f ̺ = f ̺ ′ δλ . By b), there is a σ := ( x, y, z ) ∈ ( U n E c ) with f ̺ = f ̺ ′ g σ . We get ε = ε ′ and αα ′∗ = x ∗ yz , ββ ′∗ = xy ∗ z , γγ ′∗ = xyz ∗ . It follows xyz = αα ′∗ ββ ′∗ γγ ′∗ so x = ββ ′∗ γγ ′∗ , y = αα ′∗ γγ ′∗ , z = αα ′∗ ββ ′∗ . ⇒ c is trivial. c ⇒ c If we put z := xyγ ∗ γ ′ then z = ββ ′∗ γγ ′∗ αα ′∗ γγ ′∗ γ ∗ γ ′ = αα ′∗ ββ ′∗ .c ⇒ c follows from b) and Proposition 2.2.2 a ⇒ a .d) follows from Corollary 2.1.24 b).e) follows from Corollary 2.1.24 c).f) follows from Theorem 2.1.9 c),g).g) follows from f). COROLLARY 3.2.2
We use the notation of
Proposition 3.2.1 and take ̺ :=( α, β, γ, ε ) ∈ A .a) Assume ε = 1 E and there are x, y ∈ U n E with x = βγ , y = αγ . Put z := xyγ ∗ . a ) x, y, z ∈ U n E c , z = αβ . a ) For every λ, µ ∈ {− , } the map ϕ λ,µ : S ( f ̺ ) −→ E , X X + λxX a + µyX b + λµzX c is an E -C*-homomorphism. a ) The map S ( f ̺ ) −→ E , X ( ϕ , X, ϕ , − X, ϕ − , X.ϕ − , − X ) is an E -C*-isomorphism.b) Assume IK := IR , ε = 1 E , and there are x, y ∈ U n E with x = − βγ , y = αγ , ( resp. y = − αγ ) . ut z := xyγ ∗ . Then x, y, z ∈ U n E c , z = − αβ (resp. z = αβ ), andthe maps S ( f ̺ ) −→ ( ◦ E ) , X ( X + ixX a + yX b + izX c , X + ixX a − yX b − izX c ) S ( f ̺ ) −→ ( ◦ E ) , X ( X + ixX a + iyX b − zX c , X + ixX a − iyX b + zX c ) are respectively E -C*-isomorphisms (where ◦ E denotes the complexifica-tion of E ).c) Assume IK := IR , ε = − E , and there are x, y ∈ E c with x = − βγ , y = αγ . Put z := xyγ ∗ . Then x, y, z ∈ U n E c , z = − αβ , and the map S ( f ̺ ) −→ IH ⊗ E, X X + ixX a + jyX b + kzX c , where i, j, k are the canonical units of IH , is an E -C*-isomorphism.d) If ε = − E and there is an x ∈ U n E c with x = αβ then for every δ ∈ U n E c the map S ( f ̺ ) −→ E , , X (cid:20) X + xX c γδ ∗ ( βX a − xX b ) δ ( X a + xβ ∗ X b ) X − xX c (cid:21) is an E -C*-isomorphism. The proof is a long calculation using Proposition 3.2.1 f).
Remarks. d) is contained in Proposition 3.2.3 c). An example with ε = 1 E but different from a) is presented in Proposition 3.3.2. PROPOSITION 3.2.3
We use the notation of
Proposition 3.2.1 and take ̺ := ( α, β, γ, ε ) ∈ A .a) Let ϕ : S ( f ̺ ) → E , be an E -C*-isomorphism and put (cid:20) A t B t C t D t (cid:21) := ϕV t for every t ∈ ZZ × ZZ \ { (0 , } . Then ε = − E , A t , B t , C t , D t ∈ E c and A t + D t = 0 for every t ∈ ZZ × ZZ \ { (0 , } , and A ∗ a = β ∗ γ ∗ A a , A ∗ b = − α ∗ γ ∗ A b , A ∗ c = α ∗ β ∗ A c , ∗ a = β ∗ γ ∗ C a , B ∗ b = − α ∗ γ ∗ C b , B ∗ c = α ∗ β ∗ C c ,A a + B a C a = βγ , A b + B b C b = − αγ , A c + B c C c = αβ ,A a = βγ (1 E − | B a | ) , A b = − αγ (1 E − | B b | ) , A c = αβ (1 E − | B c | ) , A a A b + B a C b + B b C a = 0 , A b A c + B b C c + B c C b = 0 , A c A a + B c C a + B a C c = 0 ,αA a = A b A c + B b C c , αB a = A b B c − A c B b , αC a = A c C b − A b C c ,βA b = A a A c + B a C c , βB b = A a B c − A c B a , βC b = A c C a − A a C c ,γA c = A a A b + B a C b , γB c = A a B b − A b B a , γC c = A b C a − A a C b , | A a | + | A b | + | A c | 6 = 0 , | B a | + | B b | + | B c | 6 = 3 . E . b) Let ( A t ) t ∈ T , ( B t ) t ∈ T , ( C t ) t ∈ T , ( D t ) t ∈ T be families in E c satisfying theabove conditions and put X ′ := A a X a + A b X b + A c X c , X ′′ := B a X a + B b X b + B c X c ,X ′′′ := C a X a + C b X b + C c X c for every X ∈ S ( f ̺ ) . If ε = − E then the map S ( f ̺ ) −→ E , , X (cid:20) X + X ′ X ′′ X ′′′ X − X ′ (cid:21) is an E -C*-isomorphism.c) Let ε = − E and assume there is an x ∈ E c with x = βγ . Let y ∈ U n E c and put z := γ ∗ xy . Then x, y, z ∈ U n E c and the map ϕ : S ( f ̺ ) −→ E , , X (cid:20) X + xX a α ( yX b + zX c ) − γy ∗ X b + βz ∗ X c X − xX a (cid:21) is an E -C*-isomorphism such that ϕ ( 12 ( V + ( x ∗ ⊗ K ) V a )) = (cid:20) (cid:21) . In particular (by the symmetry of a,b,c), if ε = − E and if there is an x ∈ E c with x = βγ , or x = − αγ , or x = αβ then S ( f ̺ ) ≈ E E , . emark. Take ̺ := (1 E , E , E , − E ), ̺ ′ := (1 E , E , γ ′ , − E ). By c), S ( f ̺ ) ≈ E S ( f ̺ ′ ) and by Proposition 3.2.1 c ⇒ c , S ( f ̺ ) ≈ S S ( f ̺ ′ ) impliesthe existence of an x ∈ U n E c with x = γ ′ . COROLLARY 3.2.4
We use the notation of
Proposition 3.2.3 and take E :=IK , α = 1 , and β = γ = ε = − . Let S be a group, F a unital C*-algebra, g ∈ F ( S, F ) , and h : (( S × ( ZZ ) ) × ( S × ( ZZ ) )) −→ U n F c , (( s , t ) , ( s , t )) f ̺ ( t , t ) g ( s , s ) . a) h ∈ F ( S × ( ZZ ) , F ) .b) S ( h ) ≈ S ( g ) , , S k·k ( h ) ≈ S k·k ( g ) , . By Proposition 3.2.3 c), S ( f ) ≈ IK , , so by Proposition 2.2.11 c),e), S ( h ) ≈ IK , ⊗ S ( g ) ≈ S ( g ) , , S k·k ( h ) ≈ IK , ⊗ S k·k ( g ) ≈ S k·k ( g ) , . EXAMPLE 3.2.5
Let
IK := IC and E := C ( IT , IC) .a) With the notation of
Proposition 3.2.1 , if ̺ := ( α, β, γ, − ∈ A then S ( f ̺ ) ≈ E E , .b) Card ( { S ( f ) | f ∈ F ( ZZ × ZZ , E ) } / ≈ S ) = 16 . Put m := w ( α ) , n := w ( β ) , p := w ( γ ) , where w denotes the winding number. By Proposition 2.2.2 a ⇒ a , we mayassume α = z m , β = z n , γ = z p .a) If n + p is even then the assertion follows from Proposition 3.2.3 c). If n + p is odd then either m + p or m + n is even and the assertion follows againfrom Proposition 3.2.3 c). 125) follows from Proposition 2.2.2 a),c). Remark.
