COMPLETELY ISOMETRIC REPRESENTATIONS OF M cb A(G)AND UCB(ˆG)∗MATTHIAS NEUFANG1,ZHONG-JIN RUAN2,AND NICO SPRONK3
Abstract.Let G be a locally compact group.It is shown that there exists a natural completely isometric
reprentation of the completely bounded Fourier multiplier algebra M cb A(G),which is dual to the
reprentation of the measure algebra M(G),on B(L2(G)).The image algebras of M(G)and M cb A(G)
in CBσ(B(L2(G)))are intrinsically characterized,and some commutant theorems are proved.It is also
shown that for any amenable group G,there is a natural completely isometric reprentation of UCB(ˆG)∗
on B(L2(G)),which can be regarded as a duality result of Neufang’s completely isometric reprentation
theorem for LUC(G)∗.
1.introduction
In this paper we assume that G is a locally compact group with afixed left Haar measureµG.We will simply write dµG(t)=dt if there is no confusion.Ghahramani showed in[15,Theorem2]that if G contains at least two elements,the convolution algebra L1(G)(and thus the measure algebra M(G)) can not be isometrically isomorphic to a subalgebra of operators on any Hilbert space.Therefore,the reprentation of the measure algebra M(G)has to be considered on some other spaces different from Hilbert spaces.
Thefirst such reprentation result was studied by Wendel[46],in which he showed that M(G)is isometrically isomorphic to the left centralizer algebra LC(L1(G))of L1(G).More precily,Wendel showed that every measureµ∈M(G)uniquely corresponds to a bounded left centralizer
mµ:f∈L1(G)→µ∗f∈L1(G)
on L1(G).If we letΦµ=m∗µdenote the adjoint of mµ,thenΦµis a bounded weak∗continuous operator on L∞(G)commuting with left ,Φµ(l g f)=l gΦµ(f)).On the other hand,every such
(L∞(G))the operator on L∞(G)is implemented by a measure in this way.Therefore,if we denote by Bσ
l
space of all bounded weak∗continuous maps on L∞(G)commuting with left translations,then
(1.1)Φ:µ∈M(G)→Φµ∈Bσl(L∞(G))
Date:Oct26,2004,revid December22,2004.
1991Mathematics Subject Classification.Primary22D15,22D20,43A10,43A22,46L07,46L10and47L10.
1Thefirst author was partially supported by NSERC.
2The cond author was partially supported by the National Science Foundation Foundation DMS-0140067.
3The third author was partially supported by an NSERC PDF.
1
2MATTHIAS NEUF ANG1,ZHONG-JIN RUAN2,AND NICO SPRONK3
(L∞(G))(e[1,§1.6]).Let LUC(G)∗denote the dual is an isometric isomorphism from M(G)onto Bσ
l
space of all left uniformly continuous bounded functions on G.Then LUC(G)∗is a Banach algebra containing M(G)as a Banach subalgebra.It was shown by Curtis and Fig`a-Talamanca(cf.Theorem 3.3in[5])that there is a similar isometric isomorphism from LUC(G)∗onto the space of all bounded operators on L∞(G)which commute with the left convolution action of L1(G)on L∞(G).Note that the proof given in[5]assumes G to be unimodular.The general ca follows from a more general result due to Lau(e Theorem1,together with Lemma1and Remark3in[28]).
It is also known that M(G)and LUC(G)∗can be nicely reprented on the space B(L2(G))of all bounded linear operators on the Hilbert space L2(G).Størmer showed in[45]that for any abelian group
(B(L2(G))),the space of all normal G,there exists an isometric homomorphismΘl from M(G)into Bσ
R(G)
bounded R(G)-bimodule morphisms on B(L2(G)),which is given by
(1.2)Θl(µ)(a)= Gλ(s)aλ(s)∗dµ(s)
forµ∈M(G)and a∈B(L2(G)).This result was extended to general(not necessarily abelian)groups by Ghahramani[15]and was further studied by Neufang in his Ph.D.thesis[32].Neufang showed that eachΘl(µ)is actually completely bounded andΘl is an isometric homomorphism from M(G)into CBσR(G)(B(L2(G))),the space of all weak∗continuous)completely bounded R(G)-bimodule morphisms on B(L2(G)).Moreover,Neufang successfully characterized the range space of the reprenta-
network technology谢天谢地(B(L2(G)))of tion(1.2)in CBσR(G)(B(L2(G)))by showing thatΘl(M(G))is equal to the space CBσ,L∞(G)
R(G)
all normal completely bounded R(G)-bimodule morphisms on B(L2(G)),which map L∞(G)into L∞(G) (e[32]and[34]).Neufang also introduced and studied the reprentation of the Banach algebra LUC(G)∗on B(L2(G))in[32]and[33].
