On synthetic and transference properties of group homomorphisms
aa r X i v : . [ m a t h . OA ] S e p ON SYNTHETIC AND TRANSFERENCE PROPERTIES OF GROUPHOMOMORPHISMS
G. K. ELEFTHERAKIS
Abstract.
We study Borel homomorphisms θ : G → H for arbitrary locally compactsecond countable groups G and H for which the measure θ ∗ ( µ )( α ) = µ ( θ − ( α )) for α ⊆ H a Borel setis absolutely continuous with respect to ν, where µ (resp. ν ) is a Haar measure for G, (resp. H ). We define a natural mapping G from the class of maximal abelian selfadjointalgebra bimodules (masa bimodules) in B ( L ( H )) into the class of masa bimodules in B ( L ( G )) and we use it to prove that if k ⊆ G × G is a set of operator synthesis, then( θ × θ ) − ( k ) is also a set of operator synthesis and if E ⊆ H is a set of local synthesisfor the Fourier algebra A ( H ), then θ − ( E ) ⊆ G is a set of local synthesis for A ( G ) . We also prove that if θ − ( E ) is an M -set (resp. M -set), then E is an M -set (resp. M -set) and if Bim ( I ⊥ ) is the masa bimodule generated by the annihilator of the ideal I in V N ( G ), then there exists an ideal J such that G ( Bim ( I ⊥ )) = Bim ( J ⊥ ) . If thisideal J is an ideal of multiplicity then I is an ideal of multiplicity. In case θ ∗ ( µ ) is aHaar measure for θ ( G ) we show that J is equal to the ideal ρ ∗ ( I ) generated by ρ ( I ) , where ρ ( u ) = u ◦ θ, ∀ u ∈ I. Introduction
Arveson discovered the connection between spectral synthesis and operator synthesis,[2]. Froelich found the precise connection for separable abelian groups, [8], and Spronkand Turowska for separable compact groups, [17]. Ludwig and Turowska generalized theprevious results in the case of locally compact second countable groups, G. They provedthat if E ⊆ G is a closed set and E ∗ = { ( s, t ) ∈ G × G : ts − ∈ E } , then E is a set oflocal synthesis if and only if E ∗ is a set of operator synthesis, [12]. Anoussis, Katavolosand Todorov stated in [1] that given a closed ideal I of the Fourier algebra A ( G ) , where G is a locally compact second countable group, there are two natural ways to construct a w ∗ -closed maximal abelian selfadjoint (masa) bimodule:(i) Let I ⊥ be the annihilator of I in V N ( G ) and then take the masa bimodule Bim ( I ⊥ )in the space of bounded operators acting on L ( G ) , B ( L ( G )) , generated by I ⊥ . (ii) Consider the space Sat ( I ) = span { N ( I ) T ( G ) } k·k T ( G ) , where N ( u )( s, t ) = u ( ts − )for all u ∈ I and T ( G ) is the projective tensor product L ( G ) ˆ ⊗ L ( G ) and then take itsannihilator Sat ( I ) ⊥ in B ( L ( G )) . One of their main results is that
Bim ( I ⊥ ) = Sat ( I ) ⊥ . They used this in order to provethat if A ( G ) possesses an approximate identity, then E ⊆ G is a set of spectral synthesisif and only if E ∗ is a set of operator synthesis.The transference of results from Harmonic Analysis to Operator Theory and vice versais not limited to the case of synthesis. In [15], Shulman, Todorov and Turowska proved Key words and phrases.
Group homomorphism, Haar measure, Spectral synthesis, Operator synthesis,MASA bimodule.2010
Mathematics Subject Classification. that if G is a locally compact second countable group and E ⊆ G , then E is an M -set(resp. M -set) if and only if E ∗ is an M -set (resp. M -set). Subsequently, Todorov andTurowska in [18] proved that an ideal J ⊆ A ( G ) is an ideal of multiplicity if and only if Bim ( J ⊥ ) is an operator space of multiplicity.In Section 2, we consider arbitrary locally compact second countable groups G and H, Borel homomorpisms θ : G → H for which the measure θ ∗ ( µ )( α ) = µ ( θ − ( α )) for α ⊆ H a Borel setis absolutely continuous with respect to ν, ( θ ∗ ( µ ) << ν ) , where µ (resp. ν ) is a Haarmeasure for G (resp. H ). Recall that Borel measurable homomorphisms between locallycompact groups are automatically continuous, [11], [13]. We define a natural mapping G from the class of masa bimodules in B ( L ( H )) to the class of masa bimodules in B ( L ( G )) . We prove that G ( M max ( k )) = M max (( θ × θ ) − ( k )) , G ( M min ( k )) = M min (( θ × θ ) − ( k )) , where M max ( k ) is the biggest masa bimodule supported on the ω -closed k, see the definitionbelow, and M min ( k ) is the smallest. Therefore if M max ( k ) is a synthetic operator space,then M max (( θ × θ ) − ( k )) is operator synthetic. This implication can also