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

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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 ϕ: S n → C is called positive definite if cjckϕ(sjs−) 0 for all choices of n N, j,k=1 k ≥ ∈ P

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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 = ϕ(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 ϕ: S C an α-bounded, positive definite→ function.∞ Then there exists µ M (Sα) such→ that ∈ + (3) ϕ(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 ϕx : S C by ϕx(s)= (s)x, x ,fors S.Using the∈ fact that ∈is a -representation it→ follows that hU i ∈ U ∗ 2 n n c c ϕ (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, ϕx is positive definite. Moreover,{ }⊆ { }⊆ ϕ (s) (s) x 2 = x 2α(s),sS, | x |≤kU k·k k k k ∈ which shows that ϕx is α-bounded. By Proposition 1 there is a unique Radon α measure µx M+(S ) such that ϕx =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). It then follows from the definition ofµ ˆ = ϕ for each−z H− and a direct− calculation− that z z ∈ (5) µˆ (s)= (s)x, y ,sS. x,y hU i ∈ Fix B (S∗) and define ΨB : H H C by ΨB (x, y)=µx,y(B)foreach x, y H.∈B It is clear from (5) and the× linearity→ and injectivity of the map ν νˆ ∈ 7→ on the space of compactly supported (complex) Radon measures on S∗ [1, p. 96] that Ψ is a sesquilinear form on H H. We proceed to show that Ψ is bounded. B × B

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Let x1,...,xn Hand c1,...,cn Cbe finite sets. Define a Radon measure { }⊆ n { }⊆ ν on (S∗)byν= cjckµxj ,xk . Then, for finite sets d1,...,dm Cand B j,k=1 { }⊆ t ,...,t Swe have { 1 m}⊆ P 2 m d d νˆ(t t−)= d d c c (t t−)x ,x = c d (t )x 0, p q p q p q j khU p q j ki j pU p j ≥ p,q=1 j,k p,q p,j X XX X

which shows thatν ˆ: S C is positive definite. By combining Theorem 2.5 on → α p. 93 and the remarks of 2.10 on p. 96 in [1] it follows that ν 0, i.e. ν M+(S ). In particular, § ≥ ∈ n n c c Ψ (x ,x )= c c µ (B)=ν(B) 0, j k B j k j k xj ,xk ≥ j,kX=1 j,kX=1 which shows that ΨB is a positive definite kernel (in the sense of [1, p. 67], for example). By the Cauchy-Schwarz inequality for such kernels (cf. [1, p. 69], for example) we have Ψ (x, y) 2 Ψ (x, x)Ψ (y,y), that is, | B | ≤ B B µ (B) [µ (B)]1/2[µ (B)]1/2 x y ,x,yH, | x,y |≤ x,x y,y ≤k k·k k ∈ where the last inequality relies on the observation that

µ (B) µ (Sα)= ρ(e)dµ (ρ)=ˆµ (e)= (e)x, x = x 2 x,x ≤ x,x x,x x,x hU i k k ZS∗ for all x H. So, the sesquilinear form ΨB is indeed bounded and hence, there is E(B) ∈(H) satisfying ∈L (6) E(B)x, y =Ψ (x, y)=µ (B),x,yH. h i B x,y ∈ To see that E (B )∗ = E (B )wenotethat

E(B)∗x, y = x, E(B)y = E(B)y,x = µ (B)=µ (B)= E(B)x, y h i h i h i y,x x,y h i for all x, y H,whereµ (B)=µ (B) follows again from the positive definite- ∈ y,x x,y ness of the kernel ΨB. Furthermore, E(S∗)=I since, by (5), we have

E(S∗)x, y = µ (S∗)=ˆµ (e)= (e)x, y = x, y ,x,yH. h i x,y x,y hU i h i ∈ It is clear from (6) that E = µ is σ-additive for all x, y H. The identity (6) x,y x,y ∈ also shows that E : (S∗) (H) is a Radon measure and that it is compactly supported since supp(BE )→L = supp(µ ) Sα for all x, y H. x,y x,y ⊆ ∈ To see that E is multiplicative we proceed as follows. Given a Borel set A S∗ and x, y H define measures ν and λ by ν (B)=µ (B)andλ (B⊆)= ∈ A A A E(A)x,y A E(A B)x, y = µ (A B)foreachB (S∗). Using (5) we have h ∩ i x,y ∩ ∈B (7) νˆA(t)=ˆµE(A)x,y(t)= (t)E(A)x, y = E(A)x, (t)∗y = µx, (t) y(A) hU i h U i U ∗ and ˆ (8) λA(t)= ρ(t)dµx,y(ρ). ZA For a fixed t S it is clear from (7) and (8) that the mappings τ1 : A νˆA(t)and ˆ ∈ α 7→ τ2: A λA(t) are Radon measures on S∗, supported in S , and so we can also calculate7→τ ˆ andτ ˆ . Indeed, for s S, we have (by (5) and (7)) that 1 2 ∈

