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B ULLETIN DE LA S. M. F.

M. T AKESAKI

A generalized commutation relation for the regular representation

Bulletin de la S. M. F., tome 97 (1969), p. 289-297

<http://www.numdam.org/item?id=BSMF_1969__97__289_0>

© Bulletin de la S. M. F., 1969, tous droits réservés.

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97, 1969, p. 289 a 297.

A GENERALIZED COMMUTATION RELATION FOR THE REGULAR REPRESENTATION

MASAMICHI TAKESAKI (*).

Introduction. — To a locally compact group G with a left invariant Haar measure ds, there correspond three von Neumann algebras on the Hilbert space L^(G) of all square integrable functions on G with respect to ds. Namely, the first one is the algebra A(G) of all multiplication operators 7i(f) defined by functions f of L°°(G); the second one is the von Neumann algebra M(G) generated by the left regular represen- tation U of G; the last one is the von Neumann algebra M'(G) gene- rated by the right regular representation V of G. These three algebras satisfy the following commutation relation :

A ( G / = A ( G ) ; M(GY==Mf(G).

Usually one's attention has been concentrated mainly on M(G) or M' (G).

But, as the author pointed out in the previous paper [17], it is very important, for the study of the group, to attack the relation of these three algebras. Indeed, a great deal of work has been done consciously or unconsciously on this subject. We can pursue its history back to the study of the Heisenberg commutation relation :

[p,q]=pq—qp=—ii.

Among various results, von Neumann's unicity theorem for Schrodinger operators (see [8], [9], [13] and [14]) and Mackey's theory of induced representations (see [6], [10], [11] and [12]) are remarkable.

In this paper, we shall show a firm interdependence of these three alge- bras by proving the following generalized commutation relation

M(G/H, U(K))'=M(K\G, V(H))

(*) The author was supported in part by NSF GP 7683.

BULL. SOC. MATH. — T. 97, PASC. 3. 19

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290 M. TAKESAKI.

for two closed subgroups H and K of a separable locally compact group G, where M{G/H, U(K)) [resp. M(K\G, V(H))] denotes the von Neumann algebra generated by the U(s), seK [resp. V(s), seK]

and the 7r(f), f^L-(G/H) [resp. 7r(f), /•eL^XVG)]. Our method of arguments requires that we restrict ourselves to separable groups.

(See the remark at the end of § 2.)

Before going into discussion, the author would like to express here his hearty thanks to Professor J. M. G. FELL who read the manuscript of this article and made several illuminating comments. Also, the author is indebted to Professor R. KADISON for his kind hospitality at the University of Pennsylvania.

1. Imprimitivity algebras.

Let r be a locally compact space and G be a locally compact trans- formation group on r. The action of G on r is denoted by ys, or sy, y € r , se G. Naturally, we call G a right or left transformation group on r according to the side of the action and write the pair as (r, G) or (G, r). In this section, suppose G is a right transformation group.

Of course, every result of this section can be translated into a result on left transformation groups. Let ds denote a left invariant Haar measure of G. Suppose ^ is a fixed quasi-invariant Radon measure on r. For the sake of simplicity, suppose there exists a continuous positive function 7 (y, s) on r x G such that

(0 ff^s) U^ s) ^(v) = ff^) d^(y), 5€ G,

</p jy

for every continuous function f on T with compact support

^(y, rs) == A(y, r) 7 (y/-, s), yeF, r, 5€ G;

(2) 7(y,e)=i.

Let JC denote the space of all continuous functions on r x G with compact support. Define a ^--algebra structure in JC as follows :

y * 9^ s) = f/-(Y, r)g^r, r-^dr;

(3) . JG

f (y, s) = ^ (s)~^(y, s^f^s, s-1),

where A^ means the modular function of G. Furthermore, we shall define

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f^^s)=^(srn^s)~^^s);

/•V (y, s) = A^ (s)~ 4 ^ (y, 5)"/-(y, s), f e JC.

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Then we have (/'A)* =: (y*)v. Now, define an inner product in X by (5) (/I 9) = (f AT, s) ^T) ^(y) ds.

^Tx <;

^rx<?

Then we can easily show that

(f\g)==(g' D;

(6) ( f ^ 9 \ K ) = (9\(f^-kh), f, g, AeJC.

The completion of the pre-Hilbert space <X is nothing but L2 (r x G, ^ 0 d^).

Then by DIXMIER'S result ([4], prop. 9), we get the following theorem.

THEOREM 1. — JC is a quasi-Hilbert algebra.

Therefore, the left von Neumann algebra ^11 (JC) and the right one ^(JC) of JC are the commutants of each other. We shall call them the left imprimitivity von Neumann algebra or the right imprimitivity von Neumann algebra of a topological transformation group (r, G) with the quasi-invariant measure |JL. For each f of JC, let u(f) and v(f) denote the operators defined by u (f) ^ == f * ^ and v(f)^ == ; * f, ^ e X , respectively.

