An Integral Representation for Semigroups of Unbounded Normal Operators

J. OPERATOR THEORY c Copyright by Theta, 2002 47(2002), 219–243

AN INTEGRAL REPRESENTATION FOR SEMIGROUPS OF UNBOUNDED NORMAL OPERATORS

P. RESSEL and W.J. RICKER

Communicated by Florian-Horia Vasilescu

Abstract. An integral representation for semigroups {Us}s∈S of unbounded normal operators in a Hilbert space H is presented which admits a signiﬁ- cantly larger class of semigroups S than usual. In particular, S need not have a topology and so the traditional assumption that the functions s 7→ hUsx, yi, for suitable elements x, y ∈ H, are continuous is no longer a requirement. The classical spectral theorem for a single (unbounded) normal or selfadjoint operator is a consequence of the main result; the point is that the techniques used do not rely on the fact that a normal operator has a spectral decomposition via its resolution of the identity. Keywords: Semigroup representation, positive deﬁniteness, normal operator, spectral theorem. MSC (2000): 47B15, 47D03.

1. INTRODUCTION

The integral representation (via spectral measures) of 1-parameter semigroups of bounded selfadjoint or normal operators on a Hilbert space is well known; see [19] or [10], Chapter XXII. The extension to semigroups of unbounded selfadjoint or normal operators is also well understood, although the technical assumptions are more involved; see [5], [6], [9], [11], [14], [15], for example, and the references there in. Such results sometimes extend to semigroups S more general than [0, ∞), although there are usually some sort of topological constraints on S. These topological requirements of S are often transferred to the semigroup of operators {Us}s∈S by requiring that the C-valued function

(1.1) s 7→ hUsx, yi , s ∈ S, is continuous, for suitable vectors x and y in the underlying Hilbert space H. The aim of this note is to exhibit an integral representation for semigroups {Us}s∈S of unbounded normal operators which admits a significantly larger class 220 P. Ressel and W.J. Ricker of semigroups S than is usually the case. In particular, S is not assumed to have any topology and so no continuity assumptions are required of the functions (1.1). The techniques used come from the theory of positive definite functions. It is time to be more precise. Let S be a commutative semigroup with a unit element e and an involution s 7→ s− (i.e. (s−)− = s and (st)− = s−t− for all s, t ∈ S). A function ρ : S → C which satisfies ρ(e) = 1 and ρ(st−) = ρ(s)ρ(t) for all s, t ∈ S is called a character of S. The set of all characters is denoted by S∗; it becomes a completely regular topological space when equipped with the pointwise convergence topology inherited from CS. Given a Hilbert space H, let L(H) be the space of all continuous linear operators of H into itself and N (H) denote the collection of all (not necessarily bounded) normal operators in H. If T ∈ N (H), then its domain is denoted by D(T ); it is always dense. T ∗ denotes the adjoint operator of T . Each element of N (H) is a closed operator. As general references we use [3] and [17]. If T1 and T2 are linear operators in H, then T1 ⊆ T2 means D(T1) ⊆ D(T2) and T1x = T2x for x ∈ D(T1). The closure of a closable operator T (see [3]) is denoted by T . A function α : S → [0, ∞) is an absolute value if α is symmetric (i.e. α(s−) = α(s) for s ∈ S), α(e) = 1 and α(st) 6 α(s)α(t) for s, t ∈ S. The family of all absolute values on S is denoted by A(S). Given a map U : S → N (H) and α ∈ A(S) define n \ o (1.2) Dα := x ∈ D(Us): kUsxk 6 α(s)kxk for all s ∈ S , s∈S S where Us denotes the value of U at s ∈ S. Then define Dc := Dα. α∈A(S) Definition 1.1. Let S be a commutative, unital semigroup with an involution. A map U : S → N (H) is called a ∗-representation if: (i) Ue = I, the identity operator on H. ∗ (ii) Us− = Us , s ∈ S. (iii) UtUs ⊆ Ust with D(UtUs) = D(Ust) ∩ D(Us), s, t ∈ S. (iv) UtUs = Ust, s, t ∈ S. S (v) Dc = Dα is dense in H. α∈A(S)

We will see in Section 2 that (i)–(v) are natural if {Us}s∈S is to have an T integral representation. From (1.2) and (v) of Deﬁnition 1 we see D(Us) is s∈S necessarily a dense subspace of H (as it contains Dc). Our main result is the following one. Theorem 1.2. Let S be a commutative, unital semigroup with involution and U : S → N (H) be a ∗-representation. Then there exists a unique selfadjoint Radon spectral measure E : B(S∗) → L(H), with supp(Ex) ⊆ S∗ compact if and only if x ∈ Dc, and Z (1.3) Usx = sb(ρ) dE(ρ)x, x ∈ D(Us), s ∈ S, S∗ An integral representation for semigroups 221

∗ where sb : S → C is deﬁned by sb(ρ) := ρ(s). Some explanation is in order. By B(S∗) is denoted the Borel σ-algebra of S∗, i.e. the smallest σ-algebra containing all the open subsets of S∗. To say the function E : B(S∗) → L(H) is a selfadjoint spectral measure means that E(S∗) = I, that each operator E(A), for A ∈ B(S∗), is selfadjoint, that E is multiplicative (i.e. E(A ∩ B) = E(A)E(B) for A, B ∈ B(S∗)), and that E is countably additive for the weak (equivalently, the strong) operator topology in ∗ L(H), that is, Ex,y : A 7→ hE(A)x, yi, for A ∈ B(S ), is a σ-additive, C-valued measure for x, y ∈ H. Since the values E(A), for A ∈ B(S∗), are orthogonal 2 projections we see Ex,x(A) = kE(A)xk is actually non-negative, for x ∈ H. To say E is Radon means Ex,x is a Radon measure (i.e. inner regular) for each x ∈ H; see [2], Chapter 2, for example. The selfadjointness of E implies