Assume IK := IR and let E be the real C*-algebra C ( IT , IC) ([C1]Theorem 4.1.1.8 a)), ε = − E , α : IT −→ IC , z z ,β : IT −→ IC , z z ,γ : IT −→ IC , z ¯ z , and ̺ := ( α, β, γ, ε ). Then by Corollary 3.2.2 c), S ( f ̺ ) ≈ IH ⊗ E . EXAMPLE 3.2.6
We put E := C ( IT , IC) , γ := 1 E , α : IT −→ IC , ( z , z ) z , β : IT −→ IC , ( z , z ) z , and (with the notation of Proposition 3.2.1 ) ̺ := ( α, β, γ, − E ) ∈ A .a) S ( f ̺ ) is not commutative and not E -C*-isomorphic to E , .b) If we put ˜ x : IT −→ IC , ( z , z ) x ( z , z ) for every x ∈ E then the map S ( f ̺ ) −→ E , , X (cid:20) ˜ X + αβ ˜ X c β ˜ X a − α ˜ X b β ˜ X a + α ˜ X b ˜ X − αβ ˜ X c (cid:21) is a C*-isomorphism.c) E , ≈ S ( f ̺ ) E E , . a) By Proposition 3.2.1 d), S ( f ̺ ) is not commutative. Assume S ( f ̺ ) ≈ E E , and let us use the notation of Proposition 3.2.3 a).Step 1 { A a = 0 } ⊂ { A b = 0 } { A a = 0 } ∩ { A b = 0 } 6 = ∅ . By Proposition 3.2.3 a),2 A a A b + B a C b + B b C a = 0 , B ∗ a = β ∗ C a , B ∗ b = − α ∗ C b so B a = 0 and B b = 0 on this set. We put A a =: | A a | e i ˜ A a , A b =: | A b | e i ˜ A b , B a =: | B a | e i ˜ B a , B b =: | B b | e i ˜ B b ,z =: e iθ , z =: e iθ , with ˜ A a , ˜ A b , ˜ B a , ˜ B b ∈ IR. By Proposition 3.2.3 a), 2 ˜ A a = θ , 2 ˜ A b = θ + π , B a C b + B b C a = − αγB a B ∗ b + βγB b B ∗ a = | B a || B b | ( e i ( θ + ˜ B b − ˜ B a ) − e i ( θ + ˜ B a − ˜ B b ) ) == | B a || B b | e i θ θ ( e i ( θ − θ + ˜ B b − ˜ B a ) − e i ( θ − θ + ˜ B a − ˜ B b ) ) == 2 | B a || B b | sin( θ − θ B b − ˜ B a ) e i θ θ π . Since 2 A a A b = − ( B a C b + B b C a ) there is a k ∈ ZZ with θ θ + π θ + θ + π k + 1) π which is a contradiction. Step 2 { A a = 0 } ⊂ { A c = 0 } The assertion follows from Step 1 by symmetry.Step 3 { A a = 0 } = { A b = A c = 0 } The assertion follows from Steps 1 and 2 and from | A a | + | A b | + | A c | 6 = 0.Step 4 The contradiction127y Step 3 and by the symmetry, the sets { A a = 0 } , { A b = 0 } , and { A c = 0 } are clopen and by | A a | + | A b | + | A c | 6 = 0 their union is equal to IT . So there isexactly one of these sets equal to IT which implies A a = z , or A b = − z or A c = z z and no one of these identities can hold.b) is a direct verification.c) follows from a) and b). T := ( ZZ ) n with n ∈ IN EXAMPLE 3.3.1
Assume f constant and put h s | t i := n Y i =1 ( − s ( i ) t ( i ) for all s, t ∈ T (where ZZ is identified with { , } ) and ϕ t : S ( f ) −→ E , X X s ∈ T h t | s i X s for all t ∈ T . Then the map ϕ : S ( f ) −→ E n , X ( ϕ t X ) t ∈ T is an E -C*-isomorphism. For r, s, t ∈ T , t + t = 0 , h s | t i = h t | s i , h r + s | t i = h r | t i h s | t i , h r | s + t i = h r | s i h r | t i . For t ∈ T and X, Y ∈ S ( f ), by Theorem 2.1.9 c),g), ϕ t ( X ∗ ) = X s ∈ T h t | s i ( X ∗ ) s = X s ∈ T h t | s i ( X s ) ∗ = ( ϕ t X ) ∗ , ϕ t X )( ϕ t Y ) = X r,s ∈ T h t | r i h t | s i X r Y s = X q,r ∈ T h t | r i h t | q − r i X r Y q − r == X q,r ∈ T h t | q i X r Y q − r = X q ∈ T h t | q i ( XY ) q = ϕ t ( XY )so ϕ t and ϕ are E -C*-homomorphisms.We have X t ∈ T h | t i = 2 n . We want to prove X t ∈ T h s | t i = 0for all s ∈ T , s = 0, by induction with respect to Card { i ∈ IN n | s ( i ) = 0 } .Let i ∈ IN n with s ( i ) = 0 and put r := s + e i , T := { t ∈ T | t ( i ) = 0 } , T := { t ∈ T | t ( i ) = 1 } . Then X t ∈ T h s | t i = X t ∈ T h r | t i , X t ∈ T h s | t i = − X t ∈ T h r | t i . But X t ∈ T h r | t i = X t ∈ T h r | t i = 2 n − if r = 0. By the hypothesis of the induction X t ∈ T h r | t i = X t ∈ T h r | t i = 0if r = 0 (with IN n replaced by IN n \ { i } , since r ( i ) = 0). This finishes the proofby induction.For r ∈ T and X ∈ S ( f ), by the above, X t ∈ T h r | t i ϕ t X = X s,t ∈ T h r | t i h t | s i X s = X s,t ∈ T h r + s | t i X s == X s ∈ T \{ r } X t ∈ T h r + s | t i X s + X t ∈ T h | t i X r = 2 n X r . Hence ϕ is bijective. 129 XAMPLE 3.3.2
Let E := C ( IT n , IC) , denote by z := ( z , z , · · · , z n ) the pointsof IT n , and put z := ( z , z , · · · , z n ) for every z ∈ IT n . We identify ( ZZ ) n with P (IN n ) by using the bijection P (IN n ) −→ ( ZZ ) n , I e I and denote by I △ J := ( I \ J ) ∪ ( J \ I ) the addition on P (IN n ) corresponding to this identification. We put λ I := Q i ∈ I z i for every I ⊂ IN n and f : P (IN n ) × P (IN n ) −→ U n E c , ( I, J ) λ I ∩ J . Then f ∈ F (( ZZ ) n , E ) and, if we put ˜ X := X I ⊂ IN n λ I ( z ) X I ( z ) ∈ E for every X ∈ S ( f ) , the map ϕ : S ( f ) −→ E , X ˜ X is an isomorphism of C*-algebras. Let
X, Y ∈ S ( f ). By Theorem 2.1.9 c),g), f X ∗ = X I ⊂ IN n λ I ( X ∗ ) I ( z ) = X I ⊂ IN n λ I λ I X ∗ I = ˜ X , g XY = X I ⊂ IN n λ I ( XY ) I ( z ) = X I ⊂ IN n λ I X J ⊂ IN n f ( J, J △ I ) X J Y J △ I == X J,K ⊂ IN n λ J △ K λ J ∩ K X J Y K = X J,K ⊂ IN n λ J λ K X J Y K = ˜ X ˜ Y so ϕ is a C*-homomorphism.We put for k ∈ IN n , i ∈ ZZ n , and I ⊂ IN n , i Ik := (cid:26) i k + 1 if k ∈ I i k if k ∈ IN n \ I , i I := ( i I , i I , · · · , i In ) ∈ ZZ n G := ( X i ∈ ZZ n a i z i z i · · · z i n n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a i ) i ∈ ZZ n ∈ IC ( ZZ n ) ) . Let x := X i ∈ ZZ n a i z i z i · · · z i n n ∈ G and for every I ⊂ IN n put X I := X i ∈ ZZ n a i I z i z i · · · z i n n , X := X I ⊂ IN n ( X I ⊗ K ) V I . Then ϕX = x and so G ⊂ ϕ ( S ( f )). Since G is dense in E , it follows that ϕ issurjective.We prove that ϕ is injective by induction with respect to n ∈ IN. The case n = 1 was proved in Example 3.1.4. Assume the assertion holds for n −
1. Let X ∈ Ker ϕ . Then X I ⊂ IN n λ I ( z ) X I ( z ) = 0 . By replacing z n by − z n in the above relation, we get X I ⊂ IN n − λ I ( z ) X I ( z ) − X n ∈ I ⊂ IN n λ I ( z ) X I ( z ) = 0and so X I ⊂ IN n − λ I ( z ) X I ( z ) = X n ∈ I ⊂ IN n λ I ( z ) X I ( z ) = 0 . By the induction hypothesis, we get X I = 0 for all I ⊂ IN n and so X = 0. Thus ϕ is injective and a C*-isomorphism. EXAMPLE 3.3.3
Let f ∈ F (( ZZ ) , E ) , put a := (0 , , , b := (0 , , , c := (0 , , , s := (1 , , , and denote by g the element of F ( ZZ , E ) defined by g (1 ,
1) := f ( s, s ) Proposi-tion 3.1.1 a) . ) There is a family ( α i , β i , γ i , ε i ) i ∈ IN in ( U n E c ) such that f is given bythe attached table and such that ε i = 1 E for every i ∈ IN and ε = ε ε , ε = ε ε , ε = ε ε , ε = ε ε ε ,α = ε ε α α ∗ α α γ ∗ , α = α β γ ∗ , α = α γ γ ∗ ,β = β γ γ ∗ , β = ε α ∗ α β , β = ε ε ε α α ∗ α γ γ ∗ ,β = ε α α ∗ α , β = ε α α ∗ α β γ ∗ , β = ε ε ε α α ∗ α ,γ = ε α α ∗ γ , γ = ε ε α α ∗ γ ∗ γ , γ = ε ε α α ∗ γ ,γ = ε α α ∗ γ , γ = ε ε ε α α ∗ β .f a b c s a + s b + s c + sa β γ γ β γ β γ β b ε γ ε α γ α γ γ β β c ε β ε α α β γ γ β β s ε γ ε γ ε γ ε α γ α α α a + s ε β ε γ ε γ ε α α β α α b + s ε γ ε γ ε γ ε α ε α ε α γ α c + s ε β ε β ε β ε β ε α ε α α β b) If ε = − E , ε = ε , γ = 1 E , and there is an x ∈ E c with x = α β ∗ then there are P ± ∈ ( E e ⊗ K ) c ∩ P r S ( f ) with P + + P − = V f and (Theorem2.2.18 b)) P + S ( f ) P + ≈ E S ( g ) ≈ E P − S ( f ) P − . c) If ε = − E , ε = ε = γ = 1 E , and there is an x ∈ E c with x = α β ∗ then S ( f ) ≈ E S ( g ) , .d) Assume ε = − E , ε = ε = α = β = γ = 1 E , γ = α ∗ , and α = α = α = 1 E and put ϕ ± : S ( f ) −→ E , , X (cid:20) X + X c ± X s ± X c + s X a − X b ± α ∗ X a + s ∓ α ∗ X b + s X a + X b ± α ∗ X a + s ± α ∗ X b + s X − X c ± X s ∓ X c + s (cid:21) . Then the map S ( f ) −→ E , × E , , X ( ϕ + X, ϕ − X ) is an E -C*-isomorphism. T := ZZ n with n ∈ IN PROPOSITION 3.4.1