be deduced fromtheorem 4.7 of [14] or from theorem 5.2 of [6]. Here we present a different proof. We alsoprove that if E ⊆ H is a set of local synthesis, then θ − ( E ) is a set of local synthesis andif U is a w ∗ -closed masa bimodule for which G ( U ) contains a non-zero compact operator(or a non-zero finite rank operator or a rank one operator), so does U . We use this resultto prove that if θ − ( E ) is an M -set (resp. M -set), then E is an M -set (resp. M -set). If I is an ideal of A ( H ), we prove that there exists an ideal J ⊆ A ( G ) such that(1.1) G ( Bim ( I ⊥ )) = Bim ( J ⊥ ) , Sat ( J ) = span { N ( ρ ( I )) T ( G ) } k·k T ( G ) ,ρ ( u ) = u ◦ θ, ∀ u ∈ I, N ( ρ ( u ))( s, t ) = ρ ( u )( ts − ) . We use equalities (1.1) to prove that if J is an ideal of multiplicity, then I is an ideal ofmultiplicity. In Section 3 we assume that the measure θ ∗ ( µ ) is a Haar measure for thegroup θ ( G ) . We prove that if I is a closed ideal of A ( H ), then ρ ( I ) ⊆ A ( G ) and so we canchoose in (1.1) as J the ideal ρ ∗ ( I ) generated by ρ ( I ) . We also prove that if A ( G ) possessesan approximate identity and E is an ultra strong Ditkin set, then θ − ( E ) is also an ultrastrong Ditkin set.We now present the definitions and notation that will be used in this paper. If S is asubset of a linear space, we denote by [ S ] its linear span. If H and K are Hilbert spaces, B ( H, K ) is the set of bounded operators from H to K. We write B ( H ) for B ( H, H ) . If X ⊆ B ( H, K ) is a subspace, we write
Ref ( X ) for the reflexive hull of X , that is, Ref ( X ) = { T ∈ B ( H, K ) :
T ξ ∈ X ξ, ∀ ξ ∈ H } . Let G be a locally compact group with Haar measure µ, and T ( G ) the projective tensorproduct L ( G ) ˆ ⊗ L ( G ) . Every element h ∈ T ( G ) is an absolutely convergent series, h = X i f i ⊗ g i , f i , g i ∈ L ( G ) , i ∈ N , where P i k f i k k g i k < + ∞ . Such an element may be considered as a function h : G × G → C , defined by h ( s, t ) = P i f i ( s ) g i ( t ) . The norm in T ( G ) is given by k h k t = inf { X i k f i k k g i k : h = X i f i ⊗ g i } . N SYNTHETIC AND TRANSFERENCE PROPERTIES OF GROUP HOMOMORPHISMS 3
The space T ( G ) is predual to B ( L ( G )) . The duality is given by(
T, h ) t = X i ( T ( f i ) , g i ) , where h is as above and ( · , · ) is the inner product of L ( G ) . A subset F ⊆ G × G is called marginally null if F ⊆ ( α × G ) ∪ ( G × β ) , where α and β are Borel sets such that µ ( α ) = µ ( β ) = 0 . In this case we write F ≃ ∅ . If F and F aresubsets of G × G , we write F ≃ F if the symmetric difference F △ F is marginally null.If F ⊆ G × G , we denote by M max ( F ) the subspace of B ( L ( G )) consisting of all thoseoperators T satisfying ( α × β ) ∩ F ≃ ∅ ⇒ P ( β ) T P ( α ) = 0 . Here P ( β ) , and similarly P ( α ) is the projection onto L ( β, µ ) . We usually identify thealgebra L ∞ ( G, µ ) with the algebra of operators M f : L ( G ) → L ( G ) , g → f g, where f ∈ L ∞ ( G, µ ) . This algebra is a maximal abelian selfadjoint algebra, referred to as“masa” in what follows. If F ⊆ G × G , then M max ( F ) is an L ∞ ( G )-bimodule. An L ∞ ( G )-bimodule will be referred to as a “masa bimodule.” The space M max ( F ) is reflexive. Also,there exists a w ∗ -closed masa bimodule U with the property that it is the smallest w ∗ -closed masa bimodule U such that Ref ( U ) = M max ( F ) . We write U = M min ( F ) . Givena reflexive masa bimodule V , there exists a set k ⊆ G × G which is marginally equal to( ∪ n ∈ N α n × β n ) c , where α n , β n are Borel subsets of G such that V = M max ( k ) . An operator T belongs to V if and only if P ( β n ) T P ( α n ) = 0 , ∀ n. A set k that is marginally equal toa complement of a countable union of Borel rectangles is called an ω -closed set.An ω -closed set k is called operator synthetic if M max ( k ) = M min ( k ) . If s ∈ G, we denote by λ s the operator given by λ s ( f )( t ) = f ( s − t ) , ∀ f ∈ L ( G ) . The homomorphism G → B ( L ( G )) : s → λ s is called the left regular representation. Wedenote by A ( G ) the set of maps u : G → C given by u ( s ) = ( λ s ( ξ ) , η ) for ξ, η ∈ L ( G ) . Forany u ∈ A ( G ), we write k u k A ( G ) = inf {k ξ k k η k : u ( s ) = ( λ s ( ξ ) , η ) ∀ s } . The set A ( G ) is an algebra under the usual multiplication, and k · k A ( G ) is a norm making A ( G ) a commutative regular semisimple Banach algebra. We