4Ex,y = Ex+y, x+y − Ex−y, x−y + iEx+iy, x+iy − iEx−iy, x−iy, x, y ∈ H, and so each complex measure Ex,y is also Radon (i.e. its variation is a Radon measure). For x ∈ H, the H-valued, σ-additive measure A 7→ E(A)x, for A ∈ B(S∗), is denoted by Ex. Its support, denoted by supp(Ex), is defined to be the support of Ex,x (see [2], p. 22 for the definition of support of a non-negative Radon measure). Finally, the integral in (1.3) exists in the usual sense of integration with respect to the vector measure Ex ([12]); this is made precise in the next section. The proof of Theorem 1.2 relies on the methods of positive definite functions, together with an earlier version of Theorem 1.2 known for ∗-representations of bounded normal operators ([16]). The reduction to the bounded case is possible because the closed subspaces Dα (that they have this property is not obvious from (1.2)) turn out to be invariant for each Us, s ∈ S, and the restriction of Us to Dα is a bounded normal operator. Moreover Dc, which also turns out to be a subspace of H, plays a crucial role because it gives a description, via intrinsic T properties of U, of the space of vectors x ∈ D(Us) for which Ex has compact s∈S support. The spectral measure E is first constructed on Dc and then extended to H by a continuity argument. Because the arguments are rather technical the proof is delayed until Section 3. In the following Section 2 we prefer to discuss Theorem 1.2 itself in combination with Definition 1.1 and some relevant examples. In particular, we indicate how Theorem 1.2 deviates significantly from other results in the literature which are of a “similar nature” yet, as will be seen, are actually quite different. In Section 4 we indicate how the classical spectral theorem for a single (unbounded) selfadjoint or normal operator is a consequence of Theorem 1.2; the point is that the proof of Theorem 1.2 nowhere uses the fact that a normal operator has a spectral decomposition via its resolution of the identity. This is an important feature in relation to “similar” results in the literature alluded to above, which actually use the spectral theorem in their proofs. k An important particular case is S = N0 (with the usual addition and the identity as involution). Theorem 1.2 then provides the answer to an operator- valued moment problem. The existence of positive definite non-moment functions 2 2 on N0 ([2], Theorem 6.3.4) leads to ∗-homomorphisms U : N0 → L(D), where L(D) is the algebra of all endomorphisms on some appropriate pre-Hilbert space D, for which there is no integral formula analogous to (1.3). For the class of perfect semigroups S this phenomenon cannot occur; see the final section. 222 P. Ressel and W.J. Ricker

2. THEOREM 1.2: DISCUSSION

Consider a selfadjoint spectral measure E : B(S∗) → L(H). Every bounded Borel function f : S∗ → C is integrable with respect to the L(H)-valued measure E in the sense of vector measures ([7], [12]). This integral, denoted by R f dE, is an S∗ element of L(H). Also, f is integrable with respect to each H-valued measure Ex, x ∈ H, and Z Z Z f dE x, y = f d(Ex), y = f dEx,y, x, y ∈ H. S∗ S∗ S∗ The map f 7→ R f dE is linear, multiplicative, i.e. R fg dE = R f dE · S∗ S∗ S∗ R R g dE , and satisfies f dE 6 |f|E where S∗ S∗ n ∗ o |f|E = inf sup | f(ρ)| : A ∈ B(S ),E(A) = I . ρ∈A R The operator f dE is normal with resolution of the identity Pf : B(C) → L(H) S∗ −1 given by Pf (A) = E(f (A)). For these facts see [8], Chapter XVII and [17], Chapter 12. ∗ R More generally, if f : S → C is merely Borel measurable, then “ f dE ” S∗ still exists, but now as an unbounded operator. Its domain D(E, f) is given by Z D(E, f) = x ∈ H : lim fχ(n) dE x exists in H , n→∞ f S∗ where χ(n) is the characteristic function of |f|−1([0, n]), for n ∈ N, and the operator f R f dE is defined by x 7→ lim R fχ(n) dE x, for x ∈ D(E, f). The existence n→∞ f S∗ S∗ of lim R fχ(n) dE x is equivalent to f being integrable with respect to the H- n→∞ f S∗ 1 valued vector measure Ex in the sense of [12], i.e. f ∈ L (Ex,y), for all y ∈ H, and for each A ∈ B(S∗) there is a vector in H, denoted by R f d(Ex), satisfying A R R R h f d(Ex), yi = f dEx,y, for y ∈ H. The unbounded operator f dE so defined A A S∗ is closed, densely defined and satisfies R f dE x = R f d(Ex), for x ∈ D(E, f). S∗ S∗ For these facts we refer to [7], Section 1; [8], Chapter XVIII. Moreover, R f dE S∗ is normal and its adjoint is R f dE. This follows from standard results in [3], S∗ Chapter 5; [8], p. 2268; [10], Section 22.2, Chapter XXII or [17], Chapter 13, once we verify that Z 2 (2.1) D(E, f) = x ∈ H : |f| dEx,x < ∞ . S∗ An integral representation for semigroups 223

R 2 1 To see this, let x ∈ H satisfy |f| dEx,x < ∞. Then f ∈ L (Ex,y), for S∗ 1/2 R R 2 y ∈ H, since |f| d|Ex,y| 6 kyk |f| dEx,x < ∞ ([17], Lemma 13.23). S∗ S∗ Since Hilbert spaces have the BP-property, f is Ex-integrable ([12], p. 31), i.e. x ∈ D(E, f). Conversely, if x ∈ D(E, f), then f is Ex-integrable. Let Fn = {ρ ∈ S∗ : |f(ρ)|2 n}, for n ∈ . Then both fχ and fχ , n ∈ , are bounded 6 N Fn Fn N Borel functions and so Z Z |f|2 dE = fχ · fχ dE x, x x,x Fn Fn ∗ Fn S Z Z = fχ dE · fχ dE x, x , Fn Fn S∗ S∗ where the last equality uses the multiplicativity of ϕ 7→ R ϕ dE on the space of S∗ ∗ all bounded Borel functions ϕ. Since R fχ dE = R fχ dE ([17], Theo- Fn Fn S∗ S∗ 2 rem 12.21), it follows that R |f|2 dE = R fχ dE x , for n ∈ . But, x,x Fn N ∗ Fn S ∞ fχ converges pointwise to f on S∗ and |fχ | |f|, n ∈ , with |f| an Fn n=1 Fn 6 N Ex-integrable function ([12], p. 27). By the dominated convergence theorem for vector measures ([12], p. 30), Z Z Z lim fχ dE x = lim fχ d(Ex) = f d(Ex). n→∞ Fn n→∞ Fn S∗ S∗ S∗ 2 Hence, R |f|2 dE = sup R |f|2 dE = sup R fχ dE x < ∞ x,x x,x Fn ∗ ∗ S n∈NFn n∈N S showing that x belongs to the right-hand-side of (2.1). So, (2.1) is indeed valid. For each s ∈ S the function s : S∗ → C is continuous and hence, Borel b R measurable. Denoting D(E, sb) simply by D(Eb(s)) and sbdE simply by Eb(s) we S∗ have the canonical representation Eb : S → N (H) given by Z (2.2) Eb(s)x = sbdEx, s ∈ S, x ∈ D(Eb(s)); S∗ moreover Eb is called the generalized Laplace transform of E (cf. Remark 2.3 below). We now summarize the basic properties of {Eb(s)}s∈S. If E is also a Radon measure it follows that Eb : S → N (H) as defined by (2.2) is a ∗-representation as in Definition 1.1. So, the requirements of Definition 1.1 are quite natural since they are necessarily satisfied by the canonical representation corresponding to any given selfadjoint, Radon spectral measure. 224 P. Ressel and W.J. Ricker