Put A := U n E c and for every α ∈ A n − put f α : ZZ n × ZZ n −→ A , ( p, q ) p + q − Y j = p α j q − Y k =1 α ∗ k ! , where ZZ n and IN n are canonically identified and α n := 1 E .a) For every f ∈ F ( ZZ n , E ) and X ∈ S ( f ) , X ∈ S ( f ) c iff X t ∈ E c for all t ∈ T . In particular, S ( f ) is commutative if E is commutative.b) f α ∈ F ( ZZ n , E ) for every α ∈ A n − and the map A n − −→ F ( ZZ n , E ) , α f α is a group isomorphism.c) The following are equivalent for all α, β ∈ A n − . c ) S ( f α ) ≈ S S ( f β ) . c ) There is a γ ∈ A such that γ n = n − Y j =1 ( α j β ∗ j ) .c ) There is a λ ∈ Λ( ZZ n , E ) such that f α = f β δλ .If these equivalent conditions are fulfilled then the map S ( f α ) −→ S ( f β ) , X U ∗ λ XU λ s an S -isomorphism and λ (1) n = n − Y j =1 ( α j β ∗ j ) = γ n , p ∈ ZZ n = ⇒ λ ( p ) = λ (1) p p − Y j =1 ( α ∗ j β j ) . d) Let α ∈ A n − and put β : IN n − −→ A , j if j < n − (cid:18) n − Q k =1 α ∗ k (cid:19) n − if j = n − . Then α and β fulfill the equivalent conditions of c).e) There is a natural bijection { S ( f ) | f ∈ F ( ZZ n , E ) } / ≈ S −→ A/ { x n | x ∈ A } . If E := C ( IT m , IC) for some m ∈ IN then Card ( { S ( f ) | f ∈ F ( ZZ n , E ) } / ≈ S ) = mn . f ) Let α ∈ A n − , β ∈ A such that β n = n − Q j =1 α j , F := ( E if IK = IC ◦ E if IK = IR , where ◦ E denotes the complexification of E , and w k : S ( f α ) −→ F , X n X j =1 β j j − Y l =1 ¯ α l ! e πijkn X j for every k ∈ IN n (= ZZ n ) . f ) If IK = IC then the map S ( f α ) −→ E n , X ( w k X ) k ∈ ZZ n is an E -C*-isomorphism. ) If IK = IR and n is odd then we may take β ∈ IR and the map S ( f α ) −→ E × ( ◦ E ) n − , X ( w n X, ( w k X ) k ∈ IN n − ) is an E -C*-isomorphism. f ) If IK = IR , n is even, and n − Q j =1 α j = − then the map S ( f α ) −→ ( ◦ E ) n , X ( w k − X ) k ∈ IN n2 is an E -C*-isomorphism. f ) If IK = IR , n is even, and n − Q j =1 α j = 1 , and β = 1 then the map S ( f α ) −→ E × E × ( ◦ E ) n − , X ( w n X, w n X, ( w k X ) k ∈ IN n2 − ) is an E -C*-isomorphism. f ) If n is even then there is a γ ∈ A such that f α ( n , n ) = γ . EXAMPLE 3.4.2
Let E := C ( IT , IC) , r ∈ ZZ n − , z : IT → IC the canonicalinclusion, and f : ZZ n × ZZ n −→ U n E c , ( p, q ) z p + q − P j = p r j − q − P j =1 r j ! , where ZZ n and IN n are canonically identified. Then f ∈ F ( ZZ n , E ) . Let further S be the subgroup of ZZ n generated by ρ ( n − P j =1 r j ) , where ρ : ZZ → ZZ n is the quotientmap, m := Card S , h := nm , ω := e πin ,σ : IN n −→ ZZ , p ph n − X j =1 r j − m p − X j =1 r j , and ϕ k : S ( f ) −→ E , X n X p =1 ( X p ◦ z m ) z σ ( p ) ω pk or every k ∈ IN h . Then the map ϕ : S ( f ) −→ E h , X ( ϕ k X ) k ∈ IN h is an E -C*-isomorphism. The next example shows that the set { S ( f ) | f ∈ F ( ZZ n , C ( IT , IC)) } is not re-duced by restricting the Schur functions to have the form indicated in Example3.4.2. EXAMPLE 3.4.3
Let E := C ( IT , IC) and g ∈ F ( ZZ n , E ) . Put ϕ : [0 , π [ −→ IR , θ log n − Y j =1 ( g ( j, e iθ ) , where we take a fixed (but arbitrary) branch of log . If we define r : IN n − −→ ZZ , j ( lim θ → π ϕ ( θ ) − ϕ (0) if j = 10 if j = 1 then there is a λ ∈ Λ( ZZ n , E ) such that g = f δλ , where f is the Schur functiondefined in Example 3.4.2 . In particular S ( f ) ≈ S S ( g ) . T := ZZ EXAMPLE 3.5.1
Let f ∈ F ( ZZ , E ) .a) S k·k ( f ) ≈ C ( IT , E ) .b) If E is a W*-algebra then S W ( f ) ≈ E ¯ ⊗ L ∞ ( µ ) ≈ L ∞ ( µ, E ) , where µ denotes the Lebesgue measure on IT . a ⇒ a , we may assume f constant. By Proposition 2.2.10 c),e), we may assume E := IC. Let α : IT → ICbe the inclusion map. Then l ( ZZ ) −→ L ( µ ) , ξ X n ∈ ZZ ξ n α n is an isomorphism of Hilbert spaces. If we identify these Hilbert spaces usingthis isomorphism then V becomes the multiplicator operator L ( µ ) −→ L ( µ ) , η αη so R ( f ) −→ L ∞ ( µ ) , X X n ∈ ZZ X n α n is an injective, involutive algebra homomorphism. The assertion follows. Throughout this subsection I is a totally ordered set, ( T i ) i ∈ I is a familyof groups, and ( f i ) i ∈ I ∈ Q i ∈ I F ( T i , E ). We put¯ t := { i ∈ I | t i = 1 i } for every t ∈ Q i ∈ I T i (where 1 i denotes the neutral element of T i ) and T := ( t ∈ Y i ∈ I T i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ t is finite ) , T ′ := (cid:8) t ∈ T | t = 1 (cid:9) . An associated f ∈ F ( T, E ) will be defined in Proposition 4.1.1 b). T is a subgroup of Q i ∈ I T i . We canonically associate to every element t ∈ T in a bijective way the ”word” t i t i · · · t i n , where { i , i , · · · , i n } = ¯ t and i < i < · · · < i n t ( to 1 ∈ T we associate the”empty word”). PROPOSITION 4.1.1 a) Let t i t i · · · t i n be a finite sequence of letters with t i j ∈ T i j \{ i j } for every j ∈ IN n and use transpositions of successive letters with distinct indices in order to bring these indices in an increasing order. If τ denotes thenumber of used transpositions then ( − τ does not depend on the mannerin which this operation was done.b) Let s, t ∈ T and let s i s i · · · s i m , t i ′ t i ′ · · · t i ′ n be the canonically associated words of s and t , respectively. We put forevery k ∈ I , ˜ s k := s i j if there is a j ∈ IN m with k = i j and ˜ s k := 1 k if theabove condition is not fulfilled and define ˜ t in a similar way. Moreoverwe put (Proposition 1.1.2 a)) f ( s, t ) := ( − τ Y k ∈ I f k (˜ s k , ˜ t k ) , where τ denotes the number of transpositions of successive letters withdistinct indices in the finite sequence of letters s i s i · · · s i m t i ′ t i ′ · · · t i ′ n in order to bring the indices in an increasing order. Then f ∈ F ( T, E ) .c) Let I be a subset of I , T the subgroup { t ∈ T | ¯ t ⊂ I } of T , and f the element of F ( T , E ) defined in a similar way as f was defined in b).Then f = f | ( T × T ) and the map S k·k ( f ) −→ S k·k ( f ) , k·k X t ∈ T ( X t e ⊗ K ) V f t k·k X t ∈ T ( X t e ⊗ K ) V ft is an injective E -C**-homomorphism with image { X ∈ S ( f ) | ( t ∈ T & X t = 0) ⇒ t ∈ T } . j, k ∈ IN n i j ≺ i k : ⇐⇒ (( i j < i k ) or ( i j = i k and j < k )) . Let P be a sequence of transpositions of successive letters in order to bring theindices in an increasing form with respect to the new order and let τ ′ be thenumber of used transpositions. Then τ − τ ′ is even and so ( − τ = ( − τ ′ .By the theory of permutations ( − τ ′ does not depend on P , which proves theassertion.b) By a), f is well-defined. Let r, s, t ∈ T and let r i r i · · · r i m , s i ′ s i ′ · · · s i ′ n , t i ′′ t i ′′ · · · t i ′′ p be the words canonically associated to r , s , and t , respectively. There are α, β ∈ {− , +1 } such that f ( r, s ) f ( rs, t ) = α Y i ∈ I f (˜ r i , ˜ s i ) f ( g r i s i , ˜ t i ) ,f ( r, st ) f ( s, t ) = β Y i ∈ I f i (˜ r i , g s i t i ) f (˜ s i , ˜ t i ) . Write the finite sequence of letters r i r i · · · r i m s i ′ s i ′ · · · s i ′ n t i ′′ t i ′′ · · · t i ′′ p and use transpositions of successive letters with distinct indices in order to bringthe indices in an increasing order. We can do this acting first on the letters of r and s only and then in a second step also on the letters of t . Then α = ( − µ ,where µ denotes the number of all performed transpositions. For β we may startfirst with the letters of s and t and then in a second step also with the lettersof r . Then β = ( − ν , where ν is the number of all effectuated transpositions.By a), α = ( − µ = ( − ν = β . The rest of the proof is obvious.c) follows from Corollary 2.1.17 d). COROLLARY 4.1.2 If I := IN then for all s, t ∈ T , f ( s, t ) = f ( s , t ) if s = 1 f ( s , t ) if t = 1 − f ( s , t ) f ( s , t ) if s = 1 , t = 1 . ROPOSITION 4.1.3
