aa r X i v : . [ m a t h . OA ] F e b Dedicated to the memory of Uffe V. Haagerup
C*-DYNAMICAL RAPID DECAY
ERIK CHRISTENSEN
Abstract.
Some well known results by Haagerup, Jolissaint andde la Harpe may be extended to the setting of a reduced crossedproduct of a C*-algebra A by a discrete group G. We show that formany discrete groups, which include Gromov’s hyperbolic groupsand finitely generated discrete groups of polynomial growth, aninequality of the form k X k ≤ C sX g ∈ G (1 + | g | ) k X g k holds for any finitely supported operator X in the reduced crossedproduct. Introduction
Any result in classical harmonic analysis naturally raises the ques-tion, does this extend to a non commutative setting ? In the situation ofa discrete dynamical system, you may work with the group of integers Z which acts on a compact space as the group of homeomorphisms gen-erated by a single homeomorphism. By Gelfand’s fundamental theoremwe know that the set of compact topological spaces correspond to uni-tal commutative C*-algebras, and then the classical discrete dynamicalsystem as described above can be made non commutative in 2 ways, ei-ther by studying a general non commutative C*-algebra equipped witha single *-automorphism or by investigating properties of an action ofa general discrete group of homeomorphisms on a compact space. Theconstruction named reduced crossed product of a C*-algebra by a dis-crete group encodes a set up for the study of the action of a generaldiscrete group on a C*-algebra.Here we will extend results, by Haagerup from [8] on free non abeliangroups F d , by Jolissaint [11] on the concept named rapid decay andby de la Harpe [10] on hyperbolic groups, from the setting of a re-duced group C*-algebra to the setting of a reduced crossed product Date : February 8, 2021.2010
Mathematics Subject Classification.
Primary: 46L55, 37A55. Secondary:22D55, 46L07.
Key words and phrases.
Haagerup, C*-algebra, discrete group, crossed product,rapid decay.
C*-algebra.
The C*-algebra A upon which the discrete group G actsby *-automorphisms α g may be abelian or non abelian, and the casewhere the algebra is the trivial A = C I is the reduced group C*-algebracase.Uffe Haagerup’s article [8] has been very influential in the study ofdiscrete groups and von Neumann algebras of type II , and his nameand results are linked to many fundamental mathematical propertiessuch as the Haagerup approximation property [3], [9], the completelybounded approximation property [5] and the
Haagerup tensor product [16]. We will not go into the study of any of these aspects but concen-trate on an extension of the first basic result from [8] and lift it fromthe reduced group case to the reduced crossed product case for discretehyperbolic groups acting on general C*-algebras. In this setting weprove that there exists an
C > , depending on the group only, suchthat for any linear combination X = P L g X g in the algebraic reducedcrossed product we have the following inequality(1.1) k X k ≤ C sX g ∈ G (1 + | g | ) k X g k which is a direct generalization of Haagerup’s estimate in the groupalgebra case, since for these groups C < . In the article [11] Jolissaintshowed that, in the group algebra case, inequalities like (1.1) may beobtained for many other discrete groups, which are not free, and heintroduced the property named rapid decay for a discrete group G with a length function | g | , if there exist positive constants C, r suchthat for any x = P x g λ g in C [ G ](1.2) k x k op ≤ C qX (1 + | g | ) r | x g | . Shortly after Jolissaint had obtained his results they were extendedto the setting of hyperbolic groups by de la Harpe [10], and it turnsout that it is possible to extend both Jolissaint’s and de la Harpe’sresults to the crossed product setting. The basic insight which makesthis possible was formulated by Jollisaint in his lemmas 3.2.2 and 3.2.3.Then de la Harpe proved that these lemmas are true in the setting ofdiscrete finitely generated hyperbolic groups, so Jolissaint’s findingscould be extended. Here we have collected the statements from Jolis-saint’s basic lemmas into a property (J) which a discrete group with alength function may have, and then we prove that the inequality (1.1)holds, in any reduced crossed product of a C*-algebra and a group sat-isfying property (J). It seems possible to us that property (J) implies
APID DECAY IN ACTIONS ON A C*-ALGEBRA 3 hyperbolicity, but we have very little experience in dealing with such aquestion.We have also considered reduced crossed products of C*-algebras bydiscrete finitely generated groups which have polynomial growth. Inthis case it is possible to get the following - much better - result. Ifa discrete finitely generated group has polynomial growth then thereexist
C > , s > X = P L g X g in a reduced crossed product C ∗ r ( A ⋊ α G ) we have k X k ≤ C k X g ∈ G (1 + | g | ) s L g X g X ∗ g L ∗ g k (1.3) k X k ≤ C k X g ∈ G (1 + | g | ) s X ∗ g X g k The advantage of (1.3) over (1.2) is that usually k X g ∈ C k X ∗ g X g k < (cid:0) X g ∈ C k k X ∗ g X g k (cid:1) . Notation and norms
Most of the content of this section is well known in the setting ofa reduced group C*-algebra, but here we deal with a reduced crossedproduct C*-algebra. In order to be able to generalize the methodsfrom Haagerup’s paper [8], we need a characterization of the operatornorm in a reduced crossed product of a C*-algebra by a discrete group,and this result, which we think might be known but unpublished, ispresented in Proposition 2.4 in a self contained frame. We start with awell known operator theoretical version of Cauchy-Schwarz’ inequality,[16].