Proposition 2.1. Let S be a commutative, unital semigroup with an involution. Let E : B(S∗) → L(H) be a selfadjoint spectral measure and let Eb : S → N (H) be its associated canonical representation as given by (2.2). (i) E(A)D(Eb(s)) ⊆ D(Eb(s)), A ∈ B(S∗), s ∈ S. (ii) E(A)Eb(s) ⊆ Eb(s)E(A), A ∈ B(S∗), s ∈ S. 2 (iii) If x ∈ D(Eb(s)) and y = Eb(s)x, then dEy,y = |sb| dEx,x. (iv) D(Eb(s)∗) = D(Eb(s−) = D(Eb(s)) and Eb(s)∗ = Eb(s−), for each s ∈ S. (v) D(Eb(t)Eb(s)) = D(Eb(st)) ∩ D(Eb(s)) and Eb(t)Eb(s) ⊆ Eb(ts), for all s, t ∈ S. (vi) Eb(t)Eb(s) = Eb(ts), t, s ∈ S. (vii) Eb(e) = I. If, in addition, E is a Radon measure, then S (viii) Dc = Dα is dense in H and so, in particular, Eb is a ∗- α∈A(s) representation. (ix) Dc = {x ∈ H : supp(Ex) is compact}. Before proving Proposition 2.1 we require some preliminaries. A function n P − ϕ : S → C is positive definite if cjckϕ(sjsk ) > 0 for all choices of n ∈ N, j,k=1 ∗ {s1, . . . , sn} ⊆ S and {c1, . . . , cn} ⊆ C. In particular, every character ρ ∈ S is positive definite. Let α ∈ A(S). A function f : S → C is α-bounded if there exists C > 0 such that |f(s)| 6 Cα(s) for s ∈ S. If f is positive definite and α-bounded, ∗ α then C = ϕ(e). A character ρ ∈ S is α-bounded iff |ρ| 6 α. Hence, the set S of all α-bounded characters is a compact subset of S∗. For all these notions we refer to [2], Chapter 4. The space of all non-negative Radon measures on S∗ is denoted ∗ ∗ by M+(S ). Given α ∈ A(S) the subspace of M+(S ) consisting of all Radon α ∗ α measures supported in S ⊆ S is denoted by M+(S ). The following result is the Berg-Maserick theorem ([1], Theorem 2.1).

Proposition 2.2. Let α ∈ A(S) and ϕ : S → C be an α-bounded, positive deﬁnite function. Then there exists a compactly supported Radon measure µ ∈ α R ∗ M+(S ) such that ϕ(s) = sbdµ, for s ∈ S, and µ is unique within M+(S ). S∗ R Remark 2.3. The map µb : s 7→ sbdµ is the generalized Laplace transform S∗ of µ. If M c(S∗) the space of all complex Radon measures on S∗ having compact c ∗ support, then µ 7→ µb is injective on M (S ) ([2], p. 96). Proof of Proposition 2.1. Parts (i)–(iii) follow from a combination of Theo- rem 13.24 and the Remark on p. 345 of [17] together with the identity Z 2 ∗ D(Eb(s)E(A)) = x ∈ H : |sb| dEx,x < ∞ ,A ∈ B(S ), s ∈ S, A which follows from (2.1) with sb in place of f. Property (iv) is immediate from [17], Theorem 13.24 (c). Theorem 13.24 (b) of [17] yields (v), and (vi) is immediate from An integral representation for semigroups 225

[3], p. 136, Theorem 7. Property (vii) is a simple consequence of the definition of ∗ Eb(e) after noting that eb is the (bounded) function 11constantly equal to 1 on S . Suppose now, in addition, that E is Radon. Fix s ∈ S. Given a compact set K ⊆ S∗ let z ∈ E(K)H, i.e. z = E(K)x for some x ∈ H. It is routine R 2 R 2 R 2 to check that |sb| dEz,z = |sb| dEz,z = |sb| dEx,x, which is finite as K is S∗ K K compact, sb is continuous and Ez,z is a finite measure. This shows z ∈ D(Eb(s)) and hence, that E(K)H ⊆ D(Eb(s)). Since K and s are arbitrary we have shown S E(K)H ⊆ T D(Eb(s)), where K(S∗) is the family of all compact subsets K∈K(S∗) s∈S ∗ ∗ of S . For K ∈ K(S ) define αK : S → [0, ∞) by

(2.3) αK (s) := sup{|sb(ρ)| : ρ ∈ K}, s ∈ S.

Then αK ∈ A(S). If x ∈ H, it was noted above that z = E(K)x ∈ T D(Eb(s)). From (iii) with x and A replaced by z and S∗, respectively, we s∈S 2 R 2 R 2 have kEb(s)zk = |sb| dEz,z = |sb| dEz,z, for s ∈ S. But, for s ∈ S, S∗ K Z Z Z 2 2 2 2 2 |sb| dEz,z 6 (αK (s)) dEz,z = (αK (s)) 11dEz,z 6 (αK (s)) kzk . K K K