Let s, t ∈ T .a) f ( s, t ) = ( − Card (¯ s × ¯ t ) − Card (¯ s ∩ ¯ t ) f ( t, s ) .b) st = ts iff V s V t = ( − Card (¯ s × ¯ t ) − Card (¯ s ∩ ¯ t ) V t V s .c) Assume ¯ s ⊂ ¯ t . If Card ¯ s is even or if Card ¯ t is odd then f ( s, t ) = f ( t, s ) .If in addition st = ts then V s V t = V t V s .d) If Card I is an odd natural number and T is commutative then V t ∈ S ( f ) c for every t ∈ T with ¯ t = I .e) Assume t ∈ T ′ . If n := Card ¯ t and α := Q i ∈ ¯ t f i ( t i , t i ) then f ( t, t ) = ( − n ( n − α , ˜ f ( t ) = ( − n ( n − α ∗ , ( V t ) = ( − n ( n − ( α e ⊗ K ) V , V ∗ t = ( − n ( n − ( α ∗ e ⊗ K ) V t . a) For i ∈ ¯ s , f ( s i , t ) = (cid:26) ( − Card ¯ t f ( t, s i ) if i ¯ t ( − Card ¯ t − f ( t, s i ) if i ∈ ¯ t so f ( s, t ) = ( − Card (¯ s × ¯ t ) − Card (¯ s ∩ ¯ t ) f ( t, s ) . b) By Proposition 2.1.2 b), V s V t = ( f ( s, t ) e ⊗ K ) V st , V t V s = ( f ( t, s ) e ⊗ K ) V ts . Thus if st = ts then by a), V s V t = (( f ( s, t ) f ( t, s ) ∗ ) e ⊗ K ) V t V s = ( − Card (¯ s × ¯ t ) − Card (¯ s ∩ ¯ t ) V t V s . Conversely, if this relation holds then by a), V st = ( f ( s, t ) ∗ e ⊗ K ) V s V t = ( − Card (¯ s × ¯ t ) − Card (¯ s ∩ ¯ t ) ( f ( t, s ) ∗ e ⊗ K ) V s V t == ( f ( t, s ) ∗ e ⊗ K ) V t V s = V ts and we get st = ts by Theorem 2.1.9 a).140) follows from a) and b).d) follows from c) (and Proposition 2.1.2 d)).e) We have f ( t, t ) = ( − ( n − ··· +2+1 α = ( − n ( n − α . By Proposition 2.1.2 b),e),( V t ) = ( f ( t, t ) e ⊗ K ) V = ( − n ( n − ( α e ⊗ K ) V ,V ∗ t = ˜ f ( t ) V t − = f ( t, t ) ∗ V t = ( − n ( n − ( α ∗ e ⊗ K ) V t . PROPOSITION 4.1.4
Let S be a finite subset of T ′ \ { } such that st = ts and Card (¯ s × ¯ t ) − Card (¯ s ∩ ¯ t ) is odd for all distinct s, t ∈ S and for every t ∈ S let α t , ε t ∈ U n E c and X t ∈ E be such that ε t = 1 E , ( V t ) = ( α t e ⊗ K ) V , X ∗ t = α t X t , X t ∈ S | X t | = 14 1 E . a) P := 12 V + X t ∈ S (( ε t X t ) ˜ ⊗ K ) V t ∈ P r S ( f ) ,V − P = 12 V + X t ∈ S (( − ε t X t ) ˜ ⊗ K ) V t ∈ P r S ( f ) . b) If s ∈ S and β ∈ E c such that X s = 0 and β = α s then P is homotopicin P r S ( f ) to
12 ( V + (( β ∗ ε s ) e ⊗ K ) V s ) . a) By Proposition 4.1.3 b),e), P ∗ = 12 V + X t ∈ S (( ε t X ∗ t α ∗ t ) e ⊗ K ) V t = 12 V + X ∈ S (( ε t X t ) e ⊗ K ) V t = P , = 14 V + X t ∈ S ( X t e ⊗ K )( V t ) + X t ∈ S (( ε t X t ) e ⊗ K ) V t ++ X s,t ∈ Ss = t (( ε s ε t X s X t ) e ⊗ K )( V s V t + V t V s ) == 14 V + X t ∈ S (( X t α t ) e ⊗ K ) V + X t ∈ S (( ε t X t ) e ⊗ K ) V t == 14 V + X t ∈ S ( | X t | e ⊗ K ) V + X t ∈ S (( ε t X t ) e ⊗ K ) V t = P . b) Remark first that β ∈ U n E c and put Y : [0 , −→ E c + , u ( 14 1 E − u X t ∈ S | X t | ) ,Z : [0 , −→ E c , u β ∗ ε s Y ( u ) ,Q : [0 , −→ S ( f ) , u V + ( Z ( u ) e ⊗ K ) V s + X t ∈ S \{ s } (( uε t X t ) e ⊗ K ) V t . For u ∈ [0 , α s Z ( u ) = β β ∗ ε s Y ( u ) = βε s Y ( u ) = Z ( u ) ∗ , | Z ( u ) | + X t ∈ S \{ s } | uX t | = 14 1 E so by a), Q ( u ) ∈ P r S ( f ). Moreover Q (0) = 12 ( V + (( β ∗ ε s ) e ⊗ K ) V s ) , Q (1) = P .
COROLLARY 4.1.5
Let s, t ∈ T ′ \ { } , s = t , st = ts , α s , α t , ε s , ε t ∈ U n E c such that ε s = ε t = 1 E , ( V s ) = ( α s e ⊗ K ) V , ( V t ) = ( α t e ⊗ K ) V , and put P s := 12 ( V + (( ε s α ∗ s ) e ⊗ K ) V s ) , P t := 12 ( V + (( ε t α ∗ t ) e ⊗ K ) V t ) . ) P s , P t ∈ P r S ( f ) ; we denote by P s ∧ P t the infimum of P s and P t in S ( f ) + (by b) and c) it exists) .b) If V s V t = V t V s then P s ∧ P t = 0 .c) If V s V t = V t V s then P s ∧ P t = P s P t ∈ P r S ( f ) . a) follows from Proposition 2.1.20 b ⇒ a .b) By Proposition 4.1.3 b), V s V t = − V t V s . Let X ∈ S ( f ) + with X ≤ P s and X ≤ P t . By [C1] Proposition 4.2.7.1 d ⇒ c , X = P s X = 12 X + 12 (( ε s α ∗ s ) e ⊗ K ) V s X ,X = (( ε s α ∗ s ) e ⊗ K ) V s X = (( ε s ε t α ∗ s α ∗ t ) e ⊗ K ) V s V t X == − (( ε s ε t α ∗ s α ∗ t ) e ⊗ K ) V t V s X = − X so X = 0 and P s ∧ P t = 0.c) We have P s P t = P t P s so P s P t ∈ P r S ( f ) and P s P t = P s ∧ P t by [C1]Corollary 4.2.7.4 a ⇒ b & d . COROLLARY 4.1.6
Let m, n ∈ IN , IN m+n ⊂ I , ( α i ) i ∈ IN m ∈ ( U n E c ) m , andfor every i ∈ IN m let t i ∈ T ′ with ¯ t i := IN n ∪ { n + i } and t i t j = t j t i for all i, j ∈ IN m . If for every i ∈ IN m , ( V t i ) = ( α i ⊗ K ) V then V + 1 √ m X i ∈ IN m ( α ∗ i ⊗ K ) V t i ! ∈ P r S ( f ) . For distinct i, j ∈ IN m , Card (¯ t i × ¯ t j ) − Card (¯ t i ∩ ¯ t j ) = ( n + 1) − n = n ( n + 1) + 1is odd. For every i ∈ IN m put X i := √ m α ∗ i . Then α i X i = 12 √ m α i = X ∗ i , | X i | = 14 m E , X i ∈ IN m | X i | = 14 1 E and the assertion follows from Proposition 4.1.4 a).143 HEOREM 4.1.7
Let n ∈ IN such that IN is an ordered subset of I , S := { t ∈ T | ¯ t ⊂ IN − } , g := f | ( S × S ) , a, b ∈ T such that a = b = 1 , ¯ a = IN − , ¯ b = IN − ∪ { n } , i ∈ IN − = ⇒ a i = b i ,ω : ZZ × ZZ → T the (injective) group homomorphism defined by ω (1 ,
0) := a , ω (0 ,
1) := b , α := f ( a, a ) , α := f ( b, b ) , β , β ∈ U n E c such that α β + α β = 0 , γ := 12 ( α ∗ β ∗ β − α ∗ β β ∗ ) = α ∗ β ∗ β = − α ∗ β β ∗ ,X := 12 (( β e ⊗ K ) V a + ( β e ⊗ K ) V b ) , P + := X ∗ X , P − := XX ∗ . We consider S ( g ) as an E -C**-subalgebra of S ( f ) (Corollary 2.1.17 e)) .a) ab = ba , γ = − α ∗ α ∗ . We put c := ab = ω (1 , .b) X, V c , P ± ∈ S ( g ) c .c) We have P ± = 12 ( V ± ( γ e ⊗ K ) V c ) ∈ P r S ( f ) , P + + P − = V , P + P − = 0 ,X = 0 , XP + = X, P − X = X, P + X = XP − = 0 , X + X ∗ ∈ U n S ( f ) . d) The map E −→ P ± S ( f ) P ± , x P ± ( x e ⊗ K ) P ± is an injective unital C**-homomorphism. We identify E with its imagewith respect to this map and consider P ± S ( f ) P ± as an E -C**-algebra.e) The map ϕ ± : S ( g ) −→ P ± S ( f ) P ± , Y P ± Y P ± = P ± Y = Y P ± is an injective unital C**-homomorphism. If Y , Y ∈ U n S ( g ) then ϕ + Y + ϕ − Y ∈ U n S ( f ) .f ) The map ψ : S ( f ) −→ S ( f ) , Z ( X + X ∗ ) Z ( X + X ∗ ) is an E -C**-isomorphism such that ψ − = ψ , ψ ( P + S ( f ) P + ) = P − S ( f ) P − , ψ ◦ ϕ + = ϕ − , ψ ◦ ϕ − = ϕ + . f Y , Y ∈ S ( g ) then ϕ + Y + ϕ − Y = ( ϕ + Y + ϕ − V ) ψ ( ϕ + Y + ϕ − V ) . g) If p ∈ P r S ( g ) then ( X ( ϕ + p ) ∗ ( X ( ϕ + p )) = ϕ + p , ( X ( ϕ + p ))( X ( ϕ + p )) ∗ = ϕ − p . h) Let R be the subgroup { , a, b, c } of T , h := f | ( R × R ) , d ∈ T such that ¯ d = IN − and d i = a i for every i ∈ IN − , and α := f ( d, d ) , α ′ := f n − (2 n − , n − , α ′′ := f n (2 n, n ) . Then α = αα ′ , α = αα ′′ , − α ′ α ′′ = ( α ∗ γ ∗ ) , h a b ca αα ′ α α ′ b − α αα ′′ − α ′′ c − α ′ α ′′ − α ′ α ′′ is the table of h , P ± ∈ P r S ( h ) , and the map ϕ : S ( h ) −→ E , , Z (cid:20) Z + γ ∗ Z c αα ′ Z a − αγ ∗ Z b Z a + α ′∗ γ ∗ Z b Z − γ ∗ Z c (cid:21) is an E -C**-isomorphism. In particular ϕP + = (cid:20) E