call this algebra a Fourieralgebra. We denote by V N ( G ) the following von Neumann subalgebra of B ( L ( G )): V N ( G ) = [ λ s : s ∈ G ] − w ∗ . This algebra is the dual of the Fourier algebra A ( G ) . The duality is given by ( λ s , u ) α = u ( s )for all u ∈ A ( G ) and s ∈ G. If E ⊆ G is a closed set, we write I ( E ) = { u ∈ A ( G ) : u | E = 0 } ,J ( E ) = { u ∈ A ( G ) : ∃ Ω open set, E ⊆ Ω , u | Ω = 0 } ., and J ( E ) for the closure of J ( E ) in A ( G ) . The spaces I ( E ) and J ( E ) are closed ideals of A ( G ) and J ( E ) ⊆ I ( E ) . The set E is called a set of spectral synthsesis if J ( E ) = I ( E ) . Let I c ( E ) be the set of all compactly supported functions f ∈ I ( E ) . We say that E is aset of local spectral synthesis if I c ( E ) ⊆ J ( E ) . G. K. ELEFTHERAKIS If u : G → L is an arbitrary map, we write N ( u ) for the map N ( u ) : G × G → L, ( s, t ) → u ( ts − ) . If u ∈ A ( G ), the map N ( u ) can be written as N ( u ) = X i ∈ N φ i ⊗ ψ i , where φ i , ψ i : G → C are Borel maps satisfying k X i ∈ N | φ i | k ∞ < + ∞ , k X i ∈ N | ψ i | k ∞ < + ∞ . The map N ( u ) satisfies N ( u ) T ( G ) ⊆ T ( G ) . See in [16] for more details.If I is a closed ideal of A ( G ) , I ⊥ is its annihilator in V N ( G ) : I ⊥ = { T ∈ V N ( G ) : ( T, u ) α = 0 , ∀ u ∈ I } . We also write
Sat ( I ) = [ N ( I ) T ( G )] −k·k t ⊆ T ( G ) . If X is a subspace of V N ( G ), we write Bim ( X ) for the folowing subspace of B ( L ( G )) : Bim ( X ) = [ M φ XM ψ : X ∈ X , φ, ψ ∈ L ∞ ( G )] − w ∗ . Synthetic and transference properties of group homomorphisms
In this section we assume that G and H are locally compact second countable groupswith Haar measures µ and ν respectively, θ : G → H is a continuous homomorphism and θ ∗ ( µ ) << ν. We conclude that the mapˆ θ : L ∞ ( H ) → L ∞ ( G ) , ˆ θ ( f ) = f ◦ θ is a weak*-continuous homomorphism. If α ⊆ H, (resp. β ⊆ G ) is a Borel set, we denoteby P ( α ) , (resp. Q ( β )) the projection onto L ( α, ν ) (resp. L ( β, µ )). If φ ∈ L ∞ ( H ) (resp. ψ ∈ L ∞ ( G )) , we denote by M φ (resp. M ψ ) the operator L ( H ) → L ( H ) : f → f φ (resp. L ( G ) → L ( G ) : g → gψ ). We define the following ternary ring of operators (TRO): N = { X : XP ( α ) = Q ( θ − ( α )) X, for α ⊆ H, a Borel set } . (For the definition and properties of TROs, see [3]). Observe that for every φ ∈ L ∞ ( H )and X ∈ N , we have XM φ = M φ ◦ θ X. Suppose that
Ker (ˆ θ ) = L ∞ ( α c ) , for some Borel set α c ⊆ H. Then the map L ∞ ( α ) → L ∞ ( G ) , ˆ θ ( f | α ) = f ◦ θ, f ∈ L ∞ ( H )is a one-to-one ∗ -homomorphism. We now define the following TRO: M = { X : XP ( α ) = Q ( θ − ( α )) X, α ⊆ α , Borel } ⊆ B ( L ( α ) , L ( G )) . If R is the projection onto L ( α ) , we can easily see that N = M R. Let
A ⊆ B ( L ( G )) be the commutant of the algebra { M φ ◦ θ : φ ∈ L ∞ ( H ) } . By theorem3.2 of [5], [ M ∗ M ] − w ∗ = L ∞ ( α ) , [ MM ∗ ] − w ∗ = A . For every masa bimodule
U ⊆ B ( L ( H )) , we define G ( U ) = [ N UN ∗ ] − w ∗ and for every masa bimodule U ⊆ B ( L ( α )) , we define F ( U ) = [ MUM ∗ ] − w ∗ . N SYNTHETIC AND TRANSFERENCE PROPERTIES OF GROUP HOMOMORPHISMS 5
By proposition 2.11 of [5], the map F is a biijection from the masa bimodules acting on L ( α ) onto the A -bimodules acting on L ( G ) . The inverse of F is given by F − ( V ) = [ M ∗ VM ] − w ∗ . We can easily see that G ( U ) = F ( R U R ) . Remark 2.1.
We inform the reader of the following:(i) The spaces U and F ( U ) are stably isomorphic in the sense that there exists a Hilbertspace H such that the spaces U ¯ ⊗ B ( H ) and F ( U ) ¯ ⊗ B ( H ) are isomorphic as dual operatorspaces, where ¯ ⊗ is the normal spatial tensor product, [7].(ii) The spaces U , F ( U ) are spatially Morita equivalent in the sense of [6]. Lemma 2.2.
Suppose k ⊆ α × α is an ω -closed set. Suppose further that U = M max ( k ) and V = M max ( σ ) , where σ = ( θ × θ ) − ( k ) . Then F ( U ) = V . Proof.
Suppose that α n ⊆ α and β n ⊆ α , n ∈ N are Borel sets such that k = ( ∪ n ( α n × β n )) c . Then σ = ( ∪ n ( θ − ( α n ) × θ − ( β n ))) c . If Z ∈ U , X, Y ∈ M , then Q ( θ − ( β n )) XZY ∗ Q ( θ − ( α n )) = XP ( β n ) ZP ( α n ) Y ∗ = X Y ∗ = 0 , ∀ n. Therefore
MUM ∗ ⊆ V . Similarly, we can prove M ∗ VM ⊆ U . The above relations implythat MM ∗ VMM ∗ ⊆ MUM ∗ ⊆ V . Since [ MM ∗ ] − w ∗ is an unital algebra, V = [ MUM ∗ ] − w ∗ = F ( U ) . (cid:3) Lemma 2.3.
