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101372_6bsmf_1826.pdf BULLETIN DE LAS. M. F.MARCA.RIEFFEL
Commutationtheoremsandgeneralized
commutationrelations Bulletin de la S. M. F., tome 104 (1976), p. 205-224ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques http://www.numdam.org/Bull. Soc. math. France,
104, 1976
, p . 205-224
.
COMMUTATION
THEOREMS AND GENERALIZED COMMUTATION RELATIONS B Y MARC A. RIEFFEL [Berkeley ]RESUME.
- O n demontre un theoreme de commutation pour une structure qui generalis e le s algebres quasi hilbertiennes d e Dixmier de tell e facon qu'elle peut manier de s paire s d'algebre d e von Neumann d e tallie s differentes. O n s e ser t de c e theoreme de commuta- tion pour donner une demonstration directe d e l a relation d e commutation generalisee , d e Takesaki pour l a representation reguliere d'u n groupe (etendue au ca s non separable par NIELSEN), evitant de s reductions par representations induites e t integrale s directes .SUMMARY.
- A commutation theorem i s proved for a structure which generalize sDixmier's
quasi-Hilbert algebras in such a way that it can handle pairs o f von Neumann algebras o f different size . This commutation theorem i s then applie d to giv e a direc t proof o f Takesaki's generalize d commutation relation for the regular representations o f groups (a s extended to the non-separable cas e by NIELSEN), thus avoiding reductions by induced representations and direct integrals. Let G be a locall y compact group, and le t H and K be close d subgroups o f G. Let L 2 (G) be the usual Hilbert spac e o f square-integrable functions on G with respect to left Haar measure. Let M(K, G/H) be the vonNeumann
algebra o f operators on L 2 (G) generated by the left translations by elements o f K together with the pointwise multiplications by bounded continuous functions on G which are constant on left cosets o f H. Similarly, le t M(H, K\G) be the von Neumann algebra generated by the right translations by elements o f H together with the pointwise multiplications by bounded continuous functions which are constant on right cosets o f K.TAKESAKI
[19 ] showed that, when G i s separable, these two von Neumann algebras are eac h other's commutants. He calle d this theorem a generalized commutation relation, since , a s he showed, i t i s closel y related to theHeisenberg
commutation relations.Takesaki's
proof was restricted to the separable cas e because o f the use he made o f direct integral theory. NIELSEN [10] , by using the theory o f liftings o f measures, extended Takesaki's theorem to non-separable groups. In both papers, the strategy o f the proof consists o f using theBULLETIN
DE LA SOCIET E MATHEMATIQUE DE FRANCE206 M. A. RIEFFEL
theory o f induced representations to make a reduction to a situation in which Dixmier's commutation theorem for quasi-Hilbert algebras [5 ] can b e invoked. In the present paper, we give a proof o f Takesaki's theorem (i n the general case ) whose strategy consists o f first proving a commutation theorem for a structure which generalizes Dixmier's quasi-Hilbert algebras, and then showing that this commutation theorem i s directly applicabl e to Takesaki's situation., Our commutation theorem is directly applicable in part because, unlike Dixmier's, it is able to handle pairs ofvo n Neumann algebras which are o f different size , such a s those which arise in Takesaki's situation. For this reason, our theorem may also be useful in other contexts. Because our commutation theorem i s basically algebraic, and applies directly, our proof o f Takesaki's theorem avoids both induced representations and most o f the measure-theoretic difficulties in which the proofs o f Takesaki and Nielsen become embroiled. In the course o f Nielsen's proof, he obtained a generalization o f a theorem o f MACKEY [8 ] and BLATTNER [2 ] concerning intertwining operators for induced representations. We will show elsewhere [15 ] that this genera- lization has a natural interpretation and proof within the version o f the theory o f induced representations