00 0 (cid:21) , ϕP − = (cid:20) E (cid:21) and E , is E -C**-isomorphic to an E -C**-subalgebra of S ( f ) .i) Assume I = IN and T n − = T n = ZZ . Then T ≈ S × ZZ × ZZ , ϕ ± isan E -C*-isomorphism with inverse P ± S ( f ) P ± −→ S ( f ) , Z X u ∈ T ( Z u ⊗ K ) V u , and S ( f ) ≈ E S ( g ) , PROPOSITION 4.1.8
We use the notation and the hypotheses of
Theorem4.1.7 and assume I := IN , T := ZZ , and T := ZZ with m ∈ IN .a) a = (1 , , b = (0 , m ) , c = (1 , m ) , α = 1 E , α ′ = α = f (1 , , α ′′ = α = f ( m, m ) , and P ± S ( f ) P ± = (cid:8) ( x e ⊗ K ) P ± (cid:12)(cid:12) x ∈ E (cid:9) . b) If m = 1 then there are α, β, γ, δ ∈ U n E c such that f is given by thefollowing table: f (0 ,
1) (0 ,
2) (0 ,
3) (1 ,
0) (1 ,
1) (1 ,
2) (1 , , α β γ − E − α − β − γ (0 , β α ∗ βγ α ∗ γ − E − β − α ∗ βγ − α ∗ γ (0 , γ α ∗ γ β ∗ γ − E − γ − α ∗ γ − β ∗ γ (1 ,
0) 1 E E E δ δ δ δ (1 , α β γ − δ − αδ − βδ − γδ (1 , β α ∗ βγ α ∗ γ − δ − βδ − α ∗ βγδ − α ∗ γδ (1 , γ α ∗ γ β ∗ γ − δ − γδ − α ∗ γδ − β ∗ γδ . ) We assume IK := IC and m := 1 and put for all j, k ∈ { , } ϕ j,k : S ( f ) −→ E , Z Z + ( − j Z b + i j Z ( k, − i j Z ( k, ,φ : S ( f ) −→ E , Z ( ϕ , Z, ϕ , Z, ϕ , Z, ϕ , Z ) . If we take α := β := γ := − δ := β := β := 1 E in b) then the map S ( f ) −→ E , × E , Z (cid:18)(cid:20) Z + Z (1 , Z (1 , − Z b Z (1 , + Z b Z − Z (1 , (cid:21) , φZ (cid:19) is an E -C**-isomorphism. a) Use Corollary 4.1.2 and Proposition 2.1.2 b).b) Use Proposition 3.4.1 a) and Proposition 4.1.1.c) follows from b) and Proposition 3.4.1 f . Throughout this subsection we denote by S a totally ordered set, put T := ( ZZ ) ( S ) , and fix a map ρ : S → U n E c . We define for every s ∈ S , f s ∈ F ( ZZ , E ) by putting f s (1 ,
1) = ρ ( s ) ( Proposition 3.1.1 a)).Moreover we denote by f ρ the Schur function f defined in Proposition4.1.1 b) (with I replaced by S ) and put C l ( ρ ) := S ( f ρ ). Remark. If S := IN then T = ZZ × ZZ so C l ( ρ ) is a special case of theexample treated in subsection 3.2. With the notation used in the left table ofProposition 3.2.1 this case appears for a := (1 ,
0) and b := (0 .
1) exactly when ε = − E , α = − ρ ( b ), β = ρ ( a ), and γ = 1 E . LEMMA 4.2.1 P f ( S ) endowed with the composition law P f ( S ) × P f ( S ) −→ P f ( S ) , ( A, B ) A △ B := ( A \ B ) ∪ ( B \ A ) is a locally finite commutative group ( Definition 2.1.18 ) with ∅ as neutral ele-ment and the map P f ( S ) −→ T, A e A s a group isomorphism with inverse T −→ P f ( S ) , x s ∈ S | x ( s ) = 1 } . We identify T with P f ( S ) by using this isomorphism and write s instead of { s } for every s ∈ S . For A, B ∈ T , f ρ ( A, B ) = ( − τ Y s ∈ A ∩ B ρ ( s ) , where τ is defined in Proposition 4.1.1 b) . PROPOSITION 4.2.2
Assume S finite and let F be an E -C*-algebra. Letfurther ( x s ) s ∈ S be a family in F such that for all distinct s, t ∈ S and for every y ∈ E , x s x t = − x t x s , x s = ρ ( s )1 F , x ∗ s = ρ ( s ) ∗ x s , x s y = yx s . Then there is a unique E -C*-homomorphism ϕ : C l ( ρ ) → F such that ϕV s = x s for all s ∈ S . If the family (cid:18) Q s ∈ A x s (cid:19) A ⊂ S is E -linearly independent (resp.generates F as an E -C*-algebra) then ϕ is injective (resp. surjective). Put ϕV A := x s x s · · · x s m for every A := { s , s , · · · , s m } , where s < s < · · · < s m , and ϕ : C l ( ρ ) −→ F , X X A ⊂ S X A ϕV A . It is easy to see that ( ϕV s )( ϕV t ) = ϕ ( V s V t ) and y ϕV s = ( ϕV s ) y for all s, t ∈ S and y ∈ E (Proposition 2.1.2 b)). Let A := { s , s , · · · , s m } ⊂ S , B := { t , t , · · · , t n } ⊂ S , { r , r , · · · , r p } := A △ B , where the letters are written in strictly increasing order. Then( ϕV A )( ϕV B ) = x s x s · · · x s m x t x t · · · x t n = f ρ ( A, B ) x r x r · · · x r p == f ρ ( A, B ) ϕV A △ B = ϕ (( f ρ ( A, B ) e ⊗ K ) V A △ B ) = ϕ ( V A V B ) , ϕV A ) ∗ = x ∗ s m · · · x ∗ s x ∗ s = ( − m ( m − x ∗ s x ∗ s · · · x ∗ s m == ( − m ( m − Y i ∈ IN m ρ ( s i ) ∗ x s x s · · · x s m = ( − m ( m − Y i ∈ IN m ρ ( s i ) ∗ ϕV A == ϕ (( − m ( m − (( Y i ∈ IN m ρ ( s i ) ∗ ) e ⊗ K ) V A ) = ϕ ( V ∗ A )by Proposition 4.1.3 e).For X, Y ∈ C l ( ρ ) (by Theorem 2.1.9 c),g)),( ϕX )( ϕY ) = X A ∈ T X A ϕV A ! X B ∈ T Y B ϕV B ! = X A,B ∈ T X A Y B ( ϕV A )( ϕV B ) == X A,B ∈ T X A Y B ϕ ( V A V B ) = X A,B ∈ T X A Y B f ρ ( A, B ) ϕV A △ B == X A,C ∈ T X A Y A △ C f ρ ( A, A △ C ) ϕV C = X C ∈ T X A ∈ T f ρ ( A, A △ C ) X A Y A △ C ! ϕV C == X C ∈ T ( XY ) C ϕV C = ϕ ( XY ) , ( ϕX ) ∗ = X A ∈ T X ∗ A ( ϕV A ) ∗ = X A ∈ T X ∗ A ϕ ( V A ) ∗ == X A ∈ T ˜ f ρ ( A ) ∗ ( X ∗ ) A ˜ f ρ ( A ) ϕV A = X A ∈ T ( X ∗ ) A ϕV A = ϕ ( X ∗ )(Proposition 4.1.3 e)) i.e. ϕ is an E -C*-homomorphism. The uniqueness andthe last assertions are obvious (by Theorem 2.1.9 a)). PROPOSITION 4.2.3
Let m, n ∈ IN ∪ { } , S := IN , S ′ := IN , K ′ := l ( P ( S ′ )) , ( α i ) i ∈ IN m ∈ ( U n E c ) m , ρ ′ : S ′ −→ U n E c , s (cid:26) ρ ( s ) if s ∈ Sα i ˜ f ρ ( S ) if s = 2 n + i with i ∈ IN m , and A i := A ∪ { n + i } for every A ⊂ S and i ∈ IN m .a) i ∈ IN m = ⇒ ˜ f ρ ′ ( S i ) = α ∗ i , ( V ρ ′ S i ) = ( α i ⊗ K ′ ) V ρ ′ ∅ . ) P := V ρ ′ ∅ + √ m P i ∈ IN m ( α ∗ i ⊗ K ′ ) V ρ ′ S i ∈ P r C l ( ρ ′ ) .c) There is a unique injective E -C*-homomorphism ϕ : C l ( ρ ) → P C l ( ρ ′ ) P such that ϕV ρs = P V ρ ′ s P = P V ρ ′ s = V ρ ′ s P for every s ∈ S .d) If m ∈ IN then ϕ is an E -C*-isomorphism. a) By Proposition 4.1.3 e),˜ f ρ ′ ( S i ) = ( − n (2 n +1) Y s ∈ S i ρ ′ ( s ) ∗ = ( − n (2 n − Y s ∈ S ρ ( s ) ∗ ! α ∗ i ˜ f ρ ( S ) ∗ = α ∗ i , ( V ρ ′ S i ) = ( α i ⊗ K ′ ) V ρ ′ ∅ . b) follows from a) and Corollary 4.1.6.c) By Proposition 4.1.3 c), for s ∈ S , V ρ ′ s V ρ ′ S i = V ρ ′ S i V ρ ′ s for every i ∈ IN m so V ρ ′ s P = P V ρ ′ s . By b), for distinct s, t ∈ S (Proposition 4.1.3 b)),( P V ρ ′ s )( P V ρ ′ t ) = P V ρ ′ s V ρ ′ t = − P V ρ ′ t V ρ ′ s = − ( P V ρ ′ t )( P V ρ ′ s ) , ( P V ρ ′ s ) = P ( V ρ ′ s ) = P ( ρ ′ ( s ) ⊗ K ′ ) V ρ ′ ∅ = ( ρ ( s ) ⊗ K ′ ) P , ( P V ρ ′ s ) ∗ = P ( V ρ ′ s ) ∗ = P ( ρ ′ ( s ) ∗ ⊗ K ′ ) V ρ ′ s = ( ρ ( s ) ⊗ K ′ ) ∗ P V ρ ′ s . By Proposition 4.2.2 there is a unique E -C*-homomorphism ϕ : C l ( ρ ) → P C l ( ρ ′ ) P with the given properties.Let X ∈ C l ( ρ ) with ϕX = 0. Then0 = X A ⊂ S ( X A ⊗ K ′ ) V ρ ′ A ! P == 12 X A ⊂ S ( X A ⊗ K ′ ) V ρ ′ A + 12 √ m X i ∈ IN m X A ⊂ S ( X A ⊗ K ′ ) f ρ ′ ( A, S i ) V ρ ′ A △ S i and this implies X A = 0 for all A ⊂ S (Theorem 2.1.9 a)). Thus ϕ is injective.d) 150he case m = 1Let Y ∈ P C l ( ρ ′ ) P . Then (by Proposition 2.1.2 b)) Y = Y P = 12 Y + 12 X A ⊂ S ′ ( α ∗ ⊗ K ′ ) V ρ ′ S Y ,Y = X A ⊂ S (( α ∗ f ρ ′ ( S , A ) Y A ) ⊗ K ′ ) V ρ ′ S △ A ++ X A ⊂ S ((( α ∗ f ρ ′ ( S , A ) Y A )) ⊗ K ′ ) V ρ ′ S △ A so (cid:26) Y A = α ∗ f ρ ′ ( S , ( S △ A ) ) Y ( S △ A ) Y A = α ∗ f ρ ′ ( S , S △ A ) Y S △ A for every A ⊂ S . If we put X := 2 X A ⊂ S ( Y A ⊗ K ) V ρA ∈ C l ( ρ )then ϕX = 12 ϕX + X A ⊂ S (( α ∗ f ρ ′ ( S , A ) Y A ) ⊗ K ′ ) V ρ ′ S △ A == X A ⊂ S ( Y A ⊗ K ′ ) V ρ ′ A + X A ⊂ S (( α ∗ f ρ ′ ( S , S △ A ) Y S △ A ) ⊗ K ′ ) V ρ ′ A == X A ⊂ S ( Y A ⊗ K ′ ) V ρ ′ A + X A ⊂ S ( Y A ⊗ K ′ ) V ρ ′ A = Y .