Accordingly, kEb(s)zk αK (s)kzk for s ∈ S. It follows from (1.2) that 6 S S z ∈ DαK . This establishes that E(K)H ⊆ DαK ⊆ Dc. So, to K∈K(S∗) K∈K(S∗) prove (viii) it suffices to verify that S E(K)H is dense. Fix x ∈ H. Since K∈K(S∗) ∗ Ex,x is Radon and the net of compact subsets of S (directed by inclusion) is ∗ ∗ 2 upwards filtering to S , it follows that lim Ex,x(K) = Ex,x(S ) = kxk . Then K∈K(S∗) the equalities c c 1/2 ∗ 1/2 kx − E(K)xk = kE(K )xk = [Ex,x(K )] = [Ex,x(S ) − Ex,x(K)] 2 1/2 = [kxk − Ex,x(K)] , valid for each K ∈ K(S∗), show that lim E(K)x = x in the norm of H. In K∈K(S∗) particular, x belongs to the closure of S E(K)H and (viii) is verified. K∈K(S∗) (ix) Suppose x ∈ H satisfies K = supp(Ex) = supp(Ex,x) is compact. Since R 2 R 2 sb is continuous, |sb| dEx,x = |sb| dEx,x < ∞ and so x ∈ D(Eb(s)), for s ∈ S. S∗ K R R By [17], Theorem 13.24 (a) we have hEb(s)x, xi = sbdEx,x = sbdEx,x, for s ∈ S. S∗ K If αK is as in (2.3), then it follows that Z Z 2 |hEb(s)x, xi| 6 |sb| dEx,x 6 αK (s) 11dEx,x 6 αK (s)kxk , s ∈ S, K K and so Proposition 3.2 (iv) of Section 3 implies that x ∈ DαK ⊆ Dc. 226 P. Ressel and W.J. Ricker

T Conversely, suppose x ∈ Dα ⊆ D(Eb(s)) for some α ∈ A(S). By the s∈S 2 Cauchy-Schwarz inequality |hEb(s)x, xi| 6 α(s)kxk , for s ∈ S, which shows that ϕx : s 7→ hEb(s)x, xi is α-bounded on S. Moreover, ϕx is also positive deﬁnite since, for any ﬁnite subsets {c1, . . . , cn} ∈ C and {s1, . . . , sn} ⊆ S we have (by (iv) and (v)) that

X − X − X − cjckϕx(sk sj) = cjckhEb(sk sj)x, xi = cjckhEb(sk )Eb(sj)x, xi j,k j,k j,k n 2 X X = cjckhEb(sj)x, Eb(sk)xi = cjEb(sj)x 0. > j,k j=1

α By the Berg-Maserick theorem there is µx ∈ M+(S ) such that hEb(s)x, xi = R R ∗ sbdµx = sbdµx, for s ∈ S, and µx is unique in M+(S ). But, also hEb(s)x, xi = S∗ Sα R ∗ sbdEx,x ([17], Theorem 13.24 (a)), with Ex,x ∈ M+(S ) and so, by uniqueness, S∗ we conclude Ex,x = µx. In particular, supp(Ex) = supp(µx) is compact. This completes the proof of Proposition 2.1.

Remark 2.4. If σ(Eb(s)) is the spectrum of Eb(s) ([17], p. 346), then σ(Eb(s)) coincides with the E-essential range of sb ([17], Theorem 13.27 (c)); see [17], p. 303 ∗ for the deﬁnition of E-essential range. Since sb : S → C is continuous, it follows that the E-essential range of sb coincides with sb(supp(E)); the bar denotes closure S in C and supp(E) := supp(Ex,x). Accordingly, the canonical ∗-representation x∈H Eb associated to E has the property, that σ(Eb(s)) = sb(supp(E)), for s ∈ S. We now show property (v) in Deﬁnition 1.1 does not follow from (i)–(iv) in general.

Example 2.5. Let S be the set of all C-valued, Borel measurable functions on [0,1]. Under pointwise multiplication and pointwise (complex) conjugation of functions, S is a commutative, unital (with e = 11)semigroup with involution. Let 2 H = L ([0, 1]). For ϕ ∈ S define the operator Uϕ on D(Uϕ) = {h ∈ H : ϕh ∈ H} by Uϕ : h 7→ ϕh. It is routine to verify the map U : ϕ 7→ Uϕ, for ϕ ∈ S, has range in N (H) and satisfies (i)–(iv) of Definition 1.1. However, it fails (v) since Dc = {0}. T T To see this it suffices to show D(Uϕ) = {0}; see (1.2). So, let g ∈ D(Uϕ) ϕ∈S ϕ∈S T and suppose g 6= 0. Since also |g| ∈ D(Uϕ), as h ∈ D(Uϕ) iff |h| ∈ D(Uϕ), for ϕ∈S −1 all ϕ ∈ S, we may assume g > 0. Hence, A = g ((0, ∞)) has positive measure. Since Lebesgue measure λ on B([0, 1]) is non-atomic there are pairwise disjoint ∞ Borel sets A ⊆ A, each with λ(A ) > 0. Then ϕ = (1/g) P (λ(A ))−1/2χ n n 0 n An n=1 belongs to S and {ϕ0 > 0} ⊆ A. A direct calculation shows gϕ0 6∈ H, that is, g 6∈ D(Uϕ0 ) which is a contradiction. Hence, {Us}s∈S fails to be a ∗-representation because it does not satisfy (v) of Definition 1.1. An integral representation for semigroups 227

Let N 0 = N ∪ {0} be the additive semigroup of non-negative integers. As involution on N 0 we take the identity function. Let N 0 × N 0 be the commutative, unital (with e = (0, 0)) semigroup whose semigroup operation is given by (m, n) + (u, v) = (m + u, n + v) and whose involution is given by (m, n)− = (n, m). n Lemma 2.6. (i) If α ∈ A(N 0), there is λ > 0 with α(n) 6 λ , for n ∈ N 0. m+n (ii) If α ∈ A(N 0 × N 0), there is λ > 0 with α(m, n) 6 λ , for m, n ∈ N 0. Proof. (i) is obvious, with λ := α(1). For (ii) put λ := max{α(1, 0), α(0, 1)}. m n Then, for m, n ∈ N 0, we have α(m, n) 6 α(m, 0)α(0, n) 6 α(1, 0) α(0, 1) 6 λm+n. For the rest of this section we discuss Theorem 1.2 in relation to some related results in the literature. Suppose S is a locally compact full semigroup (see [11], [15]) and assume that S is also commutative and unital; this is not a requirement in [11], [15]. In [15], A.E. Nussbaum considers representations U : S → N (H) which take their values in the selfadjoint operators and satisfy:

(N1) Ust ⊆ UsUt, s, t ∈ S. T (N2) s 7→ hUsx, yi is continuous for each x ∈ D(Us) and y ∈ H. s∈S (N3) There is a countable set S0 ⊆ S with S0 ∩ {ts : t ∈ S}= 6 ∅, for s ∈ S.