To each yeL^r,^.), there corresponds a bounded operator ^(f) on L^rx G, ^(g)rfs) defined by the equality

W) 0 (T» s) = f^s)^:, s), E.eL^r x G, ^ 0 d5).

Also, there is a unitary representation V of G on L2 (r x G, ^ (g) ds) defined by

(vW (T. ^) = M^nCr, ^), 5e G.

Then a direct calculation shows that ^(f) and V(s) commutes with

u(g^ ^ € X , so that ^(/") and u(s') are in ^(JC). Then we have the following by [4], prop. 10 :

THEOREM 2. — ^(JC) is generated by the ^(f) and V(s), /•eL^r, ^) and se G.

2. Generalized commutation relation.

Suppose G is a separable locally compact group with a left invariant Haar measure ds. Take and fix two arbitrary closed subgroups H and K of G, whose left invariant Haar measures are denoted by d^r and d^t respectively. Let K^ and Tn denote the right homogeneous space K\ G and the left one GjH respectively. For each se G, let ^y(s) and y^(s) denote the right coset Ks and the left one Hs respectively. Then both y^r and Tn become locally compact spaces with respect to the

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292 M. TAKESAKI.

quotient topologies. G acts on both spaces }(T and Tn as a topological transformation group. The action of G on ^T is given by

^(s)t==^(st), s,t^G;

the action of G on r^y is also given by

t^H(s)=r^(ts), s,teG.

By [1] (Chap. VII, §2, Theoreme 2), there exist essentially unique quasi- invariant measure ^ on Tn and v on ^r respectively. By [1] (Chap. VII,

§2, Theoreme 2), there exists a continuous function ^(s, y)[resp. /^(y, s)]

of G X TH (resp. ^r X G) with the properties

f /^-^M^TWT)- f /•(T)^(T)-

^r.r ^r..

resp.

resp.

f f(^)^,s)dv(^)== f /-(TWT);

J^ ^r

?i^(sf, y) == ^(s, y) ^zj(^, s-1 y); ^u {e, y) = i,

^ (y, sf) = K^ (T> 5) K^ (ys, 0; TT^ (y, e) = i, s, ^ e G.

Since for each subset 5 of r^ (resp. j^T) to be a ^-null (resp. ^-null) set it is necessary and sufficient that y7^(S) [resp. A'Y"1^)] ls a ^-null set, every function f^L^^n, ^) [resp. f^L^^T, v)] can be regarded as an element of L°°(G,ds) which is constant on each coset Hs (resp. sK);

so the representation TT of L°°(G, ds) on L^G, ds), defined by 7r(f)^s)=f(s)^(s), /-eL^G.ds) and ^eL^G, d5), can be applied to the elements of L°° (TH, ^.) and of L°° (^r, v). Namely, 7:(0, /•€L°°(r^ ^) [resp. /•eL-^r, ^], is defined by

resp.

(^(00 (^-^^(^S^);

(^(f)0 (5) = /'(A-T(5))S(^ SeL^G, ^).

Remark that L^^TH, p) [resp. L^r, v)] is the fixed algebra of the auto- morphism group H (resp. K) of L°°(G, ds) which acts on L00^(G, ds) as right (resp. left) translation. Let (7 and V be the left and right regular representations of G respectively. Now, let M^GfH, U(K)} [resp.

M(K\G, V(H)} denote thevon Neumann algebra on -L^G, ds) generated by ^(ft./'eL0 0^,^) [resp. /•eL^r, ^)], and [7(s),s€K [resp.

V(s),seJf].

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THEOREM 3. — M(G/H, U(K)y==M(K\G, V(H)), equivalency M(K\G, V(ff)y=M(G/Jf, U(K)).

Taking particular subgroups H and K, we get various commu- tation relations concerning G. For example, if G = H = K, then our theorem turns out to be the commutation relation for the left and right regular representations which was shown by DIXMIER [4]. If H = G and K == { e }, then it tells us that the co variant representation (TT, V) of (I/^G, ds), G) is irreducible.

Proof. — Since it is clear that M(G/H, U(K)y-^M(K\G, V(H)), we shall prove only the reverse inclusion.

By the stage theorem on induced representations [10], the left regular representation U of G is induced by the left regular representation Un of H. Therefore, ^(G^ds) can be regarded as the Hilbert space of all L^(H, c?^r)-valued functions ^(s) on G with properties

(0 W = ^(r)-1^), reH, se G;

(2) ( IIS(5)|WT^))<oo,

^H

where integration (2) is justified by the fact that [| ^(s) \\ can be regarded as a function on Tji because of property (i). The representations U(s), se G, and rc(f), feL^^n, ^), are defined by the equalities

(3) (U(s)^ (f) = ^(s, tY^t), te G, ^eL^G, ds);

W ^m(t)=f(t)m where i means y^(0.