developed in [12] , and that , in fact, i t is a major part o f the statement that certain C*-algebras associated with the situation are strongly Morita equivalent [13] . These results are independent o f the present paper. Nevertheless, the structure we introduce here to generalize Dixmier's quasi-Hilbert algebras was motivated by the "imprimitivity bimodules" [12 ] used in discussing Morita equivalence, and some o f the formulas used here in applying our commutation theorem to Takesaki's situation were motivated by the formulas used in [15 ] to discuss Nielsen's generalization o f the Mackey-Blattner theorem. The reader i s referred, to [19 ] for a discussion o f the connection between the generalized commutation relations and the Heisenberg commutation relations, and to [10 ] for the connection with the Takesaki-Tatsuuma duality theory for locall y compact groups [20] . Finally, we remark that, as Takesaki points out at the end o f [19] , essentially the same generalized commutation relation has been obtained by ARAKI [1 ] for the case in which G i s a Hilbert space , and H and K are close d subspaces, al l acting on the Fock representation. It would be interesting to se e how to fit Araki' s theorem into the framework o f the present paper, but s o far this has eluded me. However, it was in studying this question that I did find a fairly simple proof [14 ] o f Araki's theorem whose lines are rather parallel to those o f the present paper. TOME 10 4 - 1976 - N ° 2
COMMUTATION THEOREMS 207
Most o f the research for this paper was carried out while I was on sabbatical leave visiting at the University o f Pennsylvania. I would like to thank the members o f the Mathematics Department there for their warm hospitality during my visit. Part o f this research was supported byNational
Scienc e Foundation grant GP-3079X2. 1 . Th e Commutatio n Theorem s I n this section, we introduce a structure which generalizes the quasi-Hilbert
algebras o f DIXMIER [5] , and we then prove the analogue for this structure o f Dixmier's commutation theorem for quasi-Hilbert algebras. Th e axioms for this structure are motivated by the needs o f the next section, and by formulas and structures appearing in [12] , [13 ] and [15] . All the algebras and vector spaces we consider will be defined over the complex numbers. I f C and D are algebras (not assumed to have identity elements), then by a C-jD-bimodule we mean a vector space , X, on which C and D ac t on the left and right respectively, with the action o f C commuting with that o f D, and with both actions being compatible with the action o f the complex numbers. I f C has an involution, then by a ^-representation o f C we mean a *-homomorphism o f C into the pre-C*-algebra o f those bounded operators on a pre-Hilbert space , V, which have adjoints defined on V. We will say tha t such a representation is non-degenerate i f CV i s dense in V (using module notation). Analogous definitions are mad e for a *-representation on the right, that is , an anti-*-representation. A repre- sentation i s sai d to be faithfu l i f its kernel consists only o f 0 . We now introduce our generalization o f Dixmier's quasi-Hilbert algebras [5] . Because we will not introduce until later the analogue o f axiom (v) o f Dixmier's definition o f a quasi-Hilbert algebra, we will use the prefix "semi " instead o f "quasi". 1.1 . Definition. - Let C and. D be algebras, eac h equipped with invo- lutions, which we denote by ff and b respectively. By a Hilbert semi-C-D-birigged space we mean a C-jD-bimodule, X, equipped with an ordinary inner product, [,] , and with C and D-valued sesquilinear forms, < , >c (conjugate linear in the second variable) and < , >p (conjugate linear in the first variable), such that : 1 ° The representations o f C and D on the left and right o f X are faithfu l -^-representations. 2 ° < x, y >c z = x < y, z > p fo r all x, y, z e X.BULLETIN
DE LA SOCIETE MATHEMATIQUE DE FRANCE208 M. A. RIEFFEL