Since each Us is selfadjoint we may take the identity function on S as in- ∗ T volution, in which case Us− = Us , for s ∈ S. Condition (N3) implies D(Us) s∈S is dense ([15], Theorem 5), and (N1) implies that UsUt = Ust for s, t ∈ S ([15], p. 134). A real character of S is a continuous homomorphism of S into R and Sb is the space of all real characters with the topology of uniform convergence on compact subsets of S ([15], pp. 136–137). The main result in [15], Theorem 6 states if U : S → N (H) takes its values in the selfadjoint operators and satisﬁes (N1)–(N3), R then there exists a spectral measure E : B(Sb) → L(H) such that Usx = sbdEx, Sb for s ∈ S and x ∈ D(Us). In [11], C. Ionescu Tulcea considers semigroup representations U : S → N (H), with S a locally compact full semigroup, which satisfy the following conditions:

(IT1) Ust ⊆ UsUt, s, t ∈ S. (IT2) Us,Ut commute, for s, t ∈ S, i.e. their resolutions of the identity commute. T (IT3) s 7→ hUsx, yi is continuous for each x ∈ D(Us) and y ∈ H. s∈S T T 0 (IT4) D(Us) = D(Us) for some countable set S ⊆ S. s∈S s∈S0 It is known that (IT2) and (N3) together imply (IT4) ([11], p. 106). Any continuous homomorphism ρ 6≡ 0 of S into C is called a character ([11], p. 97), and if S happens to have a continuous involution (which we assume henceforth), then ρ is also required to satisfy ρ(s−) = ρ(s) for all s ∈ S ([11], p. 98). Let Se be the space of all characters, equipped with a suitable topology ([11], Section 2), and β(Se) be the Stone-Cechˇ compactiﬁcation of Se. The main result in [11], Theorem 3 states if U : S → N (H) satisﬁes (IT1)–(IT4), then there is a unique selfadjoint, 228 P. Ressel and W.J. Ricker

Radon spectral measure E : B(β(Se)) → L(H) which is concentrated on Se (see R [11], p. 97) such that Usx = sbdEx, for x ∈ D(Us) and s ∈ S. A similar result Se occurs in [14], Theorem 1, where S is only required to be a locally compact space (i.e. not necessarily a semigroup). The hypotheses of Definition 1.1 are different to those of Nussbaum and Ionescu Tulcea. To begin with, S need not have a topology in our setting and so (N2) and (IT3) are not available. Moreover, the arguments in [11] and [15] rely on the topology of S, the continuity requirements (N2) and (IT3) (and of the involution in the case of [11]) and the representation techniques of abelian C∗- algebras of operators via their structure space. It is clear from Definition 1.1 and our assumptions on S that such methods are not available in our setting; they will be replaced by techniques from the theory of positive definite functions. Moreover, (iii) and (iv) of Definition 1.1, which are analogues of (N1)/(IT1), are typically weaker than (N1)/(IT1). Indeed, as seen by the next example, some rather simple semigroups of normal operators (which satisfy Definition 1.1) are excluded in [11] and [15] as they do not satisfy (N1)/(IT1).

Example 2.7. Let S = R with addition as the semigroup operation and the identity function as involution. Then S is a locally compact full semigroup 2 −st with involution. Let H = L ([0, ∞)). For s ∈ R let ρs(t) = e , for t ∈ [0, ∞). Define Us to be the operator with domain D(Us) = {h ∈ H : ρsh ∈ H} given by Us : h 7→ ρsh for h ∈ D(Us). Then D(Us) = H if s > 0; otherwise D(Us) is a proper dense subspace. For s ∈ S, the operator Us is selfadjoint. Moreover, U−sUs = I whereas UsU−s is the restriction of I to D(U−s), for s > 0. Of course, U−s+s = I for s ∈ S. So, (N1)/(IT1) is not satisfied. However, U is a ∗- representation as in Definition 1.1; conditions (i)–(iv) are easily checked. To check (n) (v) of Definition 1.1 let Us be the restriction of Us to the closed, Us-invariant (n) subspace H = {h ∈ H : χ(n,∞) h = 0 a.e.}, for n ∈ N. Define αn : S → [0, ∞) by (n) (n) (n) αn(s) = kUs kL(H(n)), for s ∈ S, after noting Us ∈ L(H ). Then αn ∈ A(S) (n) (n) (n) for n ∈ N. Fix n ∈ N. If h ∈ H , then kUshk = kUs hk 6 kUs kL(H(n))khk = ∞ (n) S (n) αn(s)khk, for s ∈ S, which shows h ∈ Dαn , i.e. H ⊆ Dαn . Since H is n=1 ∞ ∞ S (n) S dense and H ⊆ Dαn ⊆ Dc we have established (v). n=1 n=1 The following ∗-representation (in our sense) fails both (N3) and (IT4).

0 Example 2.8. Let S = CN be the semigroup of all functions ϕ : N 0 → C with multiplication of functions (defined pointwise) as the semigroup operation, in which case e = 11 is the unit, and with complex conjugation ϕ 7→ ϕ (defined 2 pointwise) as involution. Let H = ` (N 0). For ϕ ∈ S, define Uϕ to be the operator with domain D(Uϕ) = {ξ ∈ H : ϕξ ∈ H} given by Uϕ : ξ 7→ ϕξ for ξ ∈ D(Uϕ). Then U : S → N (H) defined by ϕ 7→ Uϕ has properties (i)–(iv) of Definition 1.1 and \ (2.4) D(Uϕ) = {ξ ∈ H : supp(ξ) is a finite subset of N 0} . ϕ∈S An integral representation for semigroups 229