τˆ (s)= (s)x, (t)∗y = (st)x, y = ρ(s)ρ(t) dµ (ρ). 1 hU U i hU i x,y ZS∗

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But, from (8) it is also clear that

τˆ2(s)= ρ(s)dτ2(ρ)= ρ(s)ρ(t)dµx,y(ρ). ZS∗ ZS∗ By uniqueness of generalized Laplace transforms [1, p. 96], it follows that τ1 τ2 as compactly supported Radon measures (for any fixed t S). In particular,≡ for ∈ A fixed we deduce thatν ˆ (t)=λˆ (t). Since this is true for each t S it again A A ∈ follows that ν = λ as measures and hence, for any Borel set B S∗ we have A A ⊆ λA(B)=νA(B), that is, E(A B)x, y = λ (B)=ν (B)= E(B)E(A)x, y ,x,yH. h ∩ i A A h i ∈ So, E is multiplicative. α Finally,s ˆ: S∗ C is continuous and hence bounded on S , for any s S. By the theory of spectral→ integrals sˆ(ρ) dE(ρ) exists in (H). It is clear∈ from S∗ L (5) and (6) that (s)= sˆ(ρ)dE(ρ), for s S, and the proof of Theorem 2 is S∗ R complete. U ∈ R To deduce Theorem 1 from Theorem 2 we require a further result. A commuta- tive, unital complex algebra with an involution (see Definitions 10.1 and 11.14 in [5]) always induces a commutativeA semigroup with involution, namely let S = and consider just with respect to its given multiplication and involution. We thenA A define S~ := ρ S∗ : ρ is linear and, given any absolute value α: S [0, ), α { α∈ } → ∞ define S := S S~. The following result can be found in [4, Corollary 1]; we stress again that its∩ proof does not use any Banach algebra methods. Proposition 2. Let S be the semigroup induced by a unital, commutative complex algebra with an involution. Let α: S [0, ) be an absolute value. If ϕ: S C is a positive definite, α-bounded function→ which∞ is linear on S, then the unique→ α α Radon measure µ M (S ) satisfying µˆ = ϕ has its support in S . ∈ + Proof of Theorem 1. Let A (H) be an abelian C∗-algebra. Let S = ,consid- ered as a semigroup with respect⊆L to the multiplication in and with theA involution from . By Theorem 2 there exists a unique, selfadjointA Radon spectral measure A E : (S∗) (H) with compact support satisfying B →L (9) T = T (ρ)dE(ρ),TS; ∈ ZS∗ we have used the representation b(T )=T,forT S. From the proof of Theorem 2, where the absolute value α: US [0, )usedis∈ α(T)= (T ) = T ,for T S, we have that supp(E) Sα.Now,for→ ∞ x Hfixed, the positivekU k definitek k and ∈ ⊆ ∈ α-bounded function ϕx : S C defined in the proof of Theorem 2 (i.e. ϕx(T )= (T )x, x = Tx,x ,forT →S) is clearly linear on S. Then Proposition 2 implies hU i h i ∈ α α that the unique measure µx M+(S ) satisfyingµ ˆx = ϕx has its support in S . ∈ α α Since E = µ also supp(E) S .But,ifρ S ,then x,x x ⊆ ∈ ρ(T) α(T)= T ,T, | |≤ k k ∈A α which implies that ρ =1.So,S consists of all algebra homomorphisms ρ: k k S(= ) C of norm 1. It is part of the definition that each semigroup char- A → acter ρ S∗ must satisfy ρ(s−)=ρ(s) which, in the present setting, becomes ∈ ρ(T ∗)=ρ(T). But, elements of the structure space ∆ of automatically satisfy α A this condition [5, Theorem 11.18]. Hence, S is precisely ∆ and (9), which is α actually an integral over S = ∆, reduces to (1).