Take a bounded operator x on L^G) commuting with TT (L" (r/y)).

Then there exists a bounded 0 (L2 (Jf))-valued Borel function x(s) on G with the properties

(5) x(sr) == UH(r)-1 x(s) ^(r) for every reH and almost every s€ G and

(6) (^)(s)^x(s)^(s), ^L--(G).

For se G, we have

(U(s)x U(s)-^)(t) ==rc(s-10S(0. ^eL^G).

So if a; is in M(GIH, U(K)Y, then we have (7) x(tsr) = l^(r)-1 a;(s) [7^)

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294 M. TAKESAKI.

for every reH, t^K and almost every se G. Therefore, we can define a normal faithful representation a of M(GjH, U(K))' on the Hilbert space L^K^^L^H) by the direct integral

/T\

(8) o-(a-)== f :r(s)^(6-),

^r

where a;(s) is the 0 (L2 (^-valued function on ^ which is naturally defined by the fact that x(s) is constant on each right JiT-coset set Ks.

To get the conclusion, it is sufficient to show that a(M(G/H, U(K)Y)=a(M(K\G, V(H))).

First of all, let us determine O-(TT (/•)), /'eL^F), and o-(V(p)), peJf.

If x= 7T(/1), f^L^^T), then :K(s) is, for almost every se G, a multi- plication operator on L^(H) defined by the function: reH^->f(sr), so that we have

(9) (^)0(5, ^) --=f(sr)^s, r), EeL^-QJ x H^^dnr).

If x== V ( p ) , p ( E H , then ^(s) is constantly Vj/(p); so cr(.r) is simply given by

^

•i 'C/

(10) ^(x)Q(s, r) == A,,(p)2 £(5, rp), ^eL^AT X H),

where A^y means the modular function of H. Therefore, applying Theorem 2 to the topological transformation group (^r, H), the von Neumann algebra cr (M (K\ G, V (J?)), generated by cr (rr (/•)), f e L" (/, T), and cr(V(p)), pe-fiT, turns out to be the right imprimitivity von Neumann algebra of (/^r, H). Let JC denote the space of all continuous functions on /(T x H with compact support, in which a quasi-Hilbert algebra struc- ture is given as in paragraph 1. Then we have

a(M(K\G, V(H)))==V^C).

Therefore to complete the proof, it is sufficient to show that a(M(GIH, [7(X)y)c^(xy.

Regarding each element ^eL^i^TxH) as a L2 (IT)-valued function on /J, let ^ denote the value of ^ in U-(H) at s CAT. Then we have, for each jTe^C, ^eJC,

("(DO;. - f/'a p) ^(p)^..^p.

.717

/

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Take an arbitrary x of M(G{H, U(K))\ Then we have, for each

f<=x,'s,ex,

(a(a;)u(f)^.=;E(s)(u(/-)^

=x(s)j'f(s,p)UH(p)^/Hp

= ff^P)x(s)UH(p)^ d^p

J H I'

= f f(s, P) Ua(p)x(sp)^ dHp by (7),

JH I

= ff(s,p)UH(p)^W dnp

J H '

-("(/M^O-

Therefore, we have cr(x) u(f) == u(f) a(x), which implies that a(x) e 11 (cXy.

This completes the proof.

Remark. — Because of direct integral (8), we must assume the separa- bility for the given group G. However, it is very likely that Theorem 3 (and then Theorem 4) remains true without separability assumption.

Indeed if we can define the isomorphism cr of M{GfH, U (K))1 without making use of direct integral, then we can get rid of the separability assumption.

3. Connection with a generalized Heisenberg commutation relation.

In general, the Heisenberg commutation relation is presented in the form

(0 [P.q]=^pq—qp^—ii

for self-adjoint operators p, q on a Hilbert space. On taking expo- nentials, equality (i) becomes

(2) U(s) V(Q V(5)-1 V(0-1 = e-1811

for one-parameter unitary groups U(s) and V(t). In this form, we can get rid of the difficulties of unboundedness of p, q or of their domains.

Equality (2) is easily generalized to a commutation relation based on a locally compact abelian group, say G, as follows :

(3) U(s) V(t) U(sr V(tV = ^j^i,

where U(s) and V(?) are unitary representations of G and of its dual 6 on same Hilbert space and <(s, f)> means the value of the character f e Gr

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296 M. TAKESAKI.

at seG. We shall call the representations U and V covariant if U and V satisfy equality (3). As was shown by von Neumann in [14]

long ago, the covariant representations of G and 6 are essentially unique, and are given on 2^(6) by the equalities

(i) {(U(s)Q(x)=^x), s,xeG;

(OT)0(rr)=<^,?>S(^, xeG, ?eG.