Definition 2.1.
Let
H, K be Hilbert spaces, J be an index set and( a ι ) ( ι ∈ J ) a family of bounded operators in B ( H, K ) . (i) If the sum P ι a ∗ ι a ι is ultrastrongly convergent in B ( H ) wesay that the family ( a ι ) is column bounded with column norm k ( a ι ) k c := k P a ∗ ι a ι k . (ii) If the sum P ι a ι a ∗ ι is ultrastrongly convergent in B ( K ) we saythat the family ( a ι ) is row bounded with row norm k ( a ι ) k r := k P a ι a ∗ ι k . Proposition 2.2.
Let
H, K be Hilbert spaces, J be an index set and ( a ι ) ( ι ∈ J ) , ( b ι ) ( ι ∈ J ) be column bounded families of operators in B ( H, K ) . (i) The sum P ι a ∗ ι b ι is ultrastrongly convergent in B ( H ) and theoperator norm of the sum satisfies k P ι a ∗ ι b ι k ≤ k ( a ι ) k c k ( b ι ) k c . E. CHRISTENSEN (ii)
There exists a column bounded family ( e ι ) ( ι ∈ J ) of column normat most such that for the positive bounded operator h on H defined by h := ( P ι a ∗ ι a ι ) we have for each ι ∈ J, a ι = e ι h andthe sum P ι e ∗ ι e ι equals the range projection of h. (iii) Let k := ( P ι b ∗ ι b ι ) , then there exists a contraction c in B ( H ) such that P ι b ∗ ι a ι = kch. Proof.
The families ( a ι ) and ( b ι ) represent bounded column operatorsin the operator space M ( J, (cid:0) B ( H, K ) (cid:1) and the statement (i) followsfrom properties of the operator product.The statement (ii) follows from the polar decomposition applied tothe column operator ( a ι ) . The result in statement (ii) may be applied to the column operator( b ι ) such that each b ι = f ι k, we can then define a contraction c in B ( H )by c := P ι f ∗ ι e ι and statement (iii) follows. (cid:3) The rest of this article takes place in the setting of the reducedcrossed product of a C*-algebra A by a discrete group G, which actson A by the *-automorphisms α g . We made a study of the propertiesof this crossed product in the article [4] and we will use most of thenotation and several of the results of that article below. A basic point ofview in Section 2 of [4] is that there are many facts related to propertiesof the coefficients of a Fourier series which generalize to properties ofthe coefficients of an element in a reduced discrete C*-crossed product.We recall that any element X in C := C ∗ r ( A ⋊ α G ) has a Fourierseries expansion X ∼ P g ∈ G L g X g , where the sum is convergent in thenorm k . k π described in Proposition 2.2 of [4]. A simple computationshows that for X ∼ P L g X g we have that the column and row normsof the family ( L g X g ) ( g ∈ G ) may be computed as k ( L g X g ) ( g ∈ G ) k c = k π ( X ∗ X ) k = k ( X ∗ X ) e k (2.1) k ( L g X g ) ( g ∈ G ) k r = k π ( XX ∗ ) k = k ( XX ∗ ) e k . In particular we notice the following proposition.
Proposition 2.3.
Let X ∼ P g L g X g be an element in C then the sumconverges in the column norm. We will use the π − norm or column norm to estimate the operatornorm in the computations to come, so we need the following propo-sition. It may be known to several people, but may be in a slightlydifferent setting. We are not aware of an explicit formulation as theone we present, but the results of [17] have a similar flavour. We do APID DECAY IN ACTIONS ON A C*-ALGEBRA 5 also think that people who prefer to look at completely positive map-pings as correspondences or operator bimodules may know the result,but still we are missing a reference.
Proposition 2.4.
Let B be a C*-algebra, H a Hilbert space and π : B → B ( H ) a completely positive and faithful mapping then ∀ b ∈ B : k b k = p sup {k π ( x ∗ b ∗ bx ) k : k π ( x ∗ x ) k ≤ } . Proof.