Let U , be as in Lemma 2.2, Then F ( U min ) = F ( U ) min . Proof. If W = M max (Ω) is a reflexive masa bimodule, we write W min = M min (Ω) . FromLemma 2.2 F ( U ) = M max ( σ ) . Therefore F ( U ) min = M min ( σ ) . Since A contains the masa L ∞ ( G ) and F ( U ) is a reflexive A− bimodule the spaceΠ = (cid:18) A F ( U )0 A (cid:19) is a CSL algebra. From the proof of proposition 4.7 of [5], we have (cid:18) F ( U ) min (cid:19) = (cid:18) F ( U )0 0 (cid:19) min ⊆ Π min . Also the diagonal of Π , (cid:18) A A (cid:19) , belongs to Π min . Thus, (cid:18)
A F ( U ) min A (cid:19) ⊆ Π min . But
Ref (cid:18)(cid:18)
A F ( U ) min A (cid:19)(cid:19) = Π . Therefore, Π min ⊆ (cid:18) A F ( U ) min A (cid:19) . G. K. ELEFTHERAKIS
We conclude that Π min = (cid:18) A F ( U ) min A (cid:19) . Since Π is an algebra by [4, Theorem 22.19], Π min is also an algebra, which implies AF ( U ) min A ⊆ F ( U ) min . Observe that MU min M ∗ ⊆ MUM ∗ ⊆ F ( U ) . Since by Lemma 2.2 F ( U ) is a reflexive space, Ref ( MU min M ∗ ) ⊆ F ( U ) . Let Z ∈ F ( U ) and assume that Z does not belong to Ref ( MU min M ∗ ) . Thus, thereexists a ξ ∈ L ( G ) such that Zξ does not belong to [ MU min M ∗ ξ ] . Thus, there exists an ω ∈ L ( G ) such that ( XSY ∗ ξ, ω ) = 0 , ∀ X, Y ∈ M , S ∈ U min and (
Zξ, ω ) = 0 . We have( SY ∗ ξ, X ∗ ω ) = 0 , ∀ X, Y ∈ M , S ∈ U min . Since SY ∗ ξ ∈ U min Y ∗ ξ = U Y ∗ ξ , we have( SY ∗ ξ, X ∗ ω ) = 0 , ∀ X, Y ∈ M , S ∈ U ⇒ ( XSY ∗ ξ, ω ) = 0 , ∀ X, Y ∈ M , S ∈ U . Since F ( U ) = [ MUM ∗ ] − w ∗ , ( T ξ, ω ) = 0 , ∀ T ∈ F ( U ) . Therefore (
Zξ, ω ) = 0 . This contradiction shows that F ( U ) ⊆ Ref ( MU min M ∗ ) ⇒ F ( U ) = Ref ( MU min M ∗ ) . Since [ MU min M ∗ ] − w ∗ is a masa bimodule, F ( U ) min ⊆ [ MU min M ∗ ] − w ∗ . By symmetry,we have U min ⊆ [ M ∗ F ( U ) min M ] − w ∗ . Thus, F ( U ) min ⊆ [ MU min M ∗ ] − w ∗ ⊆ [ MM ∗ F ( U ) min MM ∗ ] − w ∗ ⊆ [ AF ( U ) min A ] − w ∗ ⊆ F ( U ) min . Therefore F ( U min ) = F ( U ) min . (cid:3) Theorem 2.4.
Let k ⊆ H × H be an ω -closed set. Then(i) G ( M max ( k )) = M max (( θ × θ ) − ( k )) and(ii) G ( M min ( k )) = M min (( θ × θ ) − ( k )) . Proof. (i) By Lemma 2.2, F ( M max ( k ∩ ( α × α )) = M max (( θ × θ ) − ( k ∩ ( α × α ))) . We can easily see that if k , k are ω − closed sets then M max ( k ∩ k ) = M max ( k ) ∩ M max ( k ) , thus M max (( θ × θ ) − ( k ∩ ( α × α ))) = M max (( θ × θ ) − )( k )) ∩ M max (( θ × θ ) − ( α × α )) . Since θ − ( α ) = G up to measure zero, the sets G × G, ( θ × θ ) − ( α × α ) are marginallyequal, thus M max (( θ × θ ) − ( α × α )) = B ( L ( G )) . Therefore G ( M max ( k )) = F ( M max ( k ∩ ( α × α )) = M max (( θ × θ ) − ( k )) . (ii) By Lemma 2.3, F ( M min ( k ∩ ( α × α )) = M min (( θ × θ ) − ( k ∩ ( α × α )) = M min (( θ × θ ) − ( k ) ∩ ( θ × θ ) − ( α × α )) . Since the sets G × G, ( θ × θ ) − ( α × α ) are marginally equal, we conclude that G ( M min ( k )) = F ( M min ( k ∩ ( α × α )) = M min (( θ × θ ) − )( k )) . N SYNTHETIC AND TRANSFERENCE PROPERTIES OF GROUP HOMOMORPHISMS 7 (cid:3)
Corollary 2.5. If U = M max ( k ) is a synthetic masa bimodule acting on L ( H ) , then G ( U ) = M max (( θ × θ ) − ( k )) is also synthetic. Remark 2.6.
The implication of the previous corollary was first proved in [14, Theorem4.7]. In the present paper, we have given a different proof.
Corollary 2.7.
Let E ⊆ H be a closed set. Then(i) G ( M max ( E ∗ )) = M max ( θ − ( E ) ∗ ) and G ( M min ( E ∗ )) = M min ( θ − ( E ) ∗ ); (ii) If E is a set of local synthesis, then θ − ( E ) is a set of local synthesis;(iii) If E is a set of local synthesis and A ( G ) possess an approximate identity, then θ − ( E ) is a set of spectral synthesis.Proof. (i) By Theorem 2.4, G ( M max ( E ∗ )) = M max (( θ × θ ) − ( E ∗ )) = M max ( θ − ( E ) ∗ ) . Similarly, G ( M min ( E ∗ )) = M min (( θ × θ ) − ( E ∗ )) = M min ( θ − ( E ) ∗ ) . (ii) If E is a set of local synthesis, then M max ( E ∗ ) is a masa bimodule of operatorsynthesis. By Corollary 2.5 and (i), M max ( θ − ( E ) ∗ ) is also a masa bimodule of operatorsynthesis. Thus, by [12], θ − ( E ) is a set of local synthesis.(iii) If A ( G ) possess an approximate identity, then θ − ( E ) is a set of local synthesis ifand only if θ − ( E ) is a set of spectral synthesis. Now use (ii). (cid:3) Theorem 2.8.