3 ° < X, X)c i s a self-adjoint set , that is , for any x, y e X there exis t x^ Yi e X such that < ^ , y ^ = < ^i, Yi >c- 4 ° < ^ , X >^ ^ (the linear span) i s dense in X. We will say that X i s a Hilbert C-D-birigged space i f in addition: 5 ° < x, y Yc = < V. x >c and < x, y >^ = < y, x >^ for x , y e ^ . 6 ° For any x e X both < x, x >^ and < ;c , ^ >^ ac t as non-negative operators on X. We remark that axiom 4 implies that the action o f C i s non-degenerate. From axiom 2 i t then follows that the action o f D i s non-degenerate. We further remark that i f x, y e X, ce C and d e D, then cDefine
C and Z>-valued sesquilinear forms on X by (a,bYc=ab\6 - N° 2
COMMUTATION THEOREMS 209
for a, beX. Then it i s clea r that axioms 2 and 3 are satisfied. T o se e that axiom 4 is satisfied, we note that from the hypothesis that A 2 is dense in A it follows by von Neumann's double commutant theorem [5 ] that there i s a net o f elements o f A which, as operators on the completion o f A, converge in the strong operator topology to the identity operator on A. I t follows that A 3 (a s AA 2 ) i s dense in A. But this i s axiom 4 above. I f A happens to be a Hilbert algebra, then it i s easil y see n that axioms 5 and 6 above are also satisfied. In the verification jus t sketched, no use was made o f axiom (v) o f Dixmier's definition o f a quasi-Hilbert algebra. We wil l se e that this axiom corresponds to the Coupling Condition which i s used in the commutation theorem we will consider shortly. O f course, for us the important example o f a Filbert semi-C-D-birigged spac e i s that given in the next section in connection with the generalized commutation relations. I f X i s a Hilbert semi-C-D-birigged space , we will le t X denote its Hilbert spac e completion, and our notation wil l not distinguish between the elements o f C and D viewed as operators on X and their extensions by uniform continuity acting as operators on X. We wil l le t E ' denote the commutant o f any se t o f operators, E, on X. Thus C " ^ D' (and D" c c') . We wil l say that C and D generate eac h other's commutants on X i f in fact C" = D' (s o that D" = C). The commutation theorem give s a necessary and sufficient condition, which we cal l the Coupling Condition, for C and D to generate each other's commutants. This condition i s the appropriate analogue for the present situation o f axiom (v) in Dixmier's definition o f a quasi-Hilbert algebra, and our proof that the condition i s sufficient i s obtained by trying to imitate the proof o f Dixmier's commu- tation theorem for quasi-Hilbert algebras [5] . We will se e later that theCoupling
Condition i s automatically satisfied i f X is , in fact, a HilbertC-D-birigge
d space . This generalizes the commutation theorem forHilbert
algebras ([5] , [11]) . 1.3. The commutation theorem for Hilbert semi-blrigged spaces, - Let X be a Hilbert semi-C-D-birigged space . Then C and D generate eac h other's commutants on X i f and only i f the following condition is satisfied:Coupling
Condition: Ifm, n e X and x, y e X, and if [mBULLETIN
DE LA SOCIETE MATHEMATIQUE DE FRANCE 1 4210 M. A. RIEFFEL
then for any fixed z , w e X there is a net, [ c^ }, of elements of C such that andCfe z converges to m < x, z >j), cj l ^ converges to n^y, w )>^ .Proof.
- We first prove the sufficiency o f the Coupling Condition.Imitating
the proof for quasi-Hilbert algebras, we set: 1.4. Definition. - An element, m, o f X i s said to be D-bounded i f for every x e X the linear map y \-> m < ^ , y >j) from Z to J ^ i s bounded, and s o extends by uniform continuity to a bounded operator on X, which we will denote by Z ^ ^ . We will le t X^ denote the linear manifold o f D-bounded elements in X. 1.5. LEMMA. - Let me X^. Then Z ^ ^ is in D' for every x e X.Furthermore,
ifTeD' , then TmeX^, andL^Tm,x>=
TL^^ for every xeX.Finally,
X ^ X^, and L^y^ ^ = < y, x >c for all y, x e X. Thi s lemma i s verified by routine calculations. 1.6. LEMMA. - Let J == { Z ^ ^ ; m e X^, x e X ] , and let K = J n ./* , where * denotes the adjoint operation on operators on X. Note that L^ ^ e Kfor all x, y e X. Then K" = D' .Proof.