Thus ϕ is surjective. The case m = 2Let Y ∈ P C l ( ρ ′ ) P . Then ( Y = P Y = Y + √ (( α ∗ ⊗ K ′ ) V ρ ′ S + ( α ∗ ⊗ K ′ ) V ρ ′ S ) YY = Y P = Y + √ Y (( α ∗ ⊗ K ′ ) V ρ ′ S + ( α ∗ ⊗ K ′ ) V ρ ′ S ) , √ Y = ( α ∗ ⊗ K ′ ) V ρ ′ S Y + ( α ∗ ⊗ K ′ ) V ρ ′ S Y = ( α ∗ ⊗ K ′ ) Y V ρ ′ S + ( α ∗ ⊗ K ′ ) Y V ρ ′ S . B ⊂ S put B a := B ∪ { n + 1 } , B b := B ∪ { n + 2 } , B c := B ∪ { n + 1 , n + 2 } . Then V ρ ′ S Y = X B ⊂ S (( Y B f ρ ′ ( S , B )) ⊗ K ′ ) V ρ ′ ( S △ B ) a + X B ⊂ S (( Y B a f ρ ′ ( S , B a )) ⊗ K ′ ) V ρ ′ S △ B ++ X B ⊂ S (( Y B b f ρ ′ ( S , B b )) ⊗ K ′ ) V ρ ′ ( S △ B ) c + X B ⊂ S (( Y B c f ρ ′ ( S , B c )) ⊗ K ′ ) V ( S △ B ) b ,V ρ ′ S Y == X B ⊂ S (( Y B f ρ ′ ( S , B )) ⊗ K ′ ) V ρ ′ ( S △ B ) b + X B ⊂ S (( Y B a f ρ ′ ( S , B a )) ⊗ K ′ ) V ρ ′ ( S △ B ) c ++ X B ⊂ S (( Y B b f ρ ′ ( S , B b )) ⊗ K ′ ) V ρ ′ S △ B + X B ⊂ S (( Y B c f ρ ′ ( S , B c )) ⊗ K ′ ) V ρ ′ ( S △ B ) a ,Y V ρ ′ S = X B ⊂ S (( Y B f ρ ′ ( B, S )) ⊗ K ′ ) V ρ ′ ( S △ B ) a + X B ⊂ S (( Y B a f ρ ′ ( B a , S )) ⊗ K ′ ) V ρ ′ S △ B ++ X B ⊂ S (( Y B b f ρ ′ ( B b , S )) ⊗ K ′ ) V ρ ′ ( S △ B ) c + X B ⊂ S (( Y B c f ρ ′ ( B c , S )) ⊗ K ′ ) V ρ ′ ( S △ B ) b ,Y V ρ ′ S == X B ⊂ S (( Y B f ρ ′ ( B, S )) ⊗ K ′ ) V ρ ′ ( S △ B ) b + X B ⊂ S (( Y B a f ρ ′ ( B a , S )) ⊗ K ′ ) V ρ ′ ( S △ B ) c ++ X B ⊂ S (( Y B b f ρ ′ ( B b , S )) ⊗ K ′ ) V ρ ′ S △ B + X B ⊂ S (( Y B c f ρ ′ ( B c , S )) ⊗ K ′ ) V ρ ′ ( S △ B ) a , √ Y = X B ⊂ S (( α ∗ Y B a f ρ ′ ( S , B a ) + α ∗ Y B b f ρ ′ ( S , B b )) ⊗ K ′ ) V ρ ′ S △ B ++ X B ⊂ S (( α ∗ Y B f ρ ′ ( S , B ) + α ∗ Y B c f ρ ′ ( S , B c )) ⊗ K ′ ) V ρ ′ ( S △ B ) a ++ X B ⊂ S (( α ∗ Y B c f ρ ′ ( S , B c ) + α ∗ Y B f ρ ′ ( S , B )) ⊗ K ′ ) V ρ ′ ( S △ B ) b ++ X B ⊂ S (( α ∗ Y B b f ρ ′ ( S , B b ) + α ∗ Y B a f ρ ′ ( S , B a )) ⊗ K ′ ) V ρ ′ ( S △ B ) c , √ Y = X B ⊂ S (( α ∗ Y B a f ρ ′ ( B a , S ) + α ∗ Y B b f ρ ′ ( B b , S )) ⊗ K ′ ) V ρ ′ S △ B +152 X B ⊂ S (( α ∗ Y B f ρ ′ ( B, S ) + α ∗ Y B c f ρ ′ ( B c , S )) ⊗ K ′ ) V ρ ′ ( S △ B ) a ++ X B ⊂ S (( α ∗ Y B c f ρ ′ ( B c , S ) + α ∗ Y B f ρ ′ ( B, S )) ⊗ K ′ ) V ρ ′ ( S △ B ) b ++ X B ⊂ S (( α ∗ Y B b f ρ ′ ( B b , S ) + α ∗ Y B a f ρ ′ ( B a , S )) ⊗ K ′ ) V ρ ′ ( S △ B ) c . It follows for B ⊂ S , √ Y B a = α ∗ Y S △ B f ρ ′ ( S △ B, S ) + α ∗ Y ( S △ B ) c f ρ ′ (( S △ B ) c , S ) , √ Y B b = α ∗ Y ( S △ B ) c f ρ ′ (( S △ B ) c , S ) + α ∗ Y S △ B f ρ ′ ( S △ B, S ) , √ Y B c = α ∗ Y ( S △ B ) b f ρ ′ ( S , ( S △ B ) b ) + α ∗ Y ( S △ B ) a f ρ ′ ( S , ( S △ B ) a ) == α ∗ Y ( S △ B ) b f ρ ′ (( S △ B ) b , S ) + α ∗ Y ( S △ B ) a f ρ ′ (( S △ B ) a , S ) , so by Proposition 4.1.3 a),b), Y B c = 0. If we put X := 2 X B ⊂ S ( Y B ⊗ K ) V ρB ∈ C l ( ρ )then ϕX = X B ⊂ S ( Y B ⊗ K ′ ) V ρ ′ B ! P == X B ⊂ S ( Y B ⊗ K ′ ) V ρ ′ B + 1 √ X B ⊂ S (( α ∗ Y B f ρ ′ ( B, S )) ⊗ K ′ ) V ρ ′ S △ B ++ 1 √ X B ⊂ S (( α ∗ Y B f ρ ′ ( B, S )) ⊗ K ′ ) V ρ ′ S △ B , and so for B ⊂ S ,( ϕX ) B = Y B , ( ϕX ) B a = 1 √ α ∗ Y S △ B f ρ ′ ( S △ B, S ) = Y B a , ( ϕX ) B b = 1 √ α ∗ Y S △ B f ρ ′ ( S △ B, S ) = Y B b , ( ϕX ) B c = 0 = Y B c . Thus ϕX = Y and ϕ is surjective. Remark. If m = 3 then ϕ may be not surjective.153 ROPOSITION 4.2.4
Let
IK := IR , n ∈ IN ∪ { } , S := IN , and ρ ′ : IN −→ U n E c , s (cid:26) ρ ( s ) if s ∈ S − ˜ f ρ ( S ) if s = 2 n + 1 . Let ◦ z }| { C l ( ρ ) be the complexification of C l ( ρ ) , considered as a real E -C*-algebra ([C1] Theorem 4.1.1.8 a)) by using the embedding E −→ ◦ z }| { C l ( ρ ) , x (( x ⊗ K ) V ρ ∅ , . Then there is a unique E -C*-isomorphism ϕ : C l ( ρ ′ ) → ◦ z }| { C l ( ρ ) such that ϕV ρ ′ s =( V ρs , for every s ∈ S and ϕV ρ ′ n +1 = (0 , − ( ˜ f ρ ( S ) ⊗ K ) V ρS ) . We put x s := (cid:26) ( V ρs ,
0) if s ∈ S (0 , − ( ˜ f ρ ( S ) ⊗ K ) V ρS ) if s = 2 n + 1 . For s ∈ S , by Proposition 4.1.3 b), x s x n +1 = ( V ρs , , − ( ˜ f ρ ( S ) ⊗ K ) V ρS ) = (0 , − ( ˜ f ρ ( S ) ⊗ K ) V ρs V ρS ) == (0 , ( ˜ f ρ ( S ) ⊗ K ) V ρS V ρs ) = (0 , ( ˜ f ρ ( S ) ⊗ K ) V ρs )( V ρs ,
0) = − x n +1 x s . By Proposition 2.1.2 b),e), x n +1 = ( − (( ˜ f ρ ( S ) ⊗ K ) V ρS ) ,
0) == ( − ( ˜ f ρ ( S ) ⊗ K )( f ρ ( S, S ) ⊗ K ) V ρ ∅ ,
0) = ( ρ ′ (2 n + 1) ⊗ K )( V ρ ∅ , ,x ∗ n +1 = (0 , (( ˜ f ρ ( S ) ⊗ K ) V ρS ) ∗ ) == (0 , ( ˜ f ρ ( S ) ∗ ⊗ K )( ˜ f ρ ( S ) ⊗ K ) V ρS ) = ( ρ ′ (2 n + 1) ∗ ⊗ K ) x n +1 , and the assertion follows from Proposition 4.2.2.154 ROPOSITION 4.2.5
Let n ∈ IN ∪ { } , S := IN n , S ′ := IN n+2 , K ′ := l ( P ( S ′ )) , α , α ∈ U n E c , and ρ ′ : S ′ −→ U n E c , s ρ ( s ) if s ∈ Sα if s = n + 1 − α if s = n + 2 . a) There is a unique E -C*-isomorphism ϕ : C l ( ρ ′ ) → C l ( ρ ) , such that ϕV ρ ′ s = (cid:20) V ρs − V ρs (cid:21) for every s ∈ S and ϕV ρ ′ n +1 = ( α ⊗ K ) (cid:20) V ρ ∅ V ρ ∅ (cid:21) , ϕV ρ ′ n +2 = ( α ⊗ K ) (cid:20) − V ρ ∅ V ρ ∅ (cid:21) . b) ϕ