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In conclusion we indicate how Theorem 2 also follows from Theorem 1. So, let S and : S (H) be as in Theorem 2. Define (H)tobetheC∗-algebra generatedU by→L the -closed, commutative family ofA⊆L operators = (s): s S . Of course, is the∗ operator norm closure in (H) of the linearM span{U of .Denote∈ } A L M the structure space of by ∆. For each δ ∆ define δ˜: S C by δ˜(s):= δ( (s)), s S. From theA homomorphism properties∈ of δ and the→ fact that is U ∈ ˜ ˜ ˜ U a -representation it is easily seen that δ S∗; the property δ(s−)=δ(s) again follows∗ from the comments at the end of the∈ previous paragraph. So, we can define ˜ Λ: ∆ S∗ by Λ(δ):=δ,forδ ∆, in which cases ˆ Λ=( (s)) as functions on ∆, for→ each s S.IfΛ(δ)=Λ(∈δ ), then δ ( (s)) =◦ δ ( (sU)) for all s S and ∈ 1 2 1 U 2 U ∈ hence, δ1 and δ2 agree on the linear span of . Since the span of b is dense in and both δ and δ are continuous on , it followsM that δ = δ . ThisM shows thatA Λ 1 2 A 1 2 is injective. Since ∆ (resp. S∗) is equipped with the pointwise convergence topology on (resp. S), it is clear that Λ is continuous. Then the compactness of ∆ ensures A that Λ is a topological homeomorphism of ∆ onto its range K := Λ(∆) S∗.By Theorem 1 there is a unique selfadjoint Radon spectral measure F : (∆)⊆ (H) satisfying B →L

T = T (δ) dF (δ),T. ∈A Z∆ b 1 Define E : (S∗) (H)byE(A)=F(Λ− (A)), for each A (S∗), in which case E is aB compactly→L supported (as supp(E) K), selfadjoint∈B Radon spectral measure such that ⊆

(10) (s)= ( (s)) (δ) dF (δ)= sˆ(ρ)dE(ρ),sS; U U ∈ Z∆ ZS∗ see [5, Theorem 13.28] for theb last equality. Suppose now that G: (S∗) (H) is another compactly supported, selfad- joint Radon spectral measureB satisfying→L

(s)= sˆ(ρ)dG(ρ),sS. U ∈ ZS∗

It follows that sˆ(ρ) dGx,x(ρ)= sˆ(ρ)dEx,x(ρ) for all s S and x H, S∗ S∗ ∈ ∈ that is, G = E for all x H.SincebothG and E are compactly x,x R x,x ∈ R x,x x,x supported elements of M+(S∗), it follows from the uniqueness of generalized Laplace transformsb that Ebx,x = Gx,x for all x H. Then the selfadjointness of both E and G and the polarization identity imply∈ that E = G for all x, y H.In x,y x,y ∈ particular, it follows that E(A)=G(A) for all A (S∗), that is, E = G. Hence, the measure E satisfying (10) is unique and the proof∈B is complete.

Remark 2. From the proof of Theorem 2 it is obvious that if all the measures Ex,x = µx,forx H, concentrate on a subset A S∗,thensodoesE.As a consequence we obtain∈ Stone’s theorem for S a locally⊆ compact abelian group 1 (which is a semigroup with the special involution s− := s− ), and where is assumed to be weak operator continuous, i.e. s (s)x, x is continuous for eachU 7→ hU i x H. Then by Bochner’s theorem all the measures µx live on the dual group (by definition∈ the set of all continuous characters), and so therefore does E.

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References

1. C. Berg, J. P. R. Christensen and P. Ressel, Harmonic analysis on semigroups, Theory of positive definite and related functions, Graduate Texts in Mathematics Vol. 100, Springer- Verlag, New York-Berlin-Heidelberg-Tokyo, 1984. MR 86b:43001 2. C. Berg and P. H. Maserick, Exponentially bounded positive definite functions, Illinois J. Math. 28 (1984), 162–179. MR 85i:43012 3. P. H. Maserick, Spectral theory of operator-valued transformations, J. Math. Anal. Appl. 41 (1973), 497–507. MR 49:7828 4. P. Ressel, Integral representations on convex semigroups, Math. Scand. 61 (1987), 93–111. MR 89e:43012 5. W. Rudin, Functional analysis, McGraw Hill Book Co., New York-San Francisco-St. Louis, 1973. MR 51:1315

Math.-Geogr. Fakultat,¨ Katholische Universitat¨ Eichstatt,¨ D-85071 Eichstatt,¨ Germany

School of Mathematics, University of New South Wales, Sydney, New South Wales, 2052 Australia

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