- I t i s axiom 3 which ensures that L^^ yy e K for al l x, y e X. Now le t T e D' and x, y, z, w e X. Then from Lemma 1. 5 we se e that Z* y TL^ ^ i s in J. But equally well its adjoint i s als o in J , s o that it is , in fact, in K. Thus i f we le t E be the linear span o f the range ° f < 3 )c (^ic h we saw earlier i s a 2-sided ideal in C) , then i t follows that the linear span o f K, and s o K" , contains e Tffor every e,fe E and T e D' . Now from axiom 4 we se e that EX is dense in X, s o that by von Neumann's double commutant theorem there i s a net o f elements o f E which converges in the strong operator topology to the identity operator on X. I f in the expression e Tf we le t first e, and later / , range over this net, we find that TeK\ Thus D' c K". But D' = > K" by Lemma 1.5. Thus D' = K". Q. E. D. Now le t m e X^ and x e X, and suppose that L^ ^ e K. Then fro m the definition o f K it follows that there exis t n e X^ and y e X such that TOME 10 4 - 1976 - N° 2
COMMUTATION THEOREMS 211
^m, x^ = L ^, x> » tha t i s [ m < .x , z >^ , w] = [z , n(y , w >^] for al l z , wef . But then the Coupling Condition i s applicable , and s o we can find, for any given z , w e X, a net { Cj,} having the convergence properties o f tha t condition. Suppose now that S e C\ Then for the given z and w, we haveCoupling
Condition has been shown. We now prove the necessity o f the Coupling Condition. Most likely a proof o f this which uses only bounded operators can be given, along the lines o f the necessity proofs for the commutation theorems in [14 ] and [16] . Partly because o f the connections with the developments in the next section, we prefer to giv e here a proof which i s shorter but which involves unbounded operators. It i s easily see n that a proof along th e lines presented here can also b e used in [14 ] and [16] .Assume
that C = D\ and le t m, n e X and x, y e X be such that \m < x, z >^, w~\ = [z , n < y, w >p] for al l z , w e X. Define (possibly unbounded) operators L^^y and L^^yy on X, with domain X, by L ^m, x > z = m < x ? z YD ^ or zeX, and similarly for Z^ yy . Then the above relation says that each o f these operators i s contained in the adjoint o f the other. It follows that both operators are closeable. We will denote their closures by L^^^ and L^ ^ . Now routine calculations show that the domains o f L^ ^ and L^ yy are invariant under the action o f D, and that these operators commute with the action o f D. Further routine calculations show that this i s also true with respect to the action o f the strong operator closure, D", o f D. In other words, these operators are affiliated [5 ] with D' . Now le t L^ ^ = PT be the polar decomposition o f L^ ^, where P i s a partial isometry, and Ti s a positive self-adjoint operator (see p. 1249 o f [6]) . Then it is not difficult to show (see lemma 4.4. 1 o f [9] ) that P e D' and that T is affiliated wUh D\ Let { E (r) } be the spectral resolution for T , s o tha t E(r) e D' for each r. Then PTE(r) e D' for each r.
BULLETIN
DE LA SOCIETE MATHEMATIQUE DE FRANCE212 M. A. RIEFFEL
Now let z , w e X be given. Sinc e C i s assumed to generate Z)' , it i s dense in D' in the strong-*-operator topology [17 ] by the Kaplansky density theorem [5] , and s o for eac h r > 0 we can find Cy e C such that ||c,z-Pr£(r)z||^l/ r an d ||c;w-E(r ) TP*w|| ^ 1/r . But PTE (r) z = PE (r) T z, which converges to PTz==m(x , z >^.Similarly
£ ' (r) TP * w converges to TP * w, which equalsL<^y>w=n<^,w>p
, since L^. y > i s contained in the adjoint o f L<^ ^ . Thus ^ z and c^ w converge to m < x, z >^ and ^ < >» , w >p a s desired. Q . E. D. The above theorem i s indeed a generalization o f the commutation theorem for quasi-Hilbert algebras [5] , sinc e we have: 1.7. COROLLARY. - If A is a quasi-Hilbert algebra, then its left and right regular representations generate each others commutant.Proof.