We may suppose that B is a subalgebra of B ( K ) for some Hilbertspace K, then since π is completely positive and faithful there existsby Stinespring’s result [18] a faithful representation ρ of B on H and abounded operator C in B ( K, H ) such that ∀ b ∈ B : π ( b ) = C ∗ ρ ( b ) C. We may define a semi norm, say n, on B by(2.2) ∀ b ∈ B : n ( b ) := sup {k ρ ( bx ) C k : k ρ ( x ) C k ≤ } . Since k C ∗ ρ ( y ∗ y ) C k = k ρ ( y ) C k for any y in B and π is faithful we getthat n is a norm and that ∀ b ∈ B : n ( b ) = p sup {k π ( x ∗ b ∗ bx ) k : k π ( x ∗ x ) k ≤ } . On the other hand the definition (2.2) implies that for b, d in B we have n ( bd ) ≤ n ( b ) n ( d ) so n is an algebra norm and n ( b ) ≤ k b k . For any pair b, x in B with k ρ ( x ) C k = k π ( x ∗ x ) k ≤ n ( b ∗ b ) ≥ k ρ ( b ∗ bx ) C k ≥ k C ∗ ρ ( x ∗ b ∗ bx ) C k = k ρ ( bx ) C k , so n ( b ∗ ) n ( b ) ≥ n ( b ∗ b ) ≥ n ( b ) . From here it follows that n ( b ) = n ( b ∗ ) and then n ( b ∗ b ) = n ( b ) , andthe completion say ˆ B of B with respect to the C*-norm n becomes aC*-algebra such that the inclusion of B in ˆ B is a contractive faithful*-homomorphism and hence an isometry. The proposition follows. (cid:3) In Haagerup’s and Jolissaint’s articles, [8], [11] they use the symbol ∗ to denote the operator product in a group algebra, since this is reallya convolution product. In our article on crossed product C*-algebras[4] the operator product is an invisible dot and the ∗ is used to denotethe Hadamard product, which in the notation from above takes theform (cid:0) X g L g X g (cid:1) ∗ (cid:0) X h L h Y h (cid:1) := X f L f X f Y f . We will use this convention here, too.
E. CHRISTENSEN The property (J)
The basic ideas in the arguments to come are taken from Haagerup’sarticle, and in the setting of a non commutative free group it is clearthat for two group elements x, y with reduced words x = x . . . x k and y = y . . . y l the number of cancellations, say p, needed to spellto xy gives the spelling of xy directly as xy = x . . . x ( k − p ) y ( p +1) . . . y l . In Jolissaint’s article he uses this observation in a very clever originalway, and he shows that this idea may be generalized to work up toa controllable error in some groups of isometries on a Riemannianmanifold with bounded strictly negative sectional curvature. Then dela Harpe showed that Jolissaint’s method of dealing with cancellationsworks in any finitely generated discrete group which is hyperbolic, asdefined by Gromov, [7] and [6]. Here we will instead take these resultsas the basis for the definition of a property we have named (J).We will now define the setting in which we will use Haagerup’s,Jolissaint’s and de la Harpe’s ideas. We define the cancellation numberin a general group with a length function as follows.
Definition 3.1.
Let G be a group with a length function g → | g | . For g, h in G the cancellation number c ( g, h ) of the pair is defined as thenon negative integer p ( g, h ) which satisfies2 p ( g, h ) ≤ | g | + | h | − | gh | < p ( g, h ) + 2 . It follows from the properties of a length function that 0 ≤ p ( g, h ) ≤ min {| g | , | h |} , and the cancellation number divides the cartesian product G × G into a sequence of disjoint subsets ( P p ) ( p ∈ N ) defined by P p := { ( g, h ) ∈ G × G : p ( g, h ) = p. } . Following Jolissaint we define certain subsets of a group G with alength function | g | as follows Definition 3.2. (i) ∀ r ≥ B r := { g ∈ G : | g | ≤ r } , (ii) ∀ k ≥ C k := { g ∈ G : k − < | g | ≤ k } , (iii) ∀ k ≥ ∀ α ≥ C k,α := { g ∈ G : k − α ≤ | g | ≤ k + α } , We can now collect the sufficient conditions, we have have draggedout of [11], into a property we name (J). We will not focus on whichgroups that may satisfy the property (J), but we think that the surveyarticle on rapid decay [2] by Chatterji will show that many groups dohave property (J). On the other hand we will sketch arguments whichshow that Jolissaint’s and de la Harpe’s examples do have property(J). It is easy to see that free non abelian groups do have property (J)
APID DECAY IN ACTIONS ON A C*-ALGEBRA 7 with the extra very nice properties that the constants of the definitionsatisfy α = β = γ = 0 and N = 1 . Definition 3.3.