Let
U ⊆ B ( L ( H )) be a masa bimodule. If G ( U ) contains a non-zerocompact operator, then so does U . The same holds replacing compact by finite rank or byrank one operator.Proof.
We have G ( U ) = F ( R U R ) and R U R = F − ( G ( U )) = [ M ∗ G ( U ) M ] − w ∗ . If K ∈ G ( U ) is a non-zero compact operator, then M ∗ K M ⊆ R U R ⊆ U . It suffices to prove that M ∗ K M 6 = 0 . Suppose that M ∗ K M = 0 . Then [ MM ∗ ] − w ∗ K [ MM ∗ ] − w ∗ = 0 . Since [ MM ∗ ] − w ∗ = A is an unital algebra, K = 0 . This contradiction shows that U contains a non-zero compactoperator. The remaining cases are proved similarly. (cid:3) Corollary 2.9.
Let k ⊆ H × H be an ω -closed set and assume that M max (( θ × θ ) − ( k )) (resp. M min (( θ × θ ) − ( k )) contains a non-zero compact operator, then M max ( k ) , (resp. M min ( k ) , ) also contains a non-zero compact operator. The same holds replacing compactby finite rank or by rank one operator. Remark 2.10.
The implication that if G ( M max ( k )) contains a non-zero compact operator,then M max ( k ) also contains a non-zero compact operator, was first proved in [15, Corollary4.8] for some special cases of θ. Theorem 2.11.
Let I be a closed ideal of A ( H ) and U = Bim ( I ⊥ ) . Then there exists aclosed ideal J of A ( G ) such that G ( U ) = Bim ( J ⊥ ) . G. K. ELEFTHERAKIS
Proof.
Let ρ G : G → B ( L ( G )) , t → ρ Gt , be the right regular representation of G on L ( G ) , that is, the representation ρ Gt ( f )( x ) = ∆ G ( x ) f ( xt ) , t, x ∈ G, f ∈ L ( G ) , where ∆ G is the modular function of G. Similarly, we define the right regular representation ρ H : H → B ( L ( H )) of the group H. By theorem 4.3 of [1] it suffices to prove ρ Gt G ( U ) ρ Gt − ⊆ G ( U ) , ∀ t ∈ G. If P ∈ L ∞ ( H, ν ) is a projection, there exists a Borel set α such that P = P ( α ) ≡ L ( α, ν ) . If s ∈ H, we denote by P s the projection onto L ( αs ) . We can easily see that ρ Hs P ρ Hs − = P s − . Let α ⊆ H be a Borel set and t ∈ G. Thenˆ θ ( P θ ( t ) ) = ˆ θ ( P ( αθ ( t ))) = Q ( θ − ( αθ ( t ))) = Q ( θ − ( α ) t ) = Q ( θ − ( α )) t = ˆ θ ( P ( α )) t = ˆ θ ( P ) t , where Q ( β ) ≡ L ( β, µ ) . Thus if X ∈ N , P ∈ L ∞ ( H ) and t ∈ G,XP θ ( t ) = ˆ θ ( P ) t X. Therefore, Xρ Hθ ( t ) P ρ Hθ ( t ) − = XP θ ( t ) − = ˆ θ ( P ) t − X = ρ Gt ˆ θ ( P ) ρ Gθ ( t ) − X. Also, ρ Gt − Xρ Hθ ( t ) P = ˆ θ ( P ) ρ Gt − Xρ Hθ ( t ) , for all t ∈ G and P ∈ L ∞ ( H ) . We conclude that ρ Gt − N ρ Hθ ( t ) ⊆ N . Now take
X, Y ∈ N , t ∈ G and Z ∈ U . There exist X , Y ∈ N such that ρ Gt X = X ρ Hθ ( t ) and ρ Gt Y = Y ρ Hθ ( t ) . Therefore ρ Gt XZY ∗ ρ Gt − = X ρ Hθ ( t ) Zρ Hθ ( t ) − Y ∗ . By theorem 4.3 of [1], ρ Hθ ( t ) Zρ Hθ ( t ) − ∈ U . Thus ρ Gt XZY ∗ ρ Gt − ∈ N UN ∗ . We have proven ρ Gt N UN ∗ ρ Gt − ⊆ N UN ∗ , which implies ρ Gt G ( U ) ρ Gt − ⊆ G ( U ) . (cid:3) Remark 2.12. If u ∈ A ( H ) , we denote by ρ ( u ) the function u ◦ θ. There exist casesof
G, H, θ such that ρ ( A ( H )) ∩ A ( G ) = { } . For example if G is a non-compact group θ : G → H is the trivial homomorphism and u ( e H ) = 0 then ρ ( u ) is a non-zero constantmap and therefore doesn’t belong to A ( G ) . Therefore in case ρ ( A ( H )) ∩ A ( G ) = { } if I is a closed ideal of A ( H ) , then ρ ( I ) is not contained in A ( G ) . Nevertheless, by Theorem2.11, if U = Bim ( I ⊥ ) , there is a closed ideal J ⊆ A ( G ) such that G ( U ) = Bim ( J ⊥ ) . Weare going to prove that
Sat ( J ) = [ N ( ρ ( I )) T ( G )] −k·k t . In the sequel we fix a closed ideal I ⊆ A ( H ) , and write U = Bim ( I ⊥ ) and Ξ =[ N ( ρ ( I )) T ( G )] −k·k t . Let J ⊆ A ( G ) be a closed ideal such that G ( U ) = Bim ( J ⊥ ) . Lemma 2.13.
The space Ξ ⊥ is a A -bimodule. N SYNTHETIC AND TRANSFERENCE PROPERTIES OF GROUP HOMOMORPHISMS 9
Proof.
