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Representations of a Compact Group on a Banach Space Journal of the Mathematical Society of Japan Vol. 7, No. 3, June, 1955 Representations of a compact group on a Banach space. By Koji SHIGA (Received Jan. 14, 1955) Introduction. Let G be a compact group and IYIa complex normed linear space. We mean by a representation {, T(a)} a homomorphism of G into the group of bounded linear operators on such that a --~ T(a). The following types of representations are important : i) algebraic repre- sentation, ii) bounded algebraic representation,l' iii) weakly measurable representation,2' iv) strongly measurable representation,3' v) weakly continuous representation,4' vi) strongly continuous representation.5' Clearly each condition above becomes gradually stronger when it runs from i) to vi), except iv). When the representation space 1 of {IJI,T(a)} is a Banach space, it is called a B-representation. In considering a B-representation {8, T(a)}, what we shall call the conjugate representation {8*, T *(a)} arouses our special interest. For example, we can prove, by considering {**, T **(a)} of a weakly continuous B-representation {8, T(a)}, the equivalence of all four condi- tions (Theorem 2) : 1) {l3, T(a)} is completely decomposable, 2) {8, T(a)} is strongly continuous, 3) {3, T(a)} is strongly measurable, 4) for any given x e 3, the closed invariant subspace generated by T(a)x, 1) Algebraicreps esentation such that IIT(a) II< M for a certain M(> 0). 2) Boundedalgebraic representation such that f(T(a)x) is a measurable function on G for any xe3 and f€3,*where 9* meansthe conjugatespace of . 3) Boundedalgebraic representation such that T(a)x is strongly measurable (in the sense of Bochner)for any fixedX€ l1. 4) Boundedalgebraic representation such that f(T(a)x) is a continuousfunction on G for any x€31and f€31*. 5) Algebraicrepresentation such that IIT(a)x- T(ao)x II -'0 as as -- a0, (and then necessarilybounded). Representations of a compact group on a Banach space. 225 a e G, is separable. This explains naturally the classical result of Peter and Weyl [1O].6i The content of the present paper is briefly as follows. We first give a survey of unitary representations of a compact group (~ 1). In § 2 the finite dimensionality of an irreducible algebraic representation is proved under a weak assumption. Our discussions which follow are really based on two fundamental Propositions A and B (~ 3) ; in fact, Proposition A yields the essential part of the structure Theorem 2 ( 4). On the other hand, it is possible to deduce from Proposition B a formal structure of weakly continuous representations which might be regarded as a formulation of reduction of the weak continuity to the strong one (~ 5). The relations between the weak and the strong continuity are discussed in 6. The role played by the weak and the strong measurable representations becomes clear in the last section ( 7). The principal idea used here is to reduce an arbitrary repre- sentation to a unitary one. Since we make full use of the boundedness of Haar measure, it seems difficult to generalize further this method beyond the compact case. Throughout this paper we concern ourselves with the representa- tion of a compact group G. s , 3 and denote a Hilbert space, a Banach space and a normed linear space, respectively; Hilbert space is not assumed to be separable. Linear operators are assumed to be bounded and are denoted by S(a), T(a), U(a) etc., whose adjoint opera- tors are denoted by S*(a), T *(a) etc, respectively. We denote elements of a compact group G by a, b, c, and elements of , s~ etc, by x, y, z, f, g, The integral is always taken with respect to the Haar measure on G. LA(G) (1pC C co) and C(G) (the family of con- tinuous functions on G) denote the usual Banach spaces. For a given 9, we shall denote by [x JP(x)] the closed subspace of IYI,generated by all elements x having the property P(x). The author wishes to express his deepest thanks to Professors Y. Mimura and Y. Kawada for their constant encouragements and advises during the preparation of this paper. The author is also grateful to Mr. N. Iwahori for his valuable remarks. 6) Numbers in brackets refer to the bibliography at the end of the paper, a 226 K. SHIGA 1. Unitary representations. In this section we consider a unitary representation {, U(a)}. The following three propositions are well-known.7' Since Proposition 1 plays a fundamental role for our later discussions, we give here a direct proof of Proposition 1. PROPOSITION1. {5~,U(a) } is completely reducible, i, e. S3 is a dis- crete direct sum of mutually orthogonal finite-dimensional subspaces spa's, each of which is invariant and irreducible under U(a) (a e G) : ~~.