这个杀手不太冷主题曲The aim of this paper is to investigate the corresponding reprentations of the completely bounded Fourier multiplier algebra M cb A(G)and the Banach algebra UCB(ˆG)∗introduced by Granirer[19](e §6for details)since M cb A(G)and UCB(ˆG)∗when G is amenable can be regarded as the natural dual objects of M(G)and LUC(G)∗,respectively.Our main results show that there exist natural comple
queue是什么意思
tely isometric reprentations of M cb A(G)and UCB(ˆG)∗when G is amenable on B(L2(G)).The advantage of this investigation is that it allows us to compare and study the connection of the reprentations with the corresponding reprentations of M(G)and LUC(G)∗on the same space B(L2(G)).
Since operator space techniques will play an important rˆo le,wefirst recall some necessary definitions and notations on operator spaces in§2.Readers are referred to the recent books[11],[35],and[37] for more details.In§3,we recall the reprentation theorem of M(G)by considering the weak∗-weak∗continuous completely isometric homomorphismΘr:M(G)→CBσL(G)(B(L2(G)))induced by the right
REPRESENTATIONS OF M cb A(G)AND UCB(ˆG)∗3 regular reprentation
(1.3)Θr(µ)(a)= Gρ(s)aρ(s)∗dµ(s).
With this t-up,we may significantly simplify our calculations,and we will be able to obtain some intriguing commutant theorems in§5.We provide a proof,which is simpler than Neufang’s original argument,for the following equality
(B(L2(G)))
(1.4)Θr(M(G))=CBσ,L∞(G)
L(G)
in Theorem3.2.Moreover,we show in Proposition3.3and Proposition3.4thatΘr prerves the natural involutions and matrix orders on the two spaces.We also characterize the range spaceΘr(L1(G))in Theorem3.6.
We study the reprentation of M cb A(G)in§4.Using the techniques developed in Spronk’s Ph.D. thesis[42]and published in[43],we show that M cb A(G)can be completely isometrically identified with the space V∞inv(G,m)of all left invariant measurable Schur multipliers.It follows that we obtain a weak∗-weak∗continuous completely isometric isomorphism
(B(L2(G))),
(1.5)ˆΘ:M cb A(G)∼=CBσ,L(G)
L∞(G)
which prerves the natural involutions and the matrix orders on the two spaces(e Theorem4.3a
nd Theorem4.5).In particular,if G is an abelian group,we can write L(G)=L∞(ˆG)and L∞(G)=L(ˆG). In this ca,(1.5)can be expresd in the following duality form
Θˆr(M(ˆG))=CBσ,L∞(ˆG)
(B(L2(ˆG))).
L(ˆG)
In§5,we show some commutant results forΘr(M(G))andˆΘ(M cb A(G))in CBσ(B(L2(G)))and some double commutant results forΘr(M(G))andˆΘ(M cb A(G))in CB(B(L2(G))),respectively.Finally,we show in§6that for any amenable group G,there is a natural completely isometric homomorphism of B(L2(G),which can be regarded as a duality result of Neufang’s completely
UCB(ˆG)∗into CB L(G)
L∞(G)布拉德皮特 僵尸
isometric reprentation theorem for LUC(G)∗.
2.Operator spaces and completely bounded maps
In this paper,we let X and Y be operator spaces and let CB(X,Y)denote the space of all completely bounded maps from X into Y.Then there exists a canonical operator space matrix norm on CB(X,Y) given by the identification
(2.1)M n(CB(X,Y))=CB(X,M n(Y)).