Let V , V ∈ N , X ∈ Ξ ⊥ , u ∈ I and f, g ∈ L ( G ) . If N ( u ) = P i φ i ⊗ ψ i , we have( V V ∗ XV V ∗ , N ( u ◦ θ )( f ⊗ g )) t = X i ( V V ∗ XV V ∗ , (( φ i ◦ θ ) f ) ⊗ (( ψ i ◦ θ ) g )) t = X i ( V ∗ XV , V ∗ (( φ i ◦ θ )( f )) ⊗ V ∗ (( ψ i ◦ θ )( g ))) t = X i ( V ∗ XV , V ∗ ( M φ i ◦ θ ( f )) ⊗ V ∗ ( M ψ i ◦ θ ( g ))) t = X i ( V ∗ XV , M φ i V ∗ ( f ) ⊗ M ψ i V ∗ (( g )) t = X i ( X, V M φ i V ∗ ( f ) ⊗ V M ψ i V ∗ ( g )) t = X i ( X, M φ i ◦ θ V V ∗ ( f ) ⊗ M ψ i ◦ θ V V ∗ ( g )) t = ( X, N ( u ◦ θ )( V V ∗ ( f ) ⊗ V V ∗ ( g ))) t Since N ( u ◦ θ )( V V ∗ ( f ) ⊗ V V ∗ ( g )) ∈ Ξ and X ∈ Ξ ⊥ , we have ( V V ∗ XV V ∗ , N ( u ◦ θ )( f ⊗ g )) t = 0 Thus V V ∗ XV V ∗ ∈ Ξ ⊥ . The algebra A is equal to [ N N ∗ ] − w ∗ , therefore A Ξ ⊥ A ⊆ Ξ ⊥ . (cid:3) Theorem 2.14.
The spaces Ξ and Sat ( J ) are equal.Proof. First we are going to prove that N Bim ( I ⊥ ) N ∗ ⊆ Ξ ⊥ . Let V , V ∈ N , X ∈ Ξ ⊥ , u ∈ I and f, g ∈ L ( G ) . If N ( u ) = P i φ i ⊗ ψ i , we have( V XV ∗ , N ( u ◦ θ )( f ⊗ g )) t = X i ( V XV ∗ , (( φ i ◦ θ ) f ) ⊗ (( ψ i ◦ θ ) g )) t = X i ( X, V ∗ M φ i ◦ θ ( f ) ⊗ V ∗ M ψ i ◦ θ ( g )) t = X i ( X, M φ i V ∗ ( f ) ⊗ M ψ i V ∗ ( g )) t =( X, N ( u )( V ∗ ( f ) ⊗ V ∗ ( g ))) t . Since X ∈ Bim ( I ⊥ ) and u ∈ I , we have( V XV ∗ , N ( u ◦ θ )( f ⊗ g )) t = 0 . Thus N Bim ( I ⊥ ) N ∗ ⊆ Ξ ⊥ ⇒ Bim ( J ⊥ ) ⊆ Ξ ⊥ ⇒ Ξ ⊆ Sat ( J ) . If X ∈ Ξ ⊥ and V , V , V , V ∈ N , then Lemma 2.13 implies that V V ∗ XV V ∗ ∈ Ξ ⊥ . Thus for all u ∈ I, f, g ∈ L ( G ) , we have0 = ( V V ∗ XV V ∗ , N ( u ◦ θ )( f ⊗ g )) t = ( V ∗ XV , N ( u )( V ∗ ( f ) ⊗ V ∗ ( g ))) t . Since R ( L ( H )) = [ M ∗ ( L ( G ))] , we conclude that0 = ( V ∗ XV , N ( u | α × α ( f ⊗ g )) , ∀ f, g ∈ L ( α ) , u ∈ I. Since
RBim ( I ⊥ ) R = [ N ( u | α × α ( f ⊗ g ) : u ∈ I, f, g ∈ L ( α )] ⊥ , we have that V ∗ XV ∈ RBim ( I ⊥ ) R. Therefore N ∗ Ξ ⊥ N ⊆
RBim ( I ⊥ ) R, which implies N N ∗ Ξ ⊥ N N ∗ ⊆ F ( RBim ( I ⊥ ) R ) = G ( Bim ( I ⊥ )) = Bim ( J ⊥ ) . The space A = [ N N ∗ ] − w ∗ is an unital algebra, thusΞ ⊥ ⊆ Bim ( J ⊥ ) ⇒ Bim ( J ⊥ ) ⊥ ⊆ Ξ ⇒ Sat ( J ) ⊆ Ξ . Since we have already shown Ξ ⊆ Sat ( J ), we obtain the required equality. (cid:3) For the following theorem, we recall from [18] that a closed ideal J ⊆ A ( G ) is an idealof multiplicity if J ⊥ ∩ C ∗ r ( G ) = { } , where C ∗ r ( G ) is the reduced C ∗ -algebra of G. Theorem 2.15.
Let I be a closed ideal of A ( H ) . By Theorems 2.11 and 2.14 there existsa closed ideal J ⊆ A ( G ) such that G ( Bim ( I ⊥ )) = Bim ( J ⊥ ) and Sat ( J ) = [ N ( I )( T ( G ))] −k·k t . If J is an ideal of multiplicity, then I is also an ideal of multiplicity.Proof. By [18, Corollary 1.5 ], if J is an ideal of multiplicity, then Bim ( J ⊥ ) contains a non-zero compact operator. By Theorem 2.8, Bim ( I ⊥ ) contains a non-zero compact operator.Thus, again by [18, Corollary 1.5 ], I is an ideal of multiplicity. (cid:3) A closed set E ⊆ H is called an M -set (resp. an M -set) if the ideal J H ( E ) (resp. I H ( E )) is an ideal of multiplicity. Corollaries 2.7 (i) and 2.9 together with [18, Corollary3.6] imply the following: Corollary 2.16. If E ⊆ H is a closed set such that θ − ( E ) is an M -set (resp. an M -set),then E is an M -set (resp. an M -set). Remark 2.17.