- I f we view A as a Hilbert semi-birigged space , a s in Example 1.2, then it i s easil y see n that axiom (v) in the definition o f a quasi-Hilbert algebra i s essentially the Coupling Condition. Q . E. D. We remark that the Coupling Condition can be reformulated in a more spatial form similar to the forms used in [14 ] and [16] , and about to be used in the next theorem. But the reformulation seems quite cumbersome, involving taking certain linear combinations o f the operators L^ ^ , and it does not seem to facilitate the application considered in the next section. We turn next to showing that i f X is , in fact, a Hilbert C-Z)-birigge d space , then the Coupling Condition i s automatically satisfied, s o that we obtain a generalization o f the commutation theorem for Hilbert algebras ([5] , [11]) . T o do this we first consider another coupling condi- tion which i s close r in form to the coupling condition used in [14] . This new coupling condition may conceivably als o be useful in other situations, but i t does not seem to hold in general in Hilbert semibirigged spaces in which the commutation property holds. I do not have an example to show this, but I suspec t one can be found among the Hilbert semibirigged spaces which will be considered in the next section. TOME 10 4 - 1976 - N° 2
COMMUTATION THEOREMS 213
1.8. SECOND COMMUTATION THEOREM. - Let X be a Hilbert semi-C-D-birigged
space "which also satisfies axiom 5 of Definition 1.1. Suppose that the following condition is satisfied:Second
Coupling Condition: If m, n e X and x, y e X, and if (1 ) [m, ex] =+[c^ , n] for all ceC, (2 ) [m, yd ] = -\xd\ n] for all deD , then m < x, z >^ = 0 = n < y, z >^ for all z e X. Ifm, n e X, and x, y e X, and if [m < x, z >^, w] =^[z, ^< y, w >jJ for all z, weX, then there is a sequence {d^ } of elements of D such that for all z e X: < y dk , x >c z converges to m < x, z >^, and (xd^.yYc Z converges to n<^,z>^ . In particular, the Coupling Condition of Theorem 1. 3 is satisfied, so that C and D generate each others commutants on X. We remark that the first conclusion o f this theorem says, more or less , that the (unbounded) operator ^<^,.x: > can be approximated in the strong-*-operator topology [17 ] from X (not X) by operators o f the form < y d, x y^ for d e D.Proof.
- The proof i s similar to the first part o f the proof o f the commu- tation theorem in [14] . Let X" denote X but with complex conjugate structure (p . 9 o f [5]) , and form the Hilbert spac e X © Z\ I f z e X, then z will denote z viewed as an element o f Z" . Let x and y be given a s in the statement o f the theorem, and le t G denote the se t o f pairs (m, n) in X © X~ such that [mBULLETIN
DE LA SOCIETE MATHEMATIQUE DE FRANCE214 M. A. RIEFFEL
But sinc e (mo, n^) e ^ there i s a sequence, (m^, ^) , o f elements o f G which converges to (mo, no). Then for any z , w e X a routine calculation shows that [mo ,Condition
to conclude that mo < x, z >j) = 0 = HQ < y , z >j) for al l zeX. Now suppose that (m, n) e G. Then we have (m, n) = (mo, no)+(^i » ^i ) where (mo, ^o) e Hl ^ d (m^ , ^i ) e H. In particular, there i s a sequence {^ 4 } in 2 ) such that (ydk , (x^T ) converges to (m^ , n^).Furthermore,
from the previous paragraph we have ^o < ^ z >D = 0 = ^o < ^ z >D for al l ^ e Z . It follows that m < x, z >D = m i < x, z YD = lim ^ ^ < x, z >^, ^< ^ ^>D = ^i< ^ ^>D=1™^4< ^ ^> ^ for al l z e X, which, upon using axiom 2 , gives the desired result. Q . E. D. 1.9. THEOREM. - Let X be a Hilbert D-C-birigged space. Then theSecond
Coupling Condition is automatically satisfied, so that C and D generate each others commutants.Proof.
- Let m, n e X and x, y e X, and suppose that they satisfy the two equations o f the Second Coupling Condition. In those equations se t c = < y, z >c and d = < z , x >j), where z i s an arbitrary element o f X. Then we obtain [m , 0 , z>cx ] = [6 - N ° 2
COMMUTATION THEOREMS 215
for al l z e X, Using axiom 2 and rearranging, we fin d that [z(y,yYD , ^]=[m,^Coupling
Condition i s satisfied. Q . E. D. That the question o f when two algebras o f operators generate each other's commutants can be fairl y delicate may be see n by considering the "factorizations which are not coupled factors" found by MURREY and von NEUMANN (sections 3. 1 and 13.4 o f [9]) . This question i s closely related to the subject o f normalcy in von Neumann algebras (see references in [21]) . 2. Generalize d Commutatio n Relation s In this section we show how to apply the commutation theorem o f the last section to obtain Takesaki's generalized commutation relations.BULLETIN