Let G be a discrete group with a length function g → | g | . The pair ( G, | · | ) has property, (J) if ∃ α > , β > , γ > , :(3.1)( i ) ∀ g ∈ G, ∀ ≤ s < | g | + 1 , ∃ u ( g,s ) ∈ C s,α ( ii ) v ( g,s ) := u − g,s ) g ∈ C ( | g | +1 − s ) ,β ( iii ) ∀ p ∈ N , ∀ ( a, b ) ∈ P p ∃ c ( a, b ) ∈ C p,γ s. t. a = u ( ab, ( | a |− p )) c ( a, b ) , b = c ( a, b ) − v ( ab, ( | a |− p )) . ∀ µ > , ν > ∃ N ∈ N :(3.2) ∀ b ∈ G ∀ ≤ p ≤ | b | : |{ ( c, v ) ∈ C p,µ × C ( | b |− p ) ,ν : c − v = b }| ≤ N. Remark 3.4.
It is important for the following proofs in the next sec-tion to notice that the group element u ( g,s ) is determined uniquely by g and s, and then v ( g,s ) = u − g,s ) g is also determined by g and s only.The interesting thing about the factors c ( a, b ) is that they alwaysapproximately satisfy | c ( a, b ) | = p. In Jolissaint’s proof the group element u ( g,s ) is chosen geometrically,as we sketch now. On the geodesic which connects a point m in themanifold with its image g ( m ) , one chooses the point n s which has thedistance s to g ( m ) . Then u ( g,s ) is chosen such that the distance between n s and u − g,s ) (cid:0) g ( m ) (cid:1) is minimal among the distances between n s andthe set { u ( m ) : u ∈ G } . We will not continue to quote Jolissaintsproof, but show that a group G, which is hyperbolic with respect toa given word length, has the property (J). The proof follows that ofde la Harpe [10], but in order to explicitly establish the property (J)from the definition above, we repeat part of it here. We will use thefollowing notation. For a real number s, the expression ⌊ s ⌋ means thelargest integer dominated by s. Lemma 3.5.
Let G be a finitely generated discrete group which is hy-perbolic with respect to word length. Then G has the property (J).Proof. For a group element g written in reduced form as g = g . . . g | g | and a real s, ≤ s < | g | + 1 we define u ( g,s ) := ( e if 0 ≤ s < g . . . g ⌊ s ⌋ if 1 ≤ s < | g | + 1 , E. CHRISTENSEN and we find that u ( g,s ) ∈ C s, so α = 1 may be used. Similarly we find v ( g,s ) = ( g ( ⌊ s ⌋ +1) . . . g | g | if 0 ≤ s < | g | e if | g | ≤ s < g + 1 , and we get v ( g,s ) ∈ C ( | g | +1 − s ) , , so β = 1 is possible.Given a non negative integer p and a pair of group elements ( a, b ) ∈ P p with ab = g, then for k := | a | , l := | b | there exists c ∈ { , } suchthat | g | = k + l − p − c. Then for u ( g,k − p ) we may apply the lemmaat the bottom of page 771 in [10], to see that there exists an M ≥ , independent of k, l, p, c such that for c ( a, b ) := u − g,k − p ) a the followinginequalities hold(3.3) p ≤ | c ( a, b ) | ≤ p + M, so c ( a, b ) ∈ C p,M . In order to establish the property (3.2) we remark, that in our casewe have v ( g, ( k − p )) ∈ C ( | g | +1 − k + p ) , = C ( l − p − c +1) , , which means c ( a, b ) ∈ C p,M , v g, ( k − p ) ∈ C ( | b |− p ) , and c ( a, b ) v ( g, ( k − p )) = b. The result then follows from item (ii) in the lemma of [10]. (cid:3) Rapid decay
The most basic example of the phenomenon named rapid decay byPaul Jolissaint in [11] is presented quite early in many courses onFourier series. The example tells, that if f ( t ) is a differentiable complex2 π − periodic function on R , then its Fourier series is uniformly conver-gent. This is proven via the following argument based on the Cauchy-Schwarz inequality, as follows. Let f ( t ) have the Fourier series f ( t ) ∼ P Z c n e int , then the derivative f ′ ( t ) has the Fourier series f ′ ( t ) ∼ P Z inc n e int , and the sequence of complex numbers ( nc n ) ( n ∈ Z ) is in ℓ ( Z ) . Since for n = 0 we may write c n = n ( nc n ) , we get that thesequence ( c n ) ( n ∈ Z ) is in ℓ ( Z ) with k ( c n ) k ≤ | c | + π √ k ( nc n ) k . If wetranslate this to the setting of the discrete group Z equipped with thenatural length function | n | , we find that the group algebra C ∗ r ( Z ) maybe identified with the complex continuous 2 π − periodic functions onthe real axis and an element x in the group algebra which correspondto a differentiable function has a presentation as a uniformly conver-gent sum x = P ( n ∈ Z ) x n λ n . This example may be generalized to thesetting of a discrete group with a length function when the content ofthe example is formulated as in the following proposition.