12 ( V ρ ′ ∅ + (( α ∗ α ∗ ) ⊗ K ′ ) V ρ ′ { n +1 , n +2 } ) = (cid:20) V ρ ∅
00 0 (cid:21) ,ϕ
12 ( V ρ ′ ∅ − (( α ∗ α ∗ ) ⊗ K ′ ) V ρ ′ { n +1 , n +2 } ) = (cid:20) V ρ ∅ (cid:21) . a) Put x s := (cid:20) V ρs − V ρs (cid:21) for every s ∈ S and x n +1 := ( α ⊗ K ) (cid:20) V ρ ∅ V ρ ∅ (cid:21) , x n +2 := ( α ⊗ K ) (cid:20) − V ρ ∅ V ρ ∅ (cid:21) . For distinct s, t ∈ S and i ∈ IN , x s x t = − x t x s , x s = ( ρ ′ ( s ) ⊗ K ) (cid:20) V ρ ∅ V ρ ∅ (cid:21) , x ∗ s = ( ρ ′ ( s ) ⊗ K ) ∗ x s ,x s x n + i = − x n + i x s , x n + i = ( ρ ′ ( n + i ) ⊗ K ) (cid:20) V ρ ∅ V ρ ∅ (cid:21) ,x ∗ n + i = ( ρ ′ ( n + i ) ⊗ K ) ∗ x n + i , x n +1 x n +2 = − x n +2 x n +1 . E -C*-homomorphism ϕ : C l ( ρ ′ ) →C l ( ρ ) , satisfying the given conditions.We put for every A ⊂ S and i ∈ IN | A | := Card A , A i := A ∪ { n + i } , A := A ∪ { n + 1 , n + 2 } . For A ⊂ S , ϕV ρ ′ A = ( α ⊗ K ) (cid:20) V ρA
00 ( − | A | V ρA (cid:21) (cid:20) V ρ ∅ V ρ ∅ (cid:21) == ( α ⊗ K ) (cid:20) V ρA ( − | A | V ρA (cid:21) ,ϕV ρ ′ A = ( α ⊗ K ) (cid:20) V ρA
00 ( − | A | V ρA (cid:21) (cid:20) − V ρ ∅ V ρ ∅ (cid:21) == ( α ⊗ K ) (cid:20) − V ρA ( − | A | V ρA (cid:21) ,ϕV ρ ′ A = (( α α ) ⊗ K ) (cid:20) V ρA ( − | A | V ρA (cid:21) (cid:20) − V ρ ∅ V ρ ∅ (cid:21) == (( α α ) ⊗ K ) (cid:20) V ρA − ( − | A | V ρA (cid:21) . Then for Y ∈ C l ( ρ ′ ), ( ϕY ) = P A ⊂ S (( Y A + ( α α ) Y A ) ⊗ K ) V ρA ( ϕY ) = P A ⊂ S (( α Y A − α Y A ) ⊗ K ) V ρA ( ϕY ) = P A ⊂ S ( − | A | )(( α Y A + α Y A ) ⊗ K ) V ρA ( ϕY ) = P A ⊂ S ( − | A | (( Y A − α α Y A ) ⊗ K ) V ρA . It follows from the above identities that ϕ is bijective.b) By the above, ϕV ρ ′ { n +1 , n +2 } = ϕV ρ ′ ∅ = (( α α ) ⊗ K ) (cid:20) V ρ ∅ − V ρ ∅ (cid:21) and the assertion follows. 156 OROLLARY 4.2.6
Let m, n ∈ IN ∪ { } , S := IN n , ( α i ) i ∈ IN ∈ ( U n E c ) m ,and ρ ′ : IN n+2m −→ U n E c , s (cid:26) ρ ( s ) if s ∈ S − ( − i α i if s = n + i . Then C l ( ρ ′ ) ≈ E C l ( ρ ) m , m . PROPOSITION 4.2.7
Let
IK := IR , n ∈ IN ∪ { } , S := IN , S ′ := IN , α , α ∈ U n E c , and ρ ′ : S ′ −→ U n E c , s (cid:26) ρ ( s ) if s ∈ S − α l ˜ f ρ ( S ) if s = 2 n + l with l ∈ IN . Then there is a unique E -C*-isomorphism ϕ : C l ( ρ ′ ) → C l ( ρ ) ⊗ IH such that ϕV ρ ′ s = V ρs ⊗ IH if s ∈ S ((( α ˜ f ρ ( S )) ⊗ K ) V ρS ) ⊗ i if s = 2 n + 1((( α ˜ f ρ ( S )) ⊗ K ) V ρS ) ⊗ j if s = 2 n + 2 , where i, j, k are the canonical unitaries of IH . Put x s := V ρs ⊗ IH if s ∈ S ((( α ˜ f ρ ( S )) ⊗ K ) V ρS ) ⊗ i if s = 2 n + 1((( α ˜ f ρ ( S )) ⊗ K ) V ρS ) ⊗ j if s = 2 n + 2 . For distinct s, t ∈ S and l ∈ IN , by Proposition 4.1.3 b), x s x t = − x t x s , x s = ( ρ ′ ( s ) ⊗ K )( V ρ ∅ ⊗ IH ) , x ∗ s = ( ρ ′ ( s ) ⊗ K ) ∗ x s ,x s x n + l = − x n + l x s , x n +1 x n +2 = ((( α α ˜ f ρ ( S )) ⊗ K ) V ρ ∅ ) ⊗ k = − x n +2 x n +1 , ( x n + l ) = ((( α l ˜ f ρ ( S ) ) ⊗ K )( ˜ f ρ ( S ) ∗ ⊗ K ) V ρ ∅ ) ⊗ ( − IH ) == ( ρ ′ (2 n + l ) ⊗ K )( V ρ ∅ ⊗ IH ) , ( x n + l ) ∗ = ((( α ∗ l ˜ f ρ ( S ) ∗ ) ⊗ K )( ˜ f ρ ( S ) ⊗ K ) V ρS ) ⊗ − ( i or j ) == ( ρ ′ (2 n + l ) ⊗ K ) ∗ x n + l . By Proposition 4.2.2 there is a unique E -C*-homomorphism ϕ : C l ( ρ ′ ) →C l ( ρ ) ⊗ IH satisfying the given conditions.157or X ∈ C l ( ρ ′ ), ϕX = X A ⊂ S ( X A ⊗ K ) V ρA ! ⊗ IH ++ X A ⊂ S (( X A ∪{ n +1 } α ˜ f ρ ( S ) f ρ ( A, S )) ⊗ K ) V S △ A ! ⊗ i ++ X A ⊂ S (( X A ∪{ n +2 } α ˜ f ρ ( S ) f ρ ( A, S )) ⊗ K ) V ρS △ A ! ⊗ j ++ X A ⊂ S (( X A ∪{ n +1 , n +2 } α α ˜ f ρ ( S )) ⊗ K ) V ρA ! ⊗ k and so ϕ is bijective. PROPOSITION 4.2.8
Let n ∈ IN ∪ { } , S := IN , A ′ := A ∪ { n + 1 } forevery A ⊂ S , ρ ′ : S ′ −→ U n E c , s (cid:26) ρ ( s ) if s ∈ S ˜ f ( S ) if s = 2 n + 1 ,P ± := ( V ρ ′ ∅ ± V ρ ′ S ′ ) , and θ ± : (cid:13)| A ⊂ S ˘ E → (cid:13)| A ⊂ S ′ ˘ E defined by ( θ ± ξ ) A := 1 √ ξ A , ( θ ± ξ ) A ′ := ± √ f ρ ( S △ A, S ) ξ S △ A for every ξ ∈ (cid:13)| A ⊂ S ˘ E and A ⊂ S .a) ˜ f ρ ′ ( S ′ ) = 1 E , ( V ρ ′ S ′ ) = V ρ ′ ∅ , P ± ∈ P r C l ( ρ ′ ) c ,P + + P − = V ρ ′ ∅ , V ρ ′ S ′ ∈ C l ( ρ ′ ) c , V ρ ′ S ′ P ± = ± P ± . b) For A ⊂ S , f ρ ( A, S ) ∗ = f ρ ′ ( S ′ , A ) ∗ = f ρ ′ ( S ′ , ( S △ A ) ′ ) . ) θ ± ∈ L E ( (cid:13)| A ⊂ S ˘ E, (cid:13)| A ⊂ S ′ ˘ E ) and for η ∈ (cid:13)| A ⊂ S ′ ˘ E and A ⊂ S , ( θ ∗± η ) A = 1 √ η A ± f ρ ( A, S ) ∗ η ( S △ A ) ′ ) = √ P ± η ) A . d) θ ∗± θ ± is the identity map of (cid:13)| A ⊂ S ˘ E .e) θ ± θ ∗± = P ± .f ) For every A ⊂ S , θ ± V ρA θ ∗± = V ρ ′ A P ± = P ± V ρ ′ A = P ± V ρ ′ A P ± . g) For every closed ideal F of E the map ϕ : C l ( ρ, F ) −→ P ± C l ( ρ ′ , F ) P ± , X θ ± Xθ ∗± is an E -C*-isomorphism with inverse P ± C l ( ρ ′ , F ) P ± −→ C l ( ρ, F ) , Y θ ∗± Y θ ± and