DE LA SOCIETE MATHEMATIQUE DE FRANCE216 M. A. RIEFFEL
Some o f the formulas we introduce are suggested by the developments in [12 ] and [15] . Let G be a locall y compact group, and le t H and K be close d subgroups o f G. We equip G , H and K with left Haar measures, whose modular functions we denote by A, 8 , and 5^ , respectively. Let C ^ (G) denote the spac e o f continuous complex-valued functions o f compact support on G . Let G/H and K\G denote the left and right homogeneous space s with respect to H and K respectively. Our notation will not distinguish between the points o f G and the corresponding cosets, because our notation wil l not distinguish between functions on G/H or K\G and the corresponding functions on G which are constant on cosets . Now K acts as a transfor- mation group by left translation on G/H, while H act s a s a transformation group by right translation on K\G. I f we view C ^ (G/H) and C ^ (K\G) a s pre-C*-algebras under pointwise multiplication, then the above actions define an action o f K a s a group o f automorphisms o f C ^ (G/H) and an action o f H as a group o f automorphisms o f C ^ (K\G). We can then form the corresponding transformation group algebras as in [7 ] and [18] . Specifically , le t C = C^KxG/H) with product defined a s in 3. 3 o f [7 ] byII*2:Q?,x)
= ^(q,x)^(q~ l p,q- l x)dqJK for II , S e C , p e K, XG G/H, and define an involution on C as in 3. 5 o f [7 ] by n^^nQr 1 ,?- 1 ^^- 1 ).Similarly
, le t D = Cc (Hx K\G) with product defined byQ*X(5,x)
= Q^x^Cr^.xO AJH for Q , % e Z) , s e H, xe K\G, and with involution defined by0*(s,x)==Q(s-
l ,;c5)§(s- l ) . Le t X denote Cc (G) equipped with its usual inner-product, [/^]= fJcf(x)g(x)dx for/, g e X. Left translation on G by elements o f K give s a unitary repre- sentation o f K on th e pre-Hilbert spac e X, while pointwise multiplication TOME 10 4 - 1076 - N° 2
COMMUTATION THEOREMS 217
on X by elements o f C^ (G/H) gives a representation o f C ^ (G/H) on A" . Thes e two representations together giv e a covarianf representation [18 ] o f the pair (K, C^ (G/H)), and s o define a -^-representation o f C on X, defined (following 3.20 o f [7] , or [18] ) by (II*/)(x) = f Tl(p,x)f(p- l x)dp.JK Let ^ denote the Hilbert spac e completion o f X. Then routine arguments show that the von Neumann algebra generated on X by the representation o f C i s the same a s the von Neumann algebra M (K, G/H) defined in the introduction o f this paper. Similarly we define an action o f D on the right o f X by f /*Q(x) = /(xr^AO-^QO.xr 1 )^ .JH We do not expec t this action to preserve the involutions in general, since the action o f H by right translation on X need not be unitary i f G i s not unimodular. But routine calculations show that this action i s by bounded operators (sinc e Q has compact support), and that it commutes with the left action o f C on X. Furthermore, i f the adjoint o f the action o f an element fi. ofDis calculated, it i s easily seen to be given by the action o f the element ^ o f D defined by ^(t,x)=^(t)^(t,x). Thus, i f we define a new involution, b , on D by this formula, we do obtain a *-antirepresentation o f Z) . As with C , routine arguments show that the von Neumann algebra generated on X by D i s the same as the von Neumann algebra M (H, K\G) defined in the introduction o f this paper. Finally, it i s easily see n that the representations o f C and D on X are faithful. Thus axiom 1 in the definition o f a Hilbert semi- birigged spac e i s satisfied. In 7. 8 o f [12 ] a sesquilinear form on X with values in C^(GxG/H ) was defined. I f the values o f this form are restricted to Kx G/H^e obtain a sesquilinear form on X with values in C ^ (Kx G/H), defined byBULLETIN
DE LA SOCTETE MATHEMATIQUE DE FRANCE218 M. A. RIEFFEL
tional expectation, as discussed in [15]. ) I f axiom 2 in the definition o f a Hilbert semi-birigged spac e i s to be satisfied, this determines what the Z)-valued sesquilinear form must be. A straight forward calculation shows that we must se tPQO^p^/pQ^x).