To verify (v) of Definition 1.1 define, for n ∈ N 0, an element αn ∈ A(S) (n) by αn(ϕ) = max{|ϕ(k)| : 0 6 k 6 n}, for ϕ ∈ S. Fix n ∈ N 0 and let H be the closed, Uϕ-invariant subspace of H consisting of all ξ ∈ H with supp(ξ) ⊆ n (n) 2 2 P 2 2 {0, 1, . . . , n}. Then, for ξ ∈ H , we have kUϕξk = kϕξk = |ϕ(j)| |ξ(j)| 6 j=0 ∞ 2 2 (n) S (n) (αn(ϕ)) kξk , for ϕ ∈ S, which shows ξ ∈ Dαn , i.e. H ⊆ Dαn . Since H is n=0 ∞ ∞ S (n) S dense and H ⊆ Dαn ⊆ Dc we have established (v). So, U : S → N (H) n=0 n=0 is a ∗-representation. It is routine to verify (IT2). We noted above that (IT2) and (N3) together imply (IT4). So, if we show (IT4) fails, then also (N3) fails. Proceeding by T ∞ contradiction, let D = D(Uϕ) and suppose there is a sequence {ϕn}n=1 with ϕ∈S ∞ T D = D(Uϕn ). Since D(Uϕn ) = D(U|ϕn|) we may suppose ϕn > 0, for n ∈ N. n=1 n P Also, if 0 6 ϕ 6 ψ, then D(Uψ) ⊆ D(Uϕ). Accordingly, ψn = ϕj satisfies j=1 ∞ ∞ T T D(Uψn ) ⊆ D(Uϕn ), for n ∈ N, and so D ⊆ D(Uψn ) ⊆ D(Uϕn ) = D. Hence, n=1 n=1 we may suppose 0 6 ϕ1 6 ϕ2 6 ···. Also, ϕn 6 (11 ∨ ϕn) and so D ⊆ D(U11∨ϕn ) ⊆ ∞ ∞ T T D(Uϕn ) for n ∈ N which yields D ⊆ D(U11∨ϕn ) ⊆ D(Uϕn ) = D. So, we n=1 n=1 ∞ T may suppose 11 6 ϕ1 6 ϕ2 6 ··· and D = D(Uϕn ). Define ξ : N 0 → C by n=1 ∞ ∞ P 2 P −2 ξ(n) = 1/((n + 1)ϕn+1(n)), for n ∈ N 0, in which case |ξ(n)| 6 n < n=0 n=1 ∞. Hence, ξ ∈ H. Also, (ϕnξ)(k) = ϕn(k)/((k + 1)ϕk+1(k)) for k ∈ N 0 and −1 (ϕnξ)(k) 6 (k + 1) whenever k > (n − 1). Accordingly, ϕnξ ∈ H, for n ∈ N. ∞ T That is, ξ ∈ D(Uϕn ) = D which contradicts (2.4) as ξ does not have finite n=1 ∞ support. Hence, no such sequence {ϕn}n=1 ⊆ S can exist and so (IT4) is not satisfied. Remark 2.9. (a) In the notation of Example 2.8 define ϕ, ψ ∈ S by ϕ(n) = n, for n ∈ N 0, and ψ = χF where F = {0, 2, 4,...}. Define ξ ∈ H by ξ(0) = 0 and −1 ξ(n) = n χF c (n) for n > 1. Then ξ ∈ D(UϕUψ), but ξ 6∈ D(UψUϕ). This shows (iii) and (iv) of Definition 1.1 need not imply UtUs = UsUt for s, t ∈ S, but only that UtUs = UsUt. T (b) It was noted earlier that (N3) necessarily implies the density of D(Us) s∈S T in H. Example 2.8 shows that D(Us) can be dense without (N3) being satisfied. s∈S (c) If we take S to be R N 0 rather than CN0 in Example 2.8, then the ∗- representation {Us}s∈S consists of selfadjoint operators. In this setting (a) above implies that (N1) cannot hold since, as noted earlier, (N1) always implies that UsUt = UtUs for all s, t ∈ S. 230 P. Ressel and W.J. Ricker

(d) In Example 2.8 it is clear from (2.4) and the deﬁnition of Dαn , for n ∈ N, T that Dc = D(Uϕ). This is special to this example. To see this, let S = ϕ∈S 2 N 0 and H = L ([0, ∞)). Let T be the selfadjoint operator with D(T ) = {h ∈ H : ϕh ∈ H} given by T : h 7→ ϕh, for h ∈ D(T ), where ϕ is the identity n function on [0, ∞). Then U : S → N (H) given by Un = T , for n ∈ S, has n properties (i)–(iv) of Deﬁnition 1.1. For each λ > 0 let αλ(n) = λ , for n ∈ S, S in which case αλ ∈ A(S). Lemma 2.6 (i) implies Dc = Dαλ . For λ > 0, λ>0 let Hλ = h ∈ H : χ(λ,∞) h = 0 a.e. . Then Hλ is a closed subspace contained T n n in D(Un) and, if h ∈ Hλ, then kUnhk = kϕ hk 6 λ khk, for s ∈ S. This n∈S S shows Hλ ⊆ Dαλ and, since Hλ is dense, it follows that Dc is dense. So, U λ>0 T is a ∗-representation. To show the inclusion Dc ⊆ D(Un) is strict consider n∈S −t/2 T h(t) = e , for t ∈ [0, ∞). Then khk = 1 and h ∈ D(Un). Fix λ > 0 and n∈S n n suppose h ∈ Dαλ . Then kUnhk = kϕ hk 6 αλ(n)khk = λ , for n ∈ N 0. But, n 1/2 2n kϕ hk = [(2n)!] and it follows that (2n)! 6 λ , for n ∈ N 0, which is nonsense. Hence, no such λ > 0 exists and so h 6∈ Dc.

3. PROOF OF THEOREM 1.2

The main aim of this section is to establish Theorem 1.2. This will be achieved via a series of lemmata. The following fact is straight-forward to verify. Lemma 3.1. Let X be a Hausdorﬀ topological space. Then the space M(X) of all C-valued Radon measures on X is complete with respect to the total variation norm k| · k|.

We begin by recording some basic properties of the collection {Dα}α∈A(S) given via (1.2). At this stage it is not even known that the Dα are vector spaces! Proposition 3.2. Let S be a commutative, unital semigroup with involution T and U : S → N (H) be a ∗ -representation. Denote the subspace D(Us), of H, s∈S brieﬂy by DU . (i) Us(DU ) ⊆ DU , s ∈ S. (ii) For each s, t ∈ S and z ∈ DU we have ∗ (3.1) hUstz, zi = hUsUtz, zi = hUtz, Us zi = hUtz, Us− zi .

(iii) The function ϕz(s) := hUsz, zi, s ∈ S, is positive deﬁnite, for each z ∈ DU . (iv) For each α ∈ A(S) we have 2 (3.2) Dα = {x ∈ DU : |hUsx, xi| 6 α(s)kxk , for all s ∈ S}.