APID DECAY IN ACTIONS ON A C*-ALGEBRA 9
Proposition 4.1.
Let x ∼ P Z x n λ n be an operator in C ∗ r ( Z ) . If thesequence (cid:0) (1 + | n | ) x n (cid:1) ( n ∈ Z ) is in ℓ ( Z ) , then the series is uniformlyconvergent and k x k ≤ (cid:0) π − (cid:1) (cid:0) P n ∈ Z | x n | (1 + | n | ) (cid:1) . We recall from Chatterji’s survey article [2], Definition 2.9, in a mod-ified form.
Definition 4.2.
Let G be a discrete group with a length function | g | , then G has the rapid decay property, with respect to | g | if there existspositive constants C, s such that for any operator x = P g x g λ g in thereduced group C*-algebra and with finite support we have k x k ≤ C sX g ∈ G | x g | (1 + | g | ) s . It is immediate that this definition may be extended to the setting ofa reduced crossed product of a C*-algebra by a discrete group in severalways. We have played with 3 possibilities of definition, but only beenable to obtain results for the 2 of them, which we define below. Thethird possibility is mentioned after the definition.
Definition 4.3.
Let G be a discrete group with a length function g →| g | , such that G acts on a C*-algebra A via a group of *-automorphisms α g . The reduced crossed product C := C ∗ r ( A ⋊ α G ) has rapid decay of operator type or scalar type if there exist positive constants C, s suchthat for any operator X = P L g X g with finite support:operator type: k X k ≤ C k X g ∈ G (1 + | g | ) s ( L g X g X ∗ g L ∗ g + X ∗ g X g ) k scalar type: k X k ≤ C (cid:18) X g ∈ G (1 + | g | ) s k X g k (cid:19) If one of the properties above holds for any C*-algebra A carryingan action α g of G we say that G possesses complete rapid decay ofrespectively operator type and scalar type.In the very first version of the article we thought that we could provethat free non abelian groups do have a sort of mixed rapid decay definedas mixed type: k X k ≤ C (cid:18) ∞ X k =0 (1 + | k | ) s k X g ∈ C k (cid:0) L g X g X ∗ g L ∗ g + X ∗ g X g (cid:1) k (cid:19) . Unfortunately we were wrong, but it may still be that some discretegroups with non polynomial growth satisfy such a condition, and thatwould be very helpful in the study of multipliers of the form M ϕ , as itfollows from the proof of Proposition 7.1.The complete operator type of rapid decay may be established forfinitely generated groups with polynomial growth by a simple imitationof the proofs from the group algebra case. For other groups it seemsimpossible to us to establish the operator type of rapid decay outsidethe reduced group algebra case. We establish the complete scalar typeof rapid decay for discrete groups which possess property (J) withrespect to a length function.5. complete operator rapid decay in a reduced crossedproduct of a C*-algebra by a finitely generatedgroup with polynomial growth Here we modify some of Jolissaint’s results from his section 3.1 of[11] to cover our situation.
Theorem 5.1.
Let A be a C*-algebra, G a finitely generated discretegroup with polynomial growth which has an action α g on A as a groupof *-automorphisms. There exists positive reals M, s such that for anyfinitely supported operator X = P L g X g in C := C ∗ r ( A ⋊ α G ) : k X k ≤ M k X g ∈ G (1 + | g | ) s +2 L g X g X ∗ g L ∗ g k k X k ≤ M k X g ∈ G (1 + | g | ) s +2 X ∗ g X g k . Proof.
The polynomial growth implies that there exists positive reals
C, s such that | C k | ≤ C (1 + k ) s , then the following manipulations arestandard, and the proof follows from Proposition 2.2 as follows APID DECAY IN ACTIONS ON A C*-ALGEBRA 11 X = ∞ X k =0 X g ∈ C k k ) | C k | (cid:0) (1 + k ) | C k | L g X g (cid:1) ≤√ C ∞ X k =0 X g ∈ C k k ) | C k | (cid:0) (1 + k ) (1+ s ) L g X g (cid:1) k X k ≤ π √ C √ k ∞ X k =0 X g ∈ C k (1 + k ) (2+ s ) X ∗ g X g k ≤ π √ C √ √ k X g ∈ G (1 + | g | ) (2+ s ) X ∗ g X g k so M := π √ C √ , may be used. The inequality involving L g X g X ∗ g L ∗ g follows in the same way. (cid:3) Complete scalar rapid decay for discrete groups withproperty (J)
The basic result in this section is the proposition just below, andthis is a combined extension of some of the first results in [8].