A simple calculation using this shows that P (P) 2 < g , />c (P , x) = < p g , flp > c (p, x). Thus axiom 3 holds. We remark that <,> c could have been made sym- metric by introducing the factor P into its definition, as is done in [15] , bu t this would complicate later calculations, and besides, it seem s somewhat interesting that such symmetry is not o f importance for our present purposes. We consider next axiom 4 . Now in [12 ] it was shown that the linear span o f the range o f the inner-product having values in C^ (G x G/H) i s dense in C^ (G x G/H) for the inductive limit topology. The proof consists o f lemma 7.1 0 and proposition 7.1 1 o f [12 ] together with the easily seen fact that this range i s an ideal in C^ (G x G/H), but the proof uses n o other parts o f [12] . Now <,> ^ i s obtained by restricting to Kx G/H the elements o f the range o f this inner-product from [12] . It follows immediately that the linear span o f the range o f <,> c i s dense in C in the inductive limit topology. From this it follows that this linear span contains an approximate identity for the inductive limit topology o f approximately the analogue o f that described in lemma 7.10 o f [12] . TOME 10 4 - 1976 - N° 2
COMMUTATION THEOREMS 219
From this it i s easily see n that < X, X >c X i s dense in X in the inductive limit topology, and s o in norm. Thus axiom 4 holds. We remark that a somewhat similar argument i s given in lemma 2. 5 o f [15] . We have thus verified: 2.1 . PROPOSITION . - IfG is a locally compact group, H and K are closed subgroups, and if X, C , D and their actions are defined as above, then X is a Hilbert semi-C-D-birigged space. I t i s natura l to ask at this point under what conditions X will in fact be a Hilbert C-D-birigged space . From the calculation made above in the verification o f axiom 3 it i s clear that in order for axiom 5 to hold we must have P = 1 , which i s jus t the condition under which there exists an invariant measure on K\G. A small calculation shows that similarly we need A (s)/S (s) = 1 for s e H s o that there i s an invariant measure on G/H, and that we need A (s) •= 1 for s e H, s o that the action o f H on X i s unitary. (Thes e last two requirements imply that H must be unimodular.) Under these conditions axiom 5 will hold. Axiom 6 will then hold also , as can be shown by imitating the proof o f theorem 4. 4 (due to BLATTNER) o f [12] . Thus under these conditions we can already conclude fromTheorem
1. 8 that C and D generate each other's commutants. To show that C and D generate eac h other's commutants i n general, which i s Takesaki's generalized commutation relation, i t suffices to show that the Coupling Condition o f Theorem 1. 3 holds. Thus suppose that m, n e X = L 2 (G) , tha t f,g e X, and that these satisfy [m^fe]=[^nBULLETIN
DE LA SOCIETE MATHEMATIQUE DE FRANCE220 M. A. RIEFFEL
i s integrable, and s o we can apply Fubini's theorem to conclude that xi->fe(x) Il(p,x)h{p~ l x)dpJK i s integrable. Sinc e k i s arbitrary, it follows that the function I I * h defined by (n*/i)(jc) = ^(p,x)h(p~ l x)d pJK i s determined except on a locall y null set , and i s locally integrable.Actually,
a simple calculation shows that the support o f I I * h i s o-compact. Thus it makes sense to ask whether I I * h i s in L 2 (G). For I I o f the kind we are considering, define II s by n\p, x) = ii(p- 1 , p- 1 ^^- 1 ). Then i t i s readily see n that II s is again a functio n on KxG/ H which i s locally integrable a s a function on K xG. Finally, i f we le t | I I denote the absolute value o f II , it i s readily see n that 1 1 has the same properties, and that II * = 11 1 [ ft . 2.2 . Notation. - Let Q denote the linear space o f al l those functions, II , on Kx G/H o f o-compact support, which are locally integrable on Kx G, and which have the property that for any h e Cc (G) both I I * h and | n I s * h are in L 2 (G). I t i s clea r that i f I I e 0 , then for any h e C ^ (G) both n * h and II s * h will be in L 2 (G). I f also k e C ^ (G\ then (p,x)->II(p,x)h(p~ l x)k(x) i s integrable. By a small calculation involving Fubini's theorem, we conclude that [n*A,fe]=[A,n**fe] . Thus 1 1 and II * define (probably unbounded) operators on L 2 (G) , eac h o f which is contained in the adjoint o f the other. I t is no c difficult to show tha t the closures o f these operators are affiliated with D' , although we will not need this fact. But this situation should be compared with the part o f the proof o f Theorem 1. 3 in which the necessity o f the CouplingCondition