The previous corollary was proven in [15] for some special cases of θ. The case when θ ∗ ( µ ) is a Haar measure for θ ( G )Let G and H be locally compact, second countable groups with Haar measures µ and ν respectively. Suppose that θ : G → H is a continuous homomorphism, and assume that m = θ ∗ ( µ ) << ν. Since G is a σ − compact set and θ is a continuous map then θ ( G ) is alsoa σ − compact set and hence a Borel set. Also θ ∗ ( µ ) << ν implies that ν ( θ ( G )) > . BySteinhaus theorem the group θ ( G ) contains an open set. We conclude that θ ( G ) is an openset. We note that the open subgroups of a locally compact group are closed. Using thesefacts we can easily see that ν | H is a Haar measure of H = θ ( G ) . In some cases m << ν implies that m is a Haar measure for H . Thus there exists c > m | H = cν | H . In this section we investigate this equality. We can replace the Haar measure ν with cν and thus we may assume that m ( α ) = ν ( α ) for all Borel sets α ⊆ H . For every u ∈ A ( H ) , we define ρ ( u ) = u ◦ θ. We are going to prove that ρ : A ( H ) → A ( G )is a continuous homomorphism and that if I is a closed ideal of A ( H ) and U = Bim ( I ⊥ ),then G ( U ) = Bim ( ρ ∗ ( I ) ⊥ ) , where ρ ∗ ( I ) is the closed ideal of A ( G ) generated by ρ ( I ) and G is the map defined in Section 2. For every u ∈ A ( H ) , we denote by π ( u ) the function u | H . By [10, Theorem 2.6.4], π ( u ) ∈ A ( H ) for all u ∈ A ( H ) . Thus if u ∈ A ( H ) , thereexist f, g ∈ L ( H ) such that u ( t ) = π ( u )( t ) = ( λ H t f, g ) , ∀ t ∈ H . By [10, Corollary 2.6.5], the map π : A ( H ) → A ( H ) is contractive onto homomorphism,thus π ( I ) is an ideal of A ( H ) . Also, observe that the map A : L ( H ) → L ( G ) given by A ( f ) = f ◦ θ is an isometry. N SYNTHETIC AND TRANSFERENCE PROPERTIES OF GROUP HOMOMORPHISMS 11
Lemma 3.1.
Let u ∈ A ( H ) . Then ρ ( u ) ∈ A ( G ) . Actually, if u ( t ) = π ( u )( t ) = ( λ H t f, g ) , ∀ t ∈ H , then ρ ( u )( s ) = ( λ Gs Af, Ag ) , ∀ s ∈ G. Proof.
For every s ∈ G, we have( λ Gs Af, Ag ) = Z G Af ( s − t ) Ag ( t ) dµ ( t ) = Z G ( f ◦ θ )( s − t )( g ◦ θ )( t ) dµ ( t ) = Z G ( f θ ( s − ) ◦ θ )( t )( g ◦ θ )( t ) dµ ( t ) = Z H f θ ( s − ) ( t ) g ( t ) dm ( t ) = Z H f θ ( s − ) ( t ) g ( t ) dν ( t ) = ( λ H θ ( s ) f, g ) = u ( θ ( s )) . (cid:3) Theorem 3.2.
The map ρ : A ( H ) → A ( G ) is a continuous homomorphism.Proof. If u , u ∈ A ( H ) , then ρ ( u u ) = ( u u ) ◦ θ = u ◦ θ · u ◦ θ = ρ ( u ) ρ ( u ) . Let u ∈ A ( H ) . We assume that f, g ∈ L ( H ) , π ( u )( t ) = ( λ H t f, g ) , ∀ t ∈ H . By Lemma 3.1, ρ ( u )( s ) = ( λ Gs Af, Ag ) , ∀ s ∈ G. Thus k ρ ( u ) k A ( G ) ≤ k Af k k Ag k = k f k k g k for all f and g such that π ( u )( t ) = ( λ H t f, g ) . Thus k ρ ( u ) k A ( G ) ≤ k π ( u ) k A ( H ) . Since π is a contraction, k ρ ( u ) k A ( G ) ≤ k u k A ( H ) . (cid:3) Lemma 3.3.
Let I ⊆ A ( H ) be an ideal and X ∈ B ( L ( G )) such that ( X, N ( ρ ( u )) h ) t = 0 , ∀ h ∈ T ( G ) , u ∈ I. Then ( X, N ( v ) h ) t = 0 , ∀ h ∈ T ( G ) , v ∈ ρ ∗ ( I ) . Proof.
Let K = { v ∈ ρ ∗ ( I ) : ( X, N ( v ) h ) t = 0 , ∀ h ∈ T ( G ) } . Clearly K is a closed subset of A ( G ) and ρ ( I ) ⊆ K ⊆ ρ ∗ ( I ) . If v ∈ K and v ∈ A ( G ) , wehave ( X, N ( v v ) h ) t = ( X, N ( v )( N ( v )( h )) t . Since N ( v )( h ) ∈ T ( G ) , v ∈ K , we have ( X, N ( v v ) h ) t = 0 . Thus v v ∈ K. Therefore K is an ideal. We conclude that K = ρ ∗ ( I ) . (cid:3) In the following theorem, we fix a closed ideal I ⊆ A ( H ) , we assume that U = Bim ( I ⊥ )and define G ( U ) = [ N UN ∗ ] − w ∗ , where N is the TRO defined in Section 2. We are goingto prove that G ( U ) = Bim ( ρ ∗ ( I ) ⊥ ) . Theorem 3.4.
The space G ( U ) is equal to Bim ( ρ ∗ ( I ) ⊥ ) . Proof.
