q(x) = JGX(8)(5)d5 g~(x E s?()

Transcription

1 No. 9.] On Some Representation Theorems in an Operator Algebra. II. By Hisaharu UMEGAKZ. Mathematical Institute, Kyushu University, Fukuoka. (Comm. by K. KUNUGI, M.J.A., Nov. 12, 1951.) 3. Application to a topological group. We shall first prove Bochner's Theorem for a separable locally compact group by applying the Theorem 1 and its Remark 1, next prove the decomposition into irreducible factors for arbitray two sided unitary representation of a separable unimodular locally compact group, which has been called double unitary representation by R. Godement [8]. Theorem 3. Let G be a separable locally compact group, and p(s) be a continuous positive define function on G. Then (11) ~p(s) = Rx(s' A)dcr(A) where o(a) is a suitable weight function which is an N -function of von Neumann [1], and x(s, A) are elementary continuous positive definite functions for almost all A in R. When p(s) is a central continuous p.d. (positive definite) function, these x(s, A) are also central elementary continuous p.d. functions. Proof. Let i be L'-algebra on G, then t is a complete normed*- algebra with an approximate identity. We put q(x) = JGX(8)(5)d5 g~(x E s?() the integration being by Haar measure ds on G, then clearly co(x) is a state on 2f. Therefore, by the Theorem 1 and its Remark 1 there exists a system of pure states x(x, A), 1, E Nc{A)-null set, such that p(x) = Jx(x, A)do (A). R By Riesz-Markoff-Kakutani's Theorem, there are elementary continuous p.d. functions x(s, A), A E N, such that x(x, A) =.GO( xs, A)ds. For these x(s, A), we shall prove the relation (11). Let { V} be an enumerable neighbourhoods system of the unite of G. For any t E G lim x(s, A)Ctp,3(s)ds/I V,, t = x(t, A) a.e. o(a) n~oo

2 502 H. UMEGAKI. [Vol. 27, r x(s, where C1(s) be a caracteristic function of the set tvt and ` V?, be the volume for the Haar measure of G. Hence x(t, A) are c(a)- measurable for all t E G. Let K be a subset of G and a, b E R (real number), we denote Ma,,,, k being a set of all A E R--N such that a < x(s, d) < b for all s in K. Two topologies Ti and T~ on R are defined by the families of the subsets in R{Ma.b,s I a, b E R, s E G} and {Ma,,,, k I a, b E R, K being Tuning over on the family of all compact set in G}, respectively. Since G is separable, every Borel set in RT2 is also a Borel set in Rti1, and the o(a)-measureable. Moreover x(s, A) is continuous on the product topological space G x RT, hence A) is measurable for the product measure of G and R. By Fubini's Theorem, for all x E!t ~o(x) = J x(s)x(s, A)ds dr(a) (12) '? `~ Jx(s)x(s, ~)d) ds. Therefore we obtain the relation (11). On the case of central continuous p.d. function, we may prove in another paper with the decomposition of trace in C*-algebra. Remark 2. As far as we know, the Bochner's Theorem for non-separable locally compact group has never been shown. R. Godement has given at a weak form (cf. [7]). Let G be a such group, P be a set of all elementary continuous p. d. functions and their weak limits. For any continuous p.d. function cp(s), there exists a positive Radom measure µ(.) such that Jx(s)co(s)ds/p(s)'~~ = rg x(s)x(s)dsjp(s)'~~ d(a)ds for all x E L1. The weak topology and compact open topology are coincide in I' (cf. H. Yoshizawa [11]). Then, we have p(s) = x(s)di(x). r But, on this case it is essential weak. For, as Godement has seen that P contains a p.d. function different from the elementary p.d. function. In his central group (cf. [10]), however, it may be held. The uniform closure R(G) of the collection of the operators of the form Lr (with f E L'(G)) is a C*-algebra (L,.g = f *g, g E L2(G) and its * being the convolution). We do not know that the complete relation between of the representations of R(G) and G. But we know in L'-algebra that there is a one-to-one correspondence between a continuous representation of L'(G)(= 5t say) and a continuous unitary representation of G, that is, { Ux, S~} be a continuous re-