The above covariant representations are irreducible, that is, they have no non-trivial closed invariant subspace in common. The von Neumann algebra M(G) [resp. M( G)], generated by U(s), se G [resp. V(t\ ?e G], is maximal abelian.

Now, take closed subgroups H of G and R of (?. Let M(Jf, K) denote the von Neumann algebra generated by U(s), s ^ H , and V(?),?e^. As K is the dual group of the quotient group G/J?1

of Q by the annihilator J?1 of R, the representation V(?) of & induces naturally the multiplication representation TT of the von Neumann algebra L^G/.K'1) of all bounded measurable functions in G/K1. Therefore, the von Neumann algebra M(Jf, &) turns out to be the M(G/K1, UH) of the previous section. Thus we get the following : THEOREM 4. — If G is a separable locally compact abelian group, then we have M(H, J?)' = M(J§'1, H1), for each closed subgroups H of G and K of G, where H1 is the annihilator of H in G.

Added in proof. — For the additive group of Hilbert space, ARAKI shown, in [21], the same commutation theorem for closed vector subspaces in the Fock representation of canonical commutation relation.

BIBLIOGRAPHY.

[1] BOURBAKI (N.). — Integration, chap. 7-8. — Paris, Hermann, 1968 (Act. sclent.

et ind., 1306; Bourbaki, 29).

[2] DIXMIER (J.). — Les algebres d'operateurs dans Vespace hilbertien. — Paris, Gauthier-Villars, 1957.

[3] DIXMIER (J.). — Les C*-algebres et leurs representations. — Paris, Gauthier-Villars, 1964 (Cahiers scientiflques, 29).

[4] DIXMIER (J.). — Algebres quasi-unitaires. Comment. Math. Heluet., t. 26, 1952, p. 275-322.

[5] DOPLICHER (S.), KASTLER (D.) and ROBINSON (D. W.). — Covariance algebras in field theory and statistical mechanics, Comm. Math. Phys., Berlin, t. 3, 1966, p. 1-28.

[6] FELL (J. M. G.). — An extension of Mackey's method to representations of algebraic bundles (to appear).

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[7] GLIMM (J.). — Families of induced representations. Pacific J. Math., t. 12, 1962, p. 885-91i.

[8] LOOMIS (L. H). — Note on a theorem of Mackey, Duke Math. J., t. 19, 1962, p. 64i-645.

[9] MACKEY (G. W.). — A theorem of Stone and von Neumann, Duke Math. J., t. 16, 1949, p. 3i3-326.

[10] MACKEY (G. W.). — Induced representations of locally compact groups, I, Annals of Math., Series 2, t. 55, 1962, p. 101-139.

[11] MACKEY (G. W.). — Unitary representations of group extensions, Acta Math., t. 99, 1968, p. 265-3n.

[12] MACKEY (G. W.). — Infinite-dimensional group representations, Bull. Amer.

math. Soc., t. 69, 1963, p. 628-686.

[13] NAKAMURA (M.) and UMEGAKI (H.). — Heisenberg's commutation relation and the Plancherel theorem, Proc. Japan Acad., t. 37, 1961, p. 239-242.

[14] VON NEUMANN (J.). — Die Eindeutigkeit der Schrodingerschen Operatoren, Math. Annalen, t. 104, 1931, p. 570-578.

[15] TAKESAKI (M.). — On some representations of C*-algebras, Tohoku math. J., t. 65, 1963, p. 79-95.

[16] TAKESAKI (M.). — Covariant representations of C*-algebras and their locally compact automorphism groups, Acta Math., t. 119, 1967, p. 273-3o3.

[17] TAKESAKI (M.). — A characterization of group algebra as a converse of Tanneka- Stinespring-Tatsuuma duality theorem (to appear).

[18] TAKESAKI (M.). — A liminal crossed product of a uniformly hyperfinite C*-algebra by a compact abelian automorphism group (to appear).

[19] TURUMARU (T.). — Crossed product of operator algebras, Tohoku math. J., t. 10, 1958, p. 355-365.

[20] ZELLER-MEIER (G.). — Produits croises d'une C*-algebre par un groupe d'auto- morphismes (to appear).

[21] ARAKI (H.). — A lattice of von Neumann algebras associated with the quantum theory of a free Bose field, J . math. Phys., t. 4, 1963, p. 1343-1362.

(Texte recu Ie 3i janvier 1969.) Masamichi TAKESAKI,

Department of Mathematics, University of Pennsylvania, Philadelphia, Pa i9io4-U.S.A.

(Etats-Unis).

Matematical Institute Tohoku University

Sendai, Japan.

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