the map ψ : C l ( ρ ′ , F ) −→ C l ( ρ, F ) × C l ( ρ, F ) , Y ( θ ∗ + P + Y P + θ + , θ ∗− P − Y P − θ − ) = ( θ + Y θ + , θ ∗− Y θ − ) is an E -C*-isomorphism. a) By Proposition 4.1.3 d),e), V ρ ′ S ′ ∈ C l ( ρ ′ ) c ,˜ f ρ ′ ( S ′ ) = ( − n (2 n +1) Y s ∈ S ′ ρ ′ ( s ) ∗ = ( − n (2 n − Y s ∈ S ρ ( s ) ∗ ! ρ ′ (2 n + 1) ∗ = 1 E , ( V ρ ′ S ′ ) ∗ = ˜ f ρ ′ ( S ′ ) V ρ ′ S ′ = V ρ ′ S ′ , ( V ρ ′ S ′ ) = ˜ f ( S ′ ) ∗ V ρ ′ ∅ = V ρ ′ ∅ , so P ± ∈ P r C l ( ρ ′ ) c , V ρ ′ S ′ P ± = ± P ± . b) By a), Proposition 4.1.3 c),d), Proposition 4.1.1 b), and Proposition 1.1.2b), f ρ ( A, S ) ∗ = f ρ ′ ( A, S ) ∗ = f ρ ′ ( A, S ′ ) ∗ =159 f ρ ′ ( S ′ , A ) ∗ = f ρ ′ ( S ′ , ( S △ A ) ′ ) ˜ f ρ ′ ( S ′ ) = f ρ ′ ( S ′ , ( S △ A ) ′ ) . c) For ξ ∈ (cid:13)| A ⊂ S ˘ E , h θξ | η i = X A ⊂ S η ∗ A √ ξ A ± X A ⊂ S η ∗ A ′ √ f ρ ( S △ A, S ) ξ S △ A == X A ⊂ S η ∗ A √ ξ A ± X A ⊂ S η ∗ ( S △ A ) ′ √ f ρ ( A, S ) ξ A == X A ⊂ S √ η A ± f ρ ( A, S ) ∗ η ( S △ A ) ′ ) ∗ ξ A so θ ∈ L E ( (cid:13)| A ⊂ S ˘ E, (cid:13)| A ⊂ S ′ ˘ E ) and( θ ∗ η ) A = 1 √ η A ± f ρ ( A, S ) ∗ η ( S △ A ) ′ ) . By a) and b), ( P ± η ) A = 12 η A ± f ρ ′ ( S ′ , ( S △ A ) ′ ) η ( S △ A ) ′ == 12 ( η A ± f ρ ( A, S ) ∗ η ( S △ A ) ′ ) = 1 √ θ ∗± η ) A . d) For ξ ∈ (cid:13)| A ⊂ S ˘ E and A ⊂ S , by c),( θ ∗± θ ± ξ ) A = 1 √ θξ ) A ± f ρ ( A, S ) ∗ ( θξ ) ( S △ A ) ′ ) == 12 ( ξ A + f ρ ( A, S ) ∗ f ρ ( A, S ) ξ A ) = ξ A . e) For η ∈ (cid:13)| A ⊂ S ′ ˘ E and A ⊂ S , by b) and c),( θ ± θ ∗± η ) A = 1 √ θ ∗± η ) A = ( P ± η ) A , θ ± θ ∗± η ) A ′ = ± √ f ρ ( S △ A, S )( θ ∗± η ) S △ A == ± f ρ ( S △ A, S )( η S △ A ± f ρ ( S △ A, S ) ∗ η A ′ ) = ± f ρ ( S △ A, S ) η S △ A + 12 η A ′ == 12 ( η A ′ ± f ρ ′ ( S ′ , S △ A ) η S △ A ) = 12 (( V ρ ′ ∅ η ) A ′ ± ( V ρ ′ S ′ η ) A ′ ) = ( P ± η ) A ′ , so θ ± θ ∗± = P ± .f) For η ∈ (cid:13)| B ⊂ S ′ ˘ E and B ⊂ S , by a),b),c),e) and Proposition 4.1.1 b) (andCorollary 2.1.17 e)),( V ρ ′ A P ± η ) B = f ρ ′ ( A, A △ B )( P ± η ) A △ B = f ρ ( A, A △ B )( θ ± θ ∗± η ) A △ B == 1 √ f ρ ( A, A △ B )( θ ∗± η ) A △ B = 1 √ V ρA θ ∗± η ) B = ( θ ± V ρA θ ∗± η ) B , ( θ ± V ρA θ ∗± η ) B ′ = ± √ f ρ ( S △ B, S )( V ρA θ ∗± η ) S △ B == ± √ f ρ ( S △ B, S ) f ρ ( A, S △ A △ B )( θ ∗± η ) S △ A △ B == ± f ρ ( S △ B, S ) f ρ ( A, S △ A △ B )( P ± η ) S △ A △ B = ± f ρ ( S △ B, S )( V ρ ′ A P ± η ) S △ B == ± f ρ ′ ( S ′ , S ′ △ B ′ )( V ρ ′ A P ± η ) S ′ △ B ′ = ± ( V ρ ′ S ′ V ρ ′ A P ± η ) B ′ == ± ( V ρ ′ A V ρ ′ S ′ P ± η ) B ′ = ( V ρ ′ A P ± η ) B ′ so by a), θ ± V ρA θ ∗± = V ρ ′ A P ± = P ± V ρ ′ A P ± = P ± V ρ ′ A . g) The assertion concerning ϕ as well as the identity in the definition of ψ follow from a),d),e), and f). Thus ψ is a surjective E -C*-homomorphism. For Y ∈ Ker ψ , θ ∗ + Y θ + = θ ∗− Y θ − = 0 , so by a) and e), P + Y = P − Y = 0and we get Y = P + Y + P − Y = 0i.e. ψ is injective. 161EFERENCES[C1] Corneliu Constantinescu, C*-algebras.
Elsevir, 2001.[C2] Corneliu Constantinescu,
W*-tensor products and selfdual Hilbert right W ∗ -modules. Rev. Roumaine Math. Pures Appl., : 5-6 (2006) 583-596.[C3] Corneliu Constantinescu, Selfdual Hilbert right W ∗ -modules and theirW*-tensor products. Rev. Roumaine Math. Pures Appl., : 3 (2010)159-196.[K] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory ofoperator algebras.
Academic Press, 1983-1986.[L] Christopher E. Lance,
Hilbert C*-modules. A toolkit for operatoralgebraist.
Cambridge University Press, 1995.[S] Issai Schur ¨ U ber die Darstellung der endlichen Gruppen durch gebrochenelineare Substitutionen. J. Reine Angew. Math., (1904) 20-50.[T] Masamichi Takesaki,
Theory of Operator Algebra I.
Springer, 2002.[W] N. E. Wegge-Olsen,
K-theory and C*-algebras.
Oxford University Press,1993. 162 ndex
Schur E -function for T , F ( T, E ) , ˜ f , ˆ f (Definition 1.1.1).Λ( T, E ) , ˆ λ , δλ (Definition 1.1.3). E -module, E -linear (Definition 1.2.1). E -C*-algebra, E -W*-algebra, E -C*-subalgebra, E -W*-subalgebra, E -C*-homomorphism, E -W*-homomorphism, E -C*-isomorphism, ≈ E (Definition1.2.2). C E , C E (Definition 1.2.3).adapted, ˇ F , M E (Proposition 1.2.4).ˇ ϕ (Proposition 1.2.6).Φ E (Proposition 1.2.10). e (cid:13)| , e ⊗ , fP , G T , T ¯ G , T P (Definition 1.3.1). x e ⊗ K (Lemma 1.3.2). T , T , T (Definition 1.3.3). u t , V t , V ft (Definition 2.1.1). ϕ s,t , X t (Definition 2.1.5). R ( f ) , S ( f ) , S C ( f ) , S W ( f ) , S k·k ( f ) , S ( f, F ) , S k·k ( f, F ) (Definition 2.1.7).Locally finite, S T (Definition 2.1.18). U λ (Definition 2.2.1). S -isomorphism, ≈ S (Proposition 2.2.2a)). S ( F ), V Ft (Definition 2.3.1). S ( ϕ ) (Proposition 2.3.2).IT (Definition 3.1.3). w ( x ), winding number of x (Definition 3.1.5). f ρ , C l ( ρ ) Box of subsection 4.2. P f , △△