Proposition 6.1.
Let ( G, |·| ) be a discrete group with a length functionsatisfying the property (J). There exists a positive constant N such thatfor any action of G as a group of *-automorphisms α g on a C*-algebra A , any non negative integer k and any operator X = P g ∈ G L g X g in C := C ∗ r ( A ⋊ α G ) with finite support in C k : k X k ≤ N (1 + k ) qX k X g k . Proof.
We will let Y denote any finitely supported element in C withcolumn or π − norm at most 1. By Proposition 2.4 it is sufficient tobound the π − norm of XY in order to bound the operator norm k X k . Using the cancellation numbers, the operator XY may be written as asum of k + 1 summands S p defined as follows XY = X a ∈ C k X b ∈ G L a X a L b Y b = k X p =0 X { ( a,b ) ∈ P p : a ∈ C k } L a X a L b Y b (6.1) = k X p =0 S p Then let us fix a p in the set { , , . . . , k } . By the property (J) thereexists α > , ˆ β > , M > g, each p and each pair of group elements ( a, b ) ∈ P p with ab = g and a ∈ C k there exist group elements u ( g, ( k − p )) , v ( g, ( k − p )) , c ( a, b ) such that u ( g, ( k − p )) ∈ C ( k − p ) ,α , v ( g, ( k − p )) ∈ C ( | g | +1 − k + p ) , ˆ β , c ( a, b ) ∈ C p,M , (6.2) a = u ( g, ( k − p )) c ( a, b ) , b = c ( a, b ) − v ( g, ( k − p )) . We have | g | = | a | + | b | − p − c with c ∈ { , } , hence since | a | = k weget | g | + 1 − k + p = | b | − p + 1 − c so for β := ˆ β + 1 we have(6.3) v ( g, ( k − p )) ∈ C ( | b |− p ) ,β . In Haagerup’s case with free groups we get u ( g, ( k − p )) = a . . . a ( k − p ) ,c ( a, b ) = a ( k − p +1) . . . a k , v ( g, ( k − p )) = b ( p +1) . . . b | b | , with u ( g, ( k − p )) ∈ C ( k − p ) , c ( a, b ) ∈ C p , v ( g, ( k − p )) ∈ C ( | b |− k ) . In the case of a freenon commutative group the 2 first elements u, c are determined by a, p and the last 2 elements c, v are determined by b, p, but in the generalcase a, p does not determine the pair u ( g, ( k − p )) , c ( a, b ) nor does the pair b, p determines the pair c ( a, b ) , v ( g, ( k − p )) . The condition (3.2) is designedto deal with this problem.We can now start the estimation, and we will use the results ofProposition 2.2, so for a given g we define a positive operator Q u ( g, ( k − p )) by Q u ( g, ( k − p )) : = X { ( a,b ) ∈ P p : a ∈ C k , ab = g } L a X a X ∗ a L ∗ a . (6.4)To each pair ( a, b ) in P p with a in C k , and ab = g there exists acontraction q ( a, b ) such that L a X a = Q u ( g, ( k − p )) q ( a, b ) with X { ( a,b ) ∈ P p : a ∈ C k ,ab = g } q ( a, b ) q ( a, b ) ∗ ≤ I. Analogously we define R v ( g, ( k − p )) as the positive operator which is givenby R v ( g, ( k − p )) = X { ( a,b ) ∈ P p : a ∈ C k ab = g } Y ∗ b Y b (6.5)To each group element g and each pair ( a, b ) in P p with g = ab and a in C k , there exists a contraction r ( a, b ) such that L b Y b = r ( a, b ) R v ( g, ( k − p )) with X { ( a,b ) ∈ P p : a ∈ C k ,ab = g } r ( a, b ) ∗ r ( a, b ) ≤ I, APID DECAY IN ACTIONS ON A C*-ALGEBRA 13 and according to Proposition 2.2 we may define a contraction operator m g by(6.6) ∀ g ∈ G : m g := X { ( a,b ) ∈ P p : a ∈ C k ab = g } q ( a, b ) r ( a, b ) . When combining these equations we get(6.7) S p ( g ) = Q u ( g, ( k − p )) m g R v ( g, ( k − p )) and then X g ∈ G S p ( g ) ∗ S p ( g ) = X g ∈ G R v ( g, ( k − p )) m ∗ g Q u ( g, ( k − p )) m g R v ( g, ( k − p )) by k m g k ≤ ≤ X g ∈ G k Q u ( g, ( k − p )) k R v ( g, ( k − p )) by (6 .