(v) Dα is a closed linear subspace of H, for each α ∈ A(S). (vi) For each α ∈ A(S) and s ∈ S we have Dα ⊆ D(Us) and Us(Dα) ⊆ Dα. In particular, Dc ⊆ D(Us) and Us(Dc) ⊆ Dc, for every s ∈ S. An integral representation for semigroups 231

Proof. (i) Fix s ∈ S. Given x ∈ DU we have to show y := Usx ∈ DU , i.e. y ∈ D(Ut) for t ∈ S. Since DU ⊆ D(Ust) ∩ D(Us) = D(UtUs), (see Deﬁnition 1.1 (iii)), we have

(3.3) x ∈ D(UtUs) = {h ∈ H : h ∈ D(Us) and Ush ∈ D(Ut)} , from which it follows that y ∈ D(Ut). Since t ∈ S is arbitrary we conclude that y ∈ DU . (ii) Since DU ⊆ D(Ust) ∩ D(Ut) = D(UsUt) with UsUt ⊆ Ust (see Deﬁni- tion 1.1 (iii)) we have (as z ∈ DU ) that

(3.4) hUstz, zi = hUsUtz, zi. ∗ But, also z ∈ DU ⊆ D(Us− ) = D(Us ) and so, by Definition 1.1 (ii) and the definition of adjoints, we have hUsw, zi = hw, U(s−)zi, for w ∈ D(Us). By part (i) we see w = Utz ∈ DU ⊆ D(Us) which, upon substitution into the previous ∗ identity, yields hUsUtz, zi = hUtz, Us zi = hUtz, U(s−)zi. Combining this identity with (3.4) gives (3.1). (iii) Now that (3.1) is known to be valid we can repeat the calculations (for ϕx) in the proof of Proposition 2.1 (ix) to deduce ϕz is positive definite. (iv) Fix α ∈ A(S). If x ∈ Dα, then it follows from (1.2), via the Cauchy- 2 Schwarz inequality, that |hUsx, x, i| 6 kUsxk · kxk 6 α(s)kxk , for s ∈ S. So, x belongs to the right-hand-side of (3.2). For x ∈ DU , (3.1) and Definition 1.1 (ii) imply that 2 ∗ 2 2 (3.5) hUsU(s−)x, xi = hU(s−)x, U(s−)xi = kU(s−)xk = kUs xk = kUsxk , for each s ∈ S, where the last equality in (3.5) follows from the fact that Us is normal ([17], Theorem 13.32). If, in addition, x belongs to the right-hand-side of (3.2), then − 2 − 2 2 2 |hUsU(s−)x, xi| 6 α(ss )kxk 6 α(s)α(s )kxk = (α(s)) kxk , s ∈ S.

Combining this inequality with (3.5) shows that kUsxk 6 α(s)kxk, for s ∈ S, i.e. x ∈ Dα. (v) Fix α ∈ A(S). We ﬁrst show that Dα is a vector subspace of H. It follows easily from (1.2) that λx ∈ Dα whenever λ ∈ C and x ∈ Dα. So, let x, y ∈ Dα. Using both (1.2) and (3.2) it is straightforward to check that

|hUs(x + y), (x + y)i| 6 |hUsx, xi| + |hUsx, yi| + |hUsy, xi| + |hUsy, yi| 2 6 α(s)(kxk + kyk) , for s ∈ S. This shows, with z = x + y, that ϕz in part (iii) is α-bounded. By (iii) the function ϕz is also positive definite and hence, |ϕz(s)| 6 ϕz(e)α(s) for s ∈ S 2 ([2], Chapter 4, Proposition 1.12). But, Definition 1.1 (i) implies ϕz(e) = kx + yk 2 and it follows |hUs(x + y), (x + y)i| 6 α(s)kx + yk , for s ∈ S, i.e. (x + y) ∈ Dα. So, Dα is indeed a subspace of H. ∞ To see Dα is closed, let {xn}n=1 ⊆ Dα converge to some x ∈ H. Since Dα is a subspace the differences (xn − xm) ∈ Dα, for all m, n ∈ N, and so (1.2) implies

(3.6) kUsxn − Usxmk = kUs(xn − xm)k 6 α(s)kxn − xmk, s ∈ S. 232 P. Ressel and W.J. Ricker

∞ Fix s0 ∈ S. Then (3.6) guarantees y(s0) ∈ H such that {Us0 xn}n=1 converges to y(s0). Since Dα ⊆ DU ⊆ D(Us0 ) and Us0 ∈ N (H) is a closed operator, it follows x ∈ D(Us0 ) and Us0 x = y(s0). But, s0 ∈ S is arbitrary and so x ∈ DU . Moreover, ∞ since {xn}n=1 ⊆ Dα it follows from (1.2) that kUs0 xnk 6 α(s0)kxnk, for n ∈ N. Letting n → ∞ shows that kUs0 xk 6 α(s0)kxk. Since s0 ∈ S is arbitrary it follows (from (1.2)) that x ∈ Dα. (vi) Fix α ∈ A(S). From the deﬁnition of Dα it is clear that Dα ⊆ DU ⊆ D(Us), for s ∈ S. Fix s ∈ S and x ∈ Dα. For each t ∈ S, it follows from (3.1) that

|hUt(Usx),Usxi| = |hUtsx, Usxi| 6 kUtsxk · kUsxk 6 α(st)kxk · α(s)kxk 6 Mα(t), 2 2 where M = (α(s)) kxk . This shows, with z = Usx, that the function ϕz of part (iii) is α-bounded. By (iii) it is also positive deﬁnite which, as noted above, 2 implies the inequality |hUtz, zi| = |ϕz(t)| 6 ϕz(e)α(t) = kzk α(t), for t ∈ S. Hence, z = Usx ∈ Dα. This establishes that Us(Dα) ⊆ Dα.