was shown. We remark finally that i f two functions in Q define the same operator on L 2 (G), then they differ only on a locally TOME 10 4 - 1976 - N° 2
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null set. In particular, i f 2 e Q and i f the operator defined by 2 i s contained in the adjoint o f II , then 2 agrees with II * excep t on a locall y null set . Now initially m ^ i s defined only by extending the operator p by continuity. We need to show that actually i t i s defined by "convolving" with p. For this purpose, we can use any element, 0 , o f D instead o f ^. I f A : e C ^ (G) , then the function (^ , x)\->k(xt)m(x)^l(t, x) i s easily see n to be integrable, s o that Fubini's theorem can be applie d to conclude, after making a translation, that x^k(x)\ mOcr^Acr^QO.xr 1 ) ^JH i s integrable for al l k. Thus the function m * Q defined by (m*0)(x) = mOcr^AO-^Q^xr 1 ) ^JH i s locall y integrable. I t also cleaily has a-compact support. Furthermore, another calculation using Fubinfs theorem shows that for any k e Cc (G) : (m-k^)(x)k(x)dx == [m, A^Q^ = [mQ, k],JG where w Q i s defined by extending the operator corresponding to 0 by continuity. I t follows both that m * Q i s in L 2 (G) , and that i t equals m 0 . We find next the kernel o f the operator h ^ m < / , h >^. I f F e C^ (K x G') , then it i s easily verified that th e function on GxKxH defined by (x, p, t)^F(p , xt- i )m(x)(Af)(p- l x)A(^ l ) i s integrable. Applying Fubini's theorem and translating as before, we find that the function < w,/> ^ defined on KxG/ H by (2.3)BULLETIN
DE LA SOCIETE MATHEMATIQUE DE FRANCE222 M. A. RIEFFEL
then a routine calculation using Fubini's theorem and the results o f the previous paragraph shows that "m,fyc^h)(x)k(x)dx = [m* ^ * h is in L 2 (G) and agrees with m *^ a . e . Sinc e it i s easil y see n that i, ^ >^ i s defined in the same way, and the Coupling Condition relation implies that < n, g >^ agree a . e . with < w,/>;L Since , as above, | < n, g > | * h i s in L 2 (G) for any h in C ^ (G) , it follows that [ < w,/>^ , | * h i s in L 2 (G) for any h e C^ (G) . We have thus obtained: 2.4 . LEMMA. - If m,neX and f, g e X, and satisfy the relation of the Coupling Condition, then the functions < w,/> ^ and c defined as in 2. 3 are f/2 g , a/^ each other's adjoints, and are the kernel functions for the operators h\->m(f,h)D and fe-^i
Il^-kk
converges to II**^.Proof.
- First fin d an increasing sequence, { E^ } , o f compact subsets o f KxG/ H whose unio n contains the support o f II . Define I., to have value n (p , x) i f (p , x) e E, and | n (p , x) \ ^ f , and value 0 otherwise. Thus [ S , | ^ [ I ! | , and S , converges to n a . e . on Kx G. It i s easily seen that, in addition, [ Z? [ = $ (II * [ and that £? converges to II s a . e . Then several applications o f the Lebesgue dominated convergence theorem show tha t£f*/
i converges to H-kh TOME 10 4 - 1976 - N ° 2
COMMUTATION THEOREMS 223
and£f*f
t converges to II s *f e in L 2 (G). Thus it suffices to prove the Lemma when I I i s bounded and o f compact support in Kx G/H. In this cas e we can find 2 e C such that | I I ] ^ 2 , and then we can find a sequence, { Hy } , o f elements o f C such that H, converges to I I a . e . and | Hj | ^ E for al l j. The Tl^ will have similar properties. Once again several applications o f the Lebesgue dominated convergence theorem show that and a s desired.Tlj-kh converges to n*^,I^j-kk
converges to II s * k, Q . E. D. This concludes the proof o f the following theorem. 2.6 . THEOREM (Takesakrs generalized commutation relations). - With notation as above, C and D generate each others commutants on X.REFERENCES
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. (Text e recu I e 1 5 mat 1975.
) Marc A. RIEFFEL,
Department
o f Mathematics,University
o f CaliforniaBerkeley,
Cal . 94720 (Etats-Unis). TOME 10 4 - 197
6 - N ° 2