= o ~, PROOF. Since the invariance of a closed subspace of implies that of its orthogonal complement, it suffices to prove this proposition only in the case where ~ is generated by U(a) (a e G) for some fixed ., I.H=1. For all x, y in 3, we define [x, y]= (x, U(a))(y, U(a)) da. (1) Normalizing Haar measure so that da=1, we have [x, y] < IIx I.jy. ~ Hence [x, y] defines a bounded bilinear form on ~. Let H be the Hermitian operator associated with [x, y]8': [x, y]=(Hx, y) for all x, y. Then H is a positive definite operator in the strict sense. Now, for any ao e G and for every x, y in , we have (U(ao)Hx, y) = (Hx, U(ao)*y) = (x, U(a))(y, U(aoa)) da _ (U(ao)x, U(a) -)(y, U(a)) da = (HU(a0)x, y) , whence each U(a) commutes with H. Moreover, H is completely 7) See Godement[1], Hurevitsch[5], Ito [6], Murakami[8] and Wigner L14]. 8) This Hermitianoperator H was first introducedexplicitly by S, lto [6]. Representations of a compact group on a Banach space. 227 continuous. In fact, if xn -+ xo (weakly), Hxn --~Hxo (weakly) ; on the other hand, from (1) and Lebesgue's convergence theorem follows that tHx2n j -> ~~Hxo l 2, and thus Hxn -b Hxo (strongly). Consequently, if we perform the spectral resolution of H, we have H= Jx1P~~1, where Pjj1 is the projection whose range is the finite- dimensional eigenspace 91 corresponding to the eigenvalue x1. This, together with the positive definiteness of H in the strict sense, yields => wi r• Since each finite•dimensional subspace W 1 reduces every U(a), because of the commutativity of H and U(a), the above decomposition of ~ gives a decomposition into invariant subspaces. If {JJj UjJ(z(a)}9' is still reducible, it is possible further to decompose 91 into irreducible subspaces by the use of the well-known procedures. This completes the proof. We call such a decomposition of as stated in Proposition 1 a complete decomposition of {s~,U(a)}. Now we shall establish the rela- tionship between two complete decompositions of {, U(a)}. This is easily formulated in terms of the following notion : For a complete decomposition {3, U(a)} = +0 {, Us (a)} and for a fixed equivalent a class of irreducible representations of G, we denote by S~() the closed subspace generated by all those la's for which the irreducible representation U(a) belongs to . Thus, ~ = +0a and = Gs denoting two complete decompositions of {, U(a)}, they yield another kind of decompositions such that S3= >j 0+ () and = > respectively, with respect to the class of irreducible representations. Then from the orthogonality relations we get the following result. PROPOSITION2. ~()='().`~~ PROPOSITION3. Any irreducible unitary representation of a com- pact group is equivalent to one which is obtained in a complete decom- position of the regular representation. PROOF. Let {°, U(a)} be irreducible. Then making use of nota- tions in the proof of Proposition 1, we have H=xI (x> 0), that is 9) Un1(a)denotes the representationof G obtainedby restrictingU(a) in1. Such notationis usedin § 1; in the remainingsections, if it happenssuch a case, we write U(a) for Us1(a) only for simplicity. Ifo(T(a)xo) J2da=0, and thus fo(T(a)xo)=T*(a)fo(xo)=0 for all ae G; accordingly, putting J1= {x; T*(a)fo(x)=0 for all a e G}, we have xo e 1Y1o.Buto is clearly an invariant closed subspace, whence from the irreducibility of {9, T(a)} and fo = 0 it results o= {0}, and so xo=0 as desired. (x, y) defined in (2) therefore gives an inner product of 11. ,5 denotes the completion of 1 with respect to this inner product and III III denotes 10) Irreducibilityof a representationmeans the non-existenceof a proper invariant closedsubspace. 228 K. SHIGA X!Lx 2= I(x, U(a)) I2da. Hence if we assign to every x of an element f(a)=x (x, U(a) )/j/x of L2(G), we obtain an isomorphism between 3 and a closed subspace L of L2(G), through which U(a) can be identified with the regular repre- sentation in L. REMARK. This result can be generalized to a theorem for locally compact groups which asserts that any irreducible unitary representation corresponding to an integrable positive definite function (in Godement's sense [1]) is realizable in the regular representation. 2. Irreducible representations.lo> In his paper [7] I. Kaplansky remarked the fact without giving its proof that any irreducible strongly continuous B-representation is finite- dimensional.
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