4) and (6 . ≤ X g ∈ G (cid:0) X ( a,b ) ∈ P p : ab = g k X u ( g, ( k − p )) c ( a,b ) k (cid:1) · (cid:0) X ( e,f ) ∈ P p : ef = g Y ∗ c ( e,f ) − v ( g, ( k − p )) Y c ( e,f ) − v ( g, ( k − p )) (cid:1) split to sum over h, g ≤ (cid:0) X g ∈ G X ( a,b ) ∈ P p : ab = g k X u ( g, ( k − p )) c ( a,b ) k (cid:1) · (cid:0) X h ∈ G X ( e,f ) ∈ P p : ef = h Y ∗ c ( e,f ) − v ( g, ( k − p )) Y c ( e,f ) − v ( g, ( k − p )) (cid:1) In the last 2 sums depending on g and h respectively, one can viathe property (3.2) get an upper bound on the number of times eachelement of the form k X a k or Y ∗ f Y f appears in the sum. Let a in C k be given, then the number of solutions to the equation u ∈ C ( k − p ) ,α , c ∈ C p,M : uc = a is at most N. Similarly for each f the number of solutions to the equa-tion v ∈ C ( | f |− p ) ,β , c ∈ C p,M : c − v = f is at most N, and hence by (6.8) k S p k π ≤ N (cid:0) X a ∈ C k k X a k (cid:1) k Y k π so(6.9) k XY k π ≤ ( k + 1) N (cid:0) X a ∈ C k k X a k (cid:1) k Y k π by Proposition 2 . k X k ≤ ( k + 1) N (cid:0) X a ∈ C k k X a k (cid:1) , and the proposition follows (cid:3) In the case of a free non abelian group F d with any set, finite orinfinite, of generators the proposition above holds with N = 1 . Thereason is that for a pair ( a, b ) in P p with a ∈ C k the decompositions a = u ( g, ( k − p )) c ( a, b ) and b = c ( a, b ) − v ( g, ( k − p )) are described in an exactform just below the equation (6.3), such that the constants α, β, γ inDefinition 3.3 may be used with value 0. The exact decomposition alsoshows that in this case there will only be one solution to the equation(3.2) so we get N = 1 in this case, and we may note the followingcorollary. Corollary 6.2.
Let A be a C*-algebra with an action α g of F d , the freenon commutative group with d generators, then for any k ∈ N and any X in C ∗ r ( A ) with finite support in C k : k X k ≤ ( k + 1) (cid:0) X a ∈ C k k X a k (cid:1) . It is worth to remark that the result in the corollary above in the caseof the trivial C*-algebra A = C I gives exactly the content of Lemma1.5 in [8]. The content of Lemma 1.3 in [8] is named the Haagerupproperty in [15]. We may also here add a corollary describing the formthe
Haagerup property takes in the setting of a reduced crossed productof a C*-algebra by a discrete group with the property (J) This is thenatural extension of Jolissaint’s Proposition 3.2.4 in [11].
Corollary 6.3.
For k, l, m non negative integers and any pair of finitelysupported elements X = P g L g X g with support in C k and Y = P g L g Y g with support in C l : APID DECAY IN ACTIONS ON A C*-ALGEBRA 15 k M χ m ∗ (cid:0) XY (cid:1) k ≤ N (cid:0) X a ∈ C k k X a k (cid:1) k X b ∈ C l Y ∗ b Y b k k M χ m ∗ (cid:0) XY (cid:1) k ≤ N k X a ∈ C k L a X a X ∗ a L ∗ a k (cid:0) X b ∈ C l k Y b k (cid:1) When G is a free non abelian group N = 1 . Proof.
Let a ∈ C k and b ∈ C l be such that ab ∈ C m then there existsuniquely determined c ∈ { , } and p in N such that k + l − m = 2 p + c. In particular at most one S p = 0 , and the first inequality of the corollaryfollows from the proof of the theorem. The second follows from thefirst when applied to ( Y ∗ X ∗ ) . The free group statement follows fromthe corollary just above. (cid:3)
The fact that there are 2 inequalities above indicates to us thatthere might be a hope for the desired inequality below to be true fora finitely supported X with support in C k and a finitely supported Y with support in C l we hope thatdesired inequality k M χ m ∗ (cid:0) XY (cid:1) k ≤ N k X a ∈ C k L a X a X ∗ a L ∗ a k k X b ∈ C l Y ∗ b Y b k We may then continue and consider general elemnts X with finitesupport. Theorem 6.4.