By Theorem 2.14 there exists a closed ideal J of A ( G ) such that G ( U ) = Bim ( J ⊥ )and Sat ( J ) = [ N ( u ◦ θ ) h : u ∈ I, h ∈ T ( G )] −k·k t . Clearly
Sat ( J ) ⊆ Sat ( ρ ∗ ( I )) ⇒ Bim ( ρ ∗ ( I ) ⊥ ) ⊆ Bim ( J ⊥ ) = G ( U ) . We need to prove G ( U ) ⊆ Bim ( ρ ∗ ( I ) ⊥ ) . It suffices to prove
N UN ∗ ⊆ Bim ( ρ ∗ ( I ) ⊥ ) . Let X ∈ U , V , V ∈ N , u ∈ I. Assume that N ( u ) = P i φ i ⊗ ψ i . For all f, g ∈ L ( G ) wehave ( V XV ∗ , N ( u ◦ θ )( f ⊗ g )) t = X i ( V XV ∗ , ( φ i ◦ θ ) f ⊗ ( ψ i ◦ θ ) g ) t = X i ( X, V ∗ (( φ i ◦ θ ) f ) ⊗ V ∗ ( ψ i ◦ θ ) g )) t . We have V ∗ (( φ i ◦ θ ) f ) = V ∗ M φ i ◦ θ ( f ) = M φ i V ∗ ( f ) . Similarly, V ∗ (( ψ i ◦ θ ) g ) = M ψ i V ∗ ( g ) . Therefore( V XV ∗ , N ( u ◦ θ )( f ⊗ g )) t = X i ( X, M φ i V ∗ ( f ) ⊗ M ψ i V ∗ ( g )) t = ( X, N ( u )( V ∗ ( f ) ⊗ V ∗ ( g ))) t = 0 , because X ∈ Bim ( I ⊥ ) and u ∈ I. Therefore( V XV ∗ , N ( u ◦ θ )( h )) t = 0 ⇒ ( V XV ∗ , N ( ρ ( u ))( h )) t = 0 , ∀ h ∈ T ( G ) , u ∈ I By Lemma 3.3, we have( V XV ∗ , N ( v )( h )) t = 0 , ∀ h ∈ T ( G ) , v ∈ ρ ∗ ( I ) . Thus V XV ∗ ∈ Bim ( ρ ∗ ( I ) ⊥ ) , which implies N UN ∗ ⊆ Bim ( ρ ∗ ( I ) ⊥ ) . (cid:3) Theorem 3.5.
Let I be a closed ideal of A ( H ) . If ρ ∗ ( I ) is an ideal of multiplicity then I is also an ideal of multiplicity. The proof of the above theorem is consequence of Theorems 2.15 and 3.4.In the last part of this section we will prove that if θ ∗ ( µ ) is a Haar measure for θ ( G )and A ( G ) contains a (possibly unbounded) identity and E ⊆ H is an ultra strong Ditkinset, then θ − ( E ) is also an ultra strong Ditkin set. Definition 3.1.
Let E ⊆ H be a closed set. We call E an ultra strong Ditkin set if thereexists a bounded net ( u λ ) ⊆ J H ( E ) such that u λ u → u, for every u ∈ I H ( E ) . Lemma 3.6.
Let E ⊆ H be a closed set. Then ρ ( J H ( E )) ⊆ J G ( θ − ( E )) . N SYNTHETIC AND TRANSFERENCE PROPERTIES OF GROUP HOMOMORPHISMS 13
Proof. If u ∈ J H ( E ), there is an open set Ω ⊆ H such that E ⊆ Ω and u | Ω = 0 . Weconsider the open set θ − (Ω) . Since u ◦ θ | θ − (Ω) = 0 and θ − ( E ) ⊆ θ − (Ω) , we concludethat ρ ( u ) = u ◦ θ ∈ J G ( θ − ( E )) . (cid:3) Lemma 3.7.
Let E ⊆ H be a closed set and suppose ( u λ ) ⊆ J H ( E ) is a bounded net suchthat u λ u → u for every u ∈ I H ( E ) . Then ρ ( u λ ) v → v for every v ∈ ρ ∗ ( I H ( E )) . Proof.
Define the space I = { v ∈ ρ ∗ ( I H ( E )) : v = lim λ ρ ( u λ ) v } . If u ∈ I H ( E ) we have ρ ( u λ ) ρ ( u ) → ρ ( u ) . Then ρ ( I H ( E )) ⊆ I. If v ∈ I, v ∈ A ( G ) , we have ρ ( u λ ) v → v ⇒ ρ ( u λ ) v v → v v , thus v v ∈ I. Therefore I is an ideal. Since ( ρ ( u λ )) is a bounded net we can easily seethat if ( v i ) ⊆ I is a sequence such that v i → v , then lim λ ρ ( u λ ) v = v. Thus I is a closedideal, which implies I = ρ ∗ ( I H ( E )) . The proof is complete. (cid:3)
Theorem 3.8.
Let E ⊆ H be a closed set and assume that A ( G ) has a (possibly unbounded)approximate identity. If E is an ultra strong Ditkin set, then θ − ( E ) is an ultra strongDitkin set.Proof. Theorem 5.3 of [1] implies that M min ( θ − ( E ) ∗ ) = Bim ( I G ( θ − ( E )) ⊥ ) . If E is an ultra strong Ditkin set, then by Corollary 2.5 the set θ − ( E ) ∗ is operatorsynthetic. Thus M max ( θ − ( E ) ∗ ) = M min ( θ − ( E ) ∗ ) = Bim ( I G ( θ − ( E )) ⊥ ) . From Theorem 3.4, we have M max ( θ − ( E ) ∗ ) = Bim ( ρ ∗ ( I H ( E ) ⊥ ) . Lemma 4.5 in [1] implies that(3.1) I G ( θ − ( E )) = ρ ∗ ( I H ( E )) . Now Lemmas 3.6 and 3.7 together with (3.1) imply that θ − ( E ) is also an ultra strongDitkin set. (cid:3) Acknowledgement.
We thank the anonymous referee for useful suggestions that led tothe improvement of the presentation.
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G. K. Eleftherakis, University of Patras, Faculty of Sciences, Department of Mathematics,265 00 Patra Greece
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