4MATTHIAS NEUF ANG1,ZHONG-JIN RUAN2,AND NICO SPRONK3
With this operator space structure,CB(B(H))=CB(B(H),B(H))is a completely contractive Banach algebra since the composition multiplicationΦ◦Ψon CB(B(H))satisfies
[Φij◦Ψkl] cb≤ [Φij] cb [Ψkl] cb
for all[Φij]∈M m(CB(B(H)))and[Ψkl]∈M n(CB(B(H))).There is a canonical involution on CB(B(H)) given by
Φ∗(a)=Φ(a∗)∗.
This involution is an isometrically conjugate automorphism on CB(B(H))since
(Φ◦Ψ)∗=Φ∗◦Ψ∗.
Moreover,for each n∈N there exists a natural order on the matrix space
M n(CB(B(H)))=CB(B(H),M n(B(H))
given by the positive cone CB(CB(H),M n(B(H)))+of all completely positive maps from B(H)into M n(B(H)).This determines a matrix order on CB(B(H)).
If M is a von Neumann on a Hilbert space H,we let CB M(B(H))=CB M(B(H),B(H))denote the space of all completely bounded M-bimodule morphisms on B(H)and let CBσM(B(H))denote the space of all normal completely bounded M-bimodule morphisms in CB M(B(H)).Then CBσM(B(H))⊆CB M(B(H)) are completely contractive Banach subalgebras of CB(B(H))with a natural involution and a matrix order inherited from CB(B(H)).In general,CBσM(B(H))=CB M(B(H))(e Hofmeier and Wittstock [25]).But the two spaces are equal in some special cas(trivially when H isfinite dimensional or when M=B(H)).Moreover,if G is a discrete group,then ∞(G)= ∞(G) is afinite atomic von Neumann algebra standardly reprented on 2(G).We can conclude from[25,Lemma3.5]thatfe
(2.2)CBσ
∞(G)(B( 2(G)))=CB
∞(G)
(B( 2(G))).
Inspection of the proof shows that we do not need to assume 2(G)to be parable(which is an assumption made throughout in[25]for the Hilbert spaces occurring).Similarly,we have
(2.3)CBσL(G)(B(L2(G)))=CB L(G)(B(L2(G)))
for any compact group G since in this ca,L(G)= πM n(π)⊗I n(π)and L(G) = πI n(π)⊗M n(π)are finite atomic von Neumann algebras standardly reprented on L2(G)= ⊕S2n(π),where we let S2n(π) denote the Hilbert space of all n(π)×n(π)Hilbert-Schmidt matrices and let n(π)denote the dimension of irreducible reprentationsπ:G→M n(π)of G.Then we may obtain(2.3)by considering the central projections z n(π)=I n(π)⊗I n(π)∈L(G)∩L(G) from L2(G)onto S2n(π).Again,we do not have to assume the parability of L2(G).The result is true for arbitrary compact groups.Actually,the corresponding result holds for general discrete Kac algebras.
REPRESENTATIONS OF M cb A(G)AND UCB(ˆG)∗5
It is important to note that the mapping spaces CBσM(B(H))and CB M(B(H))can be completely identi
fied with the extended(or weak∗)Haagerup tensor product M ⊗eh M and the normal Haagerup tensor product M ⊗σh M of M ,the commutant of M in B(H),respectively.We assume that readers are familiar with the Haagerup tensor product X⊗h Y(for instance,e details in[11],[35],and[37]). The definition for the extended Haagerup tensor product X⊗eh Y can be found in[8]and[12].For the convenience of the reader,let us recall that the extended Haagerup tensor norm of an element[u ij]∈M n(X⊗eh Y)is defined by
[u ij] eh,n=inf{ [x ik] M
n,I
(X) [y kj] M I,n(Y)},
where the infimum is taken over all possible reprentations[u ij]=[x ik] [y kj]with[x ik]∈M n,I(X)and [y kj]∈M I,n(Y).The index I in the above definition could be an infinite t(or a countable t if X and Y are operator spaces on parable Hilbert spaces).In this ca,the notion[u ij]=[x ik] [y kj]means that
[u ij],f⊗g =
k∈I f(x ik)g(y kj)
for all f∈X∗and g∈Y∗.For dual operator spaces X∗and Y∗,X∗⊗eh Y∗can be completely isometrically identified with the weak∗Haagerup tensor product X∗⊗w∗h Y∗introduced by Blecher and Smith[2]via the following complete isometries
X∗⊗eh Y∗=(X⊗h Y)∗=X∗⊗w∗h Y∗.