Let G be a discrete group with a length function suchthat G satisfies the condition (J). There exists an M > such that forany action α g of G on a C*-algebra A and any X = P g ∈ G L g X g offinite support in C ∗ r ( A ⋊ α G ) : k X k ≤ M sX g ∈ G (1 + | g | ) k X g k . Proof.
We may proceed as in the proof of Lemma 1.5 in [8], so k X k ≤ ∞ X k =0 k X g ∈ C k X g k≤ N ∞ X k =0 ( k + 1) − (cid:18) ( k + 1) (cid:0) X g ∈ C k k X g k (cid:1) (cid:19) ≤ N π √ (cid:18) ∞ X k =0 ( k + 1) X g ∈ C k k X g k (cid:19) ≤ N π √ √ (cid:18) X g ∈ G (1 + | g | ) k X g k (cid:19) and the theorem follows, since for g ∈ C k we have k − < | g | ≤ k. (cid:3) Again there is a sharper estimate in the case of a free non abeliangroup.
Corollary 6.5. If G is a free non abelian group the constant M maybe chosen as M = 2 . Applications
The theory of rapid decay for group C*-algebras has been applied tovarious types of approximation properties for operator algebras [2], [3][5], [9], and many research articles are based on, or inspired by theseworks.Jolissaint realized from the beginning [12] that the rapid decay prop-erty makes it possible to base some K-theoretical computations on asubalgebra of rapidly decreasing operators, and this was then used byLafforgue [14] in his fundamental work on the Baum-Connes conjecture.The construction of a spectral triple for a reduced group C*-algebraof a discrete group, which occurs in Connes’ non commutative geom-etry, has an obvious candidate if the group has a length function. Itseems natural to use the property rapid decay to get some informa-tion on the properties of this spectral triple, and such attempts haveappeared in [1] and [15]. It is interesting to see that the so-called
Haagerup condition of [15] is the content of the very basic Lemma1.3 of [8] and of Proposition 3.2.4 of [11]. Here it is contained in thecorollary 6.3.It is not our intent to pursue possible extensions of the results basedon rapid decay from the group algebra case to the crossed product
APID DECAY IN ACTIONS ON A C*-ALGEBRA 17 setting, but we have made one easy observation, which may be appliedto a possible extension of some of the approximation properties.In Haagerup’s first article [8] he shows in Lemma 1.7 that for a func-tion ϕ on a free non abelain group G the multiplier M ϕ is bounded ifsup {| ϕ ( g ) | (1+ | g | ) : g ∈ G } is finite and k M ϕ k ≤ {| ϕ ( g ) | (1+ | g | ) : g ∈ G } . This result makes it possible for him to cut the completelypositive multiplier of norm 1 given by M ϕ λ with ϕ λ ( g ) := e − λ | g | tothe subsets B n , and in this way he obtains a bounded approximatemultiplier unit consisting of functions with finite support. We can notobtain such a nice result here because Haagerup’s estimate is based onthe fact that in the group algebra case we have k λ ( f ) k ≥ k f k , andthe analogous statement for crossed products is not true. We can get aresult which is is similar to Haagerup’s Lemma 1.7 for a group actionwhich has operator rapid decay. Proposition 7.1.
Let A be a C*-algebra, G a discrete group with alength function and α g an action of the group on A such that the reducedcrossed product has operator rapid decay with coefficients C, s.
If a com-plex function ϕ on the group satisfies m := sup {| ϕ ( g ) | (2 + | g | ) ( s +1) : g ∈ G } < ∞ then the multiplier M ϕ on C ∗ r ( A ⋊ α G ) is bounded andsatisfies k M ϕ k ≤ Cm.
If the action of α g has complete operator rapiddecay, then M ϕ is completely bounded with k M ϕ k cb ≤ Cm.
Proof.
Suppose ϕ is given with m finite then for any X = P L g X g withfinite support k M ϕ ∗ X k ≤ C k X g ∈ G (1 + | g | ) s | ϕ ( g ) | ( L g X g X ∗ g L ∗ g + X ∗ g X g ) k (7.1) ≤ C k X g ∈ G (1 + | g | ) − m ( L g X g X ∗ g L ∗ g + X ∗ g X g ) k≤ C m ∞ X k =0 (1 + k ) − k X g ∈ C k L g X g X ∗ g L ∗ g + X ∗ g X g k≤ C m π k X k ≤ C m k X k . If G possesses complete operator rapid decay, the action α g on A maybe lifted to actions on M n ( A ) , which all have operator rapid decay withcoefficients C, s and the result follows. (cid:3)
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