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Semigroup Representations, Positive Definite PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2949{2955 S 0002-9939(98)04814-X SEMIGROUP REPRESENTATIONS, POSITIVE DEFINITE FUNCTIONS AND ABELIAN C∗-ALGEBRAS P. RESSEL AND W. J. RICKER (Communicated by Palle E. T. Jorgensen) Abstract. It is shown that every -representation of a commutative semi- ∗ group S with involution via operators on a Hilbert space has an integral rep- resentation with respect to a unique, compactly supported, selfadjoint Radon spectral measure defined on the Borel sets of the character space of S.The main feature is that the proof, which is based on the theory of positive defi- nite functions, makes no use what-so-ever (directly or indirectly) of the theory of C∗-algebras or more general Banach algebra arguments. Accordingly, this integral representation theorem is used to give a new proof of the Gelfand- Naimark theorem for abelian C∗-algebras. Let H be a Hilbert space and (H)anabelian C∗-algebra,thatis,the A⊆L identity operator I belongs to , the adjoint operator T ∗ whenever T , and the (commutative) algebraA is closed for the operator∈A norm topology in∈A the space (H) of all continuous linearA operators of H into itself. The Gelfand-Naimark theoremL can be formulated as follows; see [5, Theorem 12.22] for a very readable account of this result. Theorem 1. Let H be a Hilbert space and (H)an abelian C∗-algebra. Then there exists a unique selfadjoint, Radon spectralA⊆L measure F : (∆) (H)such that B →L (1) T = T (δ) dF (δ);T: ∈A Z∆ Some explanation is in order. Theb space ∆ in Theorem 1 is the structure space of ; it is a compact Hausdorff space and can be interpreted as the set of all non- A zero, complex homomorphisms δ : Cequipped with the topology of pointwise A→ convergence on .GivenT , the continuous function T :∆ Cis defined by A ∈A → T (δ):=δ(T), for δ ∆; it is called the Gelfand transform of T . Wedenoteby (∆) the Borel σ-algebra∈ of ∆, i.e. the smallest σ-algebrab containing all the open B subsetsb of ∆. To say a function F : (∆) (H)isaselfadjoint spectral measure means that F (∆) = I, that each operatorB →LF (A), for A (∆), is selfadjoint, that F is multiplicative (i.e. F (A B)=F(A)F(B) for all∈BA; B (∆)), and that F is countably additive for the∩ weak (equivalently, the strong)∈B operator topology in (H), that is, Fx;y : A F (A)x; y ,forA (∆), is a σ-additive, C-valued L 7→ h i ∈B Received by the editors March 3, 1997. 1991 Mathematics Subject Classification. Primary 47A67, 47B15, 47D03, 47D25. The support of the Deutscher Akademischer Austauschdienst (DAAD) is gratefully acknowl- edged by the second author. c 1998 American Mathematical Society 2949 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2950 P. RESSEL AND W. J. RICKER measure for each x; y H. Since all the values F (A), for A (∆), are orthogonal projections, we see that∈ F (A)= F(A)x2 is actually∈B non-negative, for each x;x k k x H.TosaythatFis Radon means that Fx;x is a Radon measure (i.e. inner regular),∈ for each x H; see [1, Ch. 2], for example. The selfadjointness of F implies that ∈ 4Fx;y = Fx+y;x+y Fx y;x y + iFx+iy;x+iy iFx iy;x iy;x;yH; − − − − − − ∈ and so each complex measure Fx;y is also Radon. Since ∆ is compact and T is continuous, it is, in particular, a bounded Borel function and so the integral in (1) defines an element of (H) via the standard theory of integration with respectb to spectral measures; seeL [5, Section 12.20]. Let S be a commutative semigroup with identity element (always denoted by e) and equipped with an involution s s− (i.e. (s−)− = s and (st)− = s−t− for 7→ all s; t S). A character of S is any function ρ: S C satisfying ρ(e)=1and ∈ → ρ(st−)=ρ(s)ρ(t) for all s; t S. The set of all characters of S is denoted by ∈ S∗; it is a completely regular space when equipped with the topology of pointwise convergence inherited from CS.IfHis a Hilbert space, then a map : S (H) U →L is called a -representation if (e)=I and (st−)= (s) (t)∗ for all s; t S. ∗ U U U U ∈ Theorem 2. Let S be a commutative semigroup with identity and an involution, and let : S (H)be a -representation. Then there exists a unique selfadjoint, U →L ∗ Radon spectral measure E : (S∗) (H)which has compact