3 No. 9.] On Some Representation Theorems. 503 presentation of!i, then there exists a continuous unitary representation {U3, of G such that (13) Use = f x(s) U eds, E where the integration is Banach space valued integral, and the converse correspondence be held by the same relation (13). Since the Theorem 2 can be applied for L1-algebraj~, we have Theorem 4. Let G be a separable unimodular locally compact group. A normal two-sided continuous unitary representation of G is a directed integral of a system of irreducible such representations of G.3) Proof. Give representation be {U3, V8, j, }. For any x e T and any e E, we put (14) U~ = Jx(s) Useds, V~ = Jx(s_1) Vseds where the integration being same way in (13). Then {U1, V1, j,,} is a two-sided continuous representation of 2. For, the conjugate linear transformation j is commute with the strong integration, that is, (juxje, '1)=(j11, Urje) = Jx(s)(iu?, US je)ds = Jx(s)UUie 1)ds = x(s)(vs,'7)ds-(vx*, ~), hence juxj=v1* and the other conditions are followed by U8Vt = VtU8 and V, = V~V8. In order to decompose { U,, Vs, j, S} into irreducible factors, first we shall prove that { U1, Vy 1 x, y E i}' _ { U,, V11 s, t E G}'. if 13U,= U,B and BV = V1B, then since bounded linear operators are commute with the strong integration, BUX = B x(s) U,ds = x(s)buseds = Jx(s) USBds a hence BU1= U1B and by the same way BVx = V,B. The converse be possible to prove by the continuity of the representation { U8, VS, j, } and similar way above one. Thus, when we consider the decomposition of the Theorem 2 for L'-algebra ~i, {U1, V1, j,. } is a directed integral of a system of the irreducible two-sided representations {U,(A), VAA), j (A), ~ }, A E Nc (A)-null set. Since {U,(A), V1(A), j(a),~} are continuous representations, there exist two-sided continuous unitary representations {U,(A), V(A), j(a), }4) of G for 1) p(s) is the measure factor of the Haar measure of G. 2) It is obvious by the same reason with the statement of Remark 1. The two-sided representation will be possible to define in an abstract * algebra, we shall discuss in another paper. 3) This theorem also holds for any such representation (being not always,normal) onto a separable Hilbert space. 4) It can be proved by the same way with (14). a

4 R 504 H. UMEGAKI. [Vol. 27, E N such that Ux(A)e~ = x(s) US(A)eads, ti z (A,; = Jx(s')vs(A)EAds y r for all x E -t and E~ E a. Then it remains to prove that the representation {U3, VS, j, S5} is a directed integral of the system of irreducible two-sided unitary representations { US(A), V8(1), j(a), We have (15) (UzE, ~) = JG' x(s}(usa~) ds, the left hand of (15)_ J(U(A)e x ~, = x(s)(us ', '7a)ds do (A). R (; Now, we can apply the proof of theoren 3 for these functions (US(A)~~,') instesd of x(s, A) (cf. (12)) and hence can use Fubini's Theorem, so (15) is equal to () R x(s)(u3(~)~, '1~)do(A) ds. As we have used sometimes (e.g. the equation (12)) (r' x(s)(use, ~) ds dr(a) = J GR((' x(s)us~)a E?.)d~(A) ds for any x(s) in ~?l. Thus, G (Use, C) = (U8(A)Sr it, C.) da(a) for any s E G. Since E.~ is arbitrary, it completes the proof. Add in proofs. From the equation (5) in the first paper I,P. 330, we have stated without proof that almost all {U(A), xs } are representations of {. Now we may prove this. Since t is separable, there exists an enumerable dense self-adjoint subset?io={x,z} of?z such that Ux?Ux,a(A) and I f U.,(A) I I ( Ux a 11, U(A) U(A) Ux~(A), U,*(A) = Ux b(a) * for A E Nm7a(o (A)-null set). Put N = Nma. For any x E i, there exists a sequence {x} nz Ao such that x; -~ x in 1. Since III Ux' -- Ux x;-x II -~ 0 and If I U(A) - U.. (A) I!I s III Ux~ Uj III for A E N, we can find a bounded operator A(A) on (A E N) which is an uniform limit of U5'(A). Hence A(A) = U(A) for A N, and it may be proved {U(A), x ~} (A E N) being representation. Next, we have concluded from (S) that ja is our j-involution a.e. Q(A), there we had omitted the precise proof of the term of a. e. o (A).

5 No. 9.J On Some Representation Theorems. 505 Throughout these papers I and II, we have described only summary notes. Their details will be descussed in more general from with other statements, it will appear elsewhere. Bibliography. 1. J, von Neumann : On Rings of Operators. Reduction Theory. Ann. of Math. 50 (1949), pp I. E. Segal : Irreducible Representations of Operator Algebra; Bull. Amer. Math. Soc. 48 (1947) M. Nakamura : The Two-side Representations of an Operator Algebra. Proc. Japan Acad. 27 (1951), 4, M. Nakamura and Y. Misono : Centering of an Operator Algebra appear in Tohoku Math. J. 5. F. I. Mautner : Unitary Representations of Locally Compact Groups. I. Ann. of Math. 51 (1950), F. I. Mautner : Unitary Representations of Locally Compact Groups. II. Ann. of Math. 52 (1951), R. Godement: Les fonctions de type positif et la theorie does groupes. Trans. Amer. Math. Soc. 63 (1948), R. Godement : Sur la theorie des representations unitaires. Ann, of Math. 53 (1951), R. Godement : Stir la theorie des caracteres. I et II, C. R. Paris 229 (1949) , R. Godement : Analyse harmonique dans les groupes centraux I, ibid., 225 (1949) H. Yoshizawa : On Some Type of Convergence of Positive Definite Functions. Osaka Math. J. 1(1949),