The normal Haagerup tensor product
我爱你韩文X∗⊗σh Y∗=(X⊗eh Y)∗
for dual operator spaces wasfirst introduced by Effros and Kishimoto[7].It was shown by Effros and Ruan[12,§5]that the identity map on X∗⊗Y∗extends to a completely isometric inclusion
(2.4)X∗⊗eh Y∗ →X∗⊗σh Y∗
and the image space X∗⊗eh Y∗is completely contractively complemented in X∗⊗σh Y∗since the adjoint map(ιX⊗Y)∗ofιX⊗Y:X⊗h Y →X⊗eh Y induces a completely contractive projection from X∗⊗σh Y∗onto X∗⊗eh Y∗.
人工翻译收费Given u= k∈I x k⊗y k∈M ⊗eh M ,we can define a normal completely bounded M-bimodule morphism
thu
(2.5)T(u)(a)= k∈I x k a y k(converging in weak∗limit)
on B(H).It was shown by Haagerup[20](also e[9]and[40])that
T:u∈M ⊗eh M →T(u)∈CBσM(B(H))
6MATTHIAS NEUF ANG1,ZHONG-JIN RUAN2,AND NICO SPRONK3
determines a weak∗-weak∗continuous completely isometric isomorphism from M ⊗eh M onto CBσM(B(H)) with respect to the completely contractive Banach algebra structure on M ⊗eh M given by
(x⊗y)◦(˜x⊗˜y)→x˜x⊗˜y y.
Moreover,there is an isometric involution
(x⊗y)∗=y∗⊗x∗.
and a matrix order on M ⊗eh M given by the positive cones
M n(M ⊗eh M )+={[u ij]∈M n(M⊗eh M ):such that[u ij]=x∗ x for some x=[x kj]∈M I,n(M )}.
It is easy to e that T prerves the involution and the matrix order on the spaces.
We can similarly define a completely contractive Banach algebra,an isometric involution,and a matrix order on the normal Haagerup tensor product M ⊗σh M .It was shown by Effros and Kishimoto[7]that there is a natural extension of T to a weak∗-weak∗continuous completely isometric isomorphism˜T from M ⊗σh M onto CB M(B(H)).Moreover,˜T prerve the involution and the matrix order on M ⊗σh M and CB M(B(H)).Therefore,we can completely identify the spaces
(2.6)M ⊗eh M ∼=CBσM(B(H))and M ⊗σh M ∼=CB M(B(H))
via T and˜T,respectively.The complement of M ⊗eh M in M ⊗σh M exactly corresponds to the space CB s M(B(H))of all singular completely bounded M-bimodule morphisms on B(H).
Finally we note that there is a commutant theorem for CBσM(B(H))(respectively,for CB M(B(H)))in CB(B(H)).If V is a subspace of CB(B(H)),we let
V c={Ψ∈CB(B(H)):Ψ◦Φ=Φ◦Ψfor allΦ∈V}
denote the commutant of V in CB(B(H)).Then we have
(2.7)CBσM(B(H))c=CB M (B(H)),
典范英语1and if,in addition,M is standardly reprented on H,then
(2.8)CB M(B(H))c=CBσM (B(H)).
Combining(2.7)and(2.8),we obtain the following double commutant theorem
(2.9)CBσM(B(H))cc=CBσM(B(H))
when M is standardly reprented on H.(2.7)is due to Effros and Exel[6,§3];(2.8)was proved by Hofmeier and Wittstock[25,Proposition3.1and Remark4.3]in ca H is parable,and was extended to the non-parable situation by Magajna[31,§2].We remark that,trivially,we also have
CB M(B(H))cc=CB M(B(H)).