support, such that B →L (2) (s)= sˆ(ρ)dE(ρ);sS: U ∈ ZS∗ Again some explanation is needed. The support of any spectral measure E : (S∗) B (H), denoted by supp(E), is defined to be the closed subset of S∗ given by →L x H supp(Ex;x), where supp(Ex;x) is the usual support of a non-negative Radon ∈ measure [1, p. 22]. Finally, given s S, the functions ˆ: S∗ C is defined by S ∈ → sˆ(ρ):=ρ(s), for ρ S∗. Sinces ˆ is continuous and supp(E) is compact, the spectral integral on the right-hand-side∈ of (2) exists and is an element of (H). A version of Theorem 2 is formulated in [2, Theorem 2.6] whereL it is indicated that the proof is a combination of an integral representation theorem for expo- nentially bounded, positive definite functions on semigroups (now-a-days referred to as the Berg-Maserick theorem) [2, Theorem 2.1], together with the method of proof given for an earlier version of Theorem 2 formulated for uniformly bounded representations [3, Theorem 3.2]. An examination of that proof (i.e. when is uniformly bounded)U shows that an essential ingredient is the use of the theoryU of abelian C∗-algebras (cf. Section 2 of [3]) together with some more general Banach algebra arguments (cf. [3, p. 501]). The aim of this note is to highlight the fact, contrary to the line of argument suggested above, that Theorems 1 and 2 are actually “independent” of one an- other. That is, we present a (new) proof of Theorem 2 which uses neither Gelfand- Naimark theory nor any Banach algebra arguments (either directly or indirectly). We then use Theorem 2 to establish Theorem 1, thereby giving a new proof of the Gelfand-Naimark theorem for abelian C∗-algebras. Conversely, via a quite different argument than that suggested by Berg and Maserick, we show that Theorem 2 also follows from Theorem 1. In order to prove Theorem 2 we require a few preliminaries. A function ': S n → C is called positive definite if cjck'(sjs−) 0 for all choices of n N, j;k=1 k ≥ ∈ P License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SEMIGROUP REPRESENTATIONS, AND ABELIAN C∗-ALGEBRAS 2951 s1;:::;sn Sand c1;:::;cn C. In particular, every character ρ S∗ is positive{ definite.}⊆ { }⊆ ∈ A function α: S [0; ) satisfying α(e) = 1 is called an absolute value if → ∞ α(s−)=α(s)andα(st) α(s)α(t) for all s; t S. A function f : S C is called α-bounded if there exists≤C>0 such that f(s)∈ Cα(s) for all s S.If→ fhappens to be positive definite and α-bounded, then| it|≤ is possible to choose∈ C = '(e). A α character ρ S∗ is α-bounded iff ρ α. Hence, the set S of all α-bounded ∈ | |≤ characters is a compact subset of S∗. For all of these notions and further properties we refer to [1, Ch. 4]. The space of all non-negative Radon measures on S∗ is denoted by M+(S∗). Given any absolute value α: S [0; ) the subspace of M+(S∗) consisting of all → ∞ α α Radon measures supported in the compact subset S S∗ is denoted by M+(S ). The following result is the Berg-Maserick theorem mentioned⊆ above. Proposition 1. Let α : S [0; ) be an absolute value on S and ': S C an α-bounded, positive definite→ function.∞ Then there exists µ M (Sα) such→ that ∈ + (3) '(s)= sˆ(ρ)dµ(ρ);sS; ∈ ZS∗ and µ is unique within M+(S∗). Remark 1. The function on the right-hand-side of (3), with domain S, is denoted byµ ˆ andiscalledthegeneralized Laplace transform of µ. It is important to note that the proof of Proposition 1 given in [1, Ch. 4, 2], which is based on the integral version of the Krein-Milman theorem, makes no§ use of any Banach algebra techniques what-so-ever. Proof of Theorem 2. Define an absolute value α: S [0; )byα(s)= (s) for → ∞ kU k s S.Forx Hfixed define 'x : S C by 'x(s)= (s)x; x ,fors S.Using the∈ fact that ∈is a -representation it→ follows that hU i ∈ U ∗ 2 n n c c ' (s s−)= c (s )x 0 j k x j k jU j ≥ j;k=1 j=1 X X for any finite sets c1;:::;cn Cand s 1;:::;sn S and hence, 'x is positive definite. Moreover,{ }⊆ { }⊆ ' (s) (s) x 2 = x 2α(s);sS; | x |≤kU k·k k k k ∈ which shows that 'x is α-bounded. By Proposition 1 there is a unique Radon α measure µx M+(S ) such that 'x =ˆµx. For x; y ∈H define a complex Radon measure µ by ∈ x;y 1 (4) µx;y = (µx+y µx y + iµx+iy iµx iy); 4 − − − − 1 in which caseµ ˆx;y = 4 (ˆµx+y µˆx y + iµˆx+iy iµˆx iy).
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