Remark 3.3. If there is at least one s0 ∈ S such that D(Us0 ) 6= H, then Dα 6= DU for every α ∈ A(S). For, suppose there is α ∈ A(S) with Dα = DU . Then DU is closed in H (cf. Proposition 3.2 (v)) and dense (as Dc ⊆ DU ), which forces DU = H. Since DU ⊆ D(Us0 ) it follows that D(Us0 ) = H, contrary to the assumption on Us0 . Proposition 3.4. ([16], Theorem 2) Let S be a commutative, unital semigroup with involution and U : S → L(H) be a map satisfying Ue = I and ∗ Ust− = UsUt , for all s, t ∈ S. Then there exists a unique selfadjoint, Radon spectral measure E : B(S∗) → L(H) which has compact support and satisfies R Us = sbdE, for s ∈ S. S∗ To begin the proof of Theorem 1.2 let U : S → N (H) be a ∗-representation. (α) Fix α ∈ A(S). By Proposition 3.2 (vi) the restriction Us := Us|Dα of Us to Dα ⊆ D(Us) is defined, for s ∈ S, as an operator from Dα into Dα. Moreover, (1.2) (α) (α) implies that Us ∈ L(Dα) since the inequalities kUs xk = kUsxk 6 α(s)kxk, for x ∈ Dα and s ∈ S, show that (α) (3.7) kUs kL(Dα) 6 α(s) < ∞, s ∈ S. Using the properties listed in Proposition 3.2 it is routine to verify that (α) (α) U : S → L(Dα) defined by s 7→ Us satisfies the assumptions of Proposition 3.4 in the Hilbert space Dα. Accordingly, by Proposition 3.4 there exists a unique (α) ∗ selfadjoint, Radon spectral measure E : B(S ) → L(Dα) which is compactly supported and satisfies Z (α) (α) (3.8) Us = sbdE , s ∈ S. S∗ An examination of the proof of Theorem 2 in [16] (i.e. of Proposition 3.4 above) shows supp(E(α)) ⊆ Sδ, where δ : S → [0, ∞) is the absolute value (α) δ(s) := kUs kL(Dα), for s ∈ S. But, (3.7) implies δ(s) 6 α(s), for s ∈ S, and so supp(E(α)) ⊆ Sα. An integral representation for semigroups 233

We note that α ∨ β ∈ A(S) whenever α, β ∈ A(S), where (α ∨ β)(s) := max{α(s), β(s)}, for s ∈ S. So, deﬁning 6 in A(S) by α 6 β iﬀ α(s) 6 β(s) for all s ∈ S, we have (A(S), 6) is a partially ordered set. It is also a directed set, since α 6 α ∨ β and β 6 α ∨ β whenever α, β ∈ A(S). It is a consequence of (1.2) that Dα ⊆ Dβ whenever α 6 β in A(S). Of course, Dα is a closed subspace of (β) Dβ. It is routine to check Dα is invariant for each operator Us , s ∈ S, and that (β) (α) Us |Dα = Us for each s ∈ S. Since {Dα}α∈A(S) is an upwards directed family S of subspaces it is clear that Dc = Dα is a vector subspace of H (contained α∈A(S) in DU , of course). Lemma 3.5. Let S be a commutative, unital semigroup with an involution and U : S → N (H) be a ∗-representation. (α) (β) (i) Let α 6 β in A(S). Then, for each x ∈ Dα ⊆ Dβ, we have Ex,x = Ex,x c ∗ ∗ c ∗ as elements of M+(S ) := M+(S ) ∩ M (S ). (α) (β) (ii) Given any α, β ∈ A(S), the equality Ex,x = Ex,x is valid as elements of c ∗ M+(S ), for every x ∈ Dα ∩ Dβ.

(α) (β) β Proof. (i) For x ∈ Dα ﬁxed, both Ex,x ,Ex,x ∈ M+(S ). So, it suﬃces to (α) ∧ (β) ∧ show their generalized Laplace transforms (Ex,x ) and (Ex,x) coincide. But, from (3.8) we see Z Z (α) ∧ (α) (α) (α) (3.9) (Ex,x ) (s) = sbdEx,x = sbdE x, x = hUs x, xi, S∗ S∗

(β) ∧ (β) for each s ∈ S. A similar calculation shows that (Ex,x) (s) = hUs x, xi, for each (β) (α) (α) ∧ (β) ∧ s ∈ S. Since Us |Dα = Us it follows that (Ex,x ) = (Ex,x) . (α) (α∨β) (ii) Since α 6 (α ∨ β) and β 6 (α ∨ β), part (i) implies that Ex,x = Ex,x (β) (α∨β) (α) (β) and Ex,x = Ex,x , whenever x ∈ Dα ∩ Dβ. In particular, Ex,x = Ex,x.

Let x ∈ Dc. Then x ∈ Dα for some α ∈ A(S). Lemma 3.5 ensures the ∗ compactly supported Radon measure µx : B(S ) → [0, ∞) given by µx(A) := (α) ∗ c ∗ Ex,x (A), for A ∈ B(S ), is well deﬁned. If x, y ∈ Dc, we deﬁne µx,y ∈ M (S ) by 1 µx,y = 4 (µx+y − µx−y + iµx+iy − iµx−iy). Choose α ∈ A(S) with both x, y ∈ Dα. Then (x ± y) ∈ Dα and (x ± iy) ∈ Dα. It follows from the selfadjointness of (α) (α) E (A) ∈ L(Dα) and the polarization identity that µx,y(A) = hE (A)x, yi, for ∗ (α) A ∈ B(S ). A calculation as in (3.9) shows µbx,y(s) = hUs x, yi, for s ∈ S. So, we have the following result.

(α) ∗ Lemma 3.6. Let x, y ∈ Dc. Then µx,y(A) = hE (A)x, yi, for A ∈ B(S ), and (α) (3.10) µbx,y(s) = hUs x, yi, s ∈ S, for every α ∈ A(S) with the property that both x, y ∈ Dα. An immediate consequence is the following fact. 234 P. Ressel and W.J. Ricker

Lemma 3.7. The map (x, y) 7→ µx,y is sesquilinear from Dc × Dc into M c(S∗).

Proof. Fix x, y ∈ Dc and λ ∈ C. Choose α ∈ A(S) with x, y ∈ Dα. It (α) (α) ∧ follows from (3.10) that µbλx,y(s) = hUs λx, yi = λhUs x, yi = (λµx,y) (s), for s ∈ S. Uniqueness of generalized Laplace transforms gives µλx,y = λµx,y. A similar calculation shows µx,λy = λµx,y. Fix x1, x2, y ∈ Dc. Choose α ∈ A(S) with x1, x2, y ∈ Dα. Again by (3.10) we have µ (s) = hU (α)(x + x ), yi = hU (α)x , yi + hU (α)x , yi = µ (s) + bx1+x2,y s 1 2 s 1 s 2 bx1,y µ (s) = (µ + µ )∧ (s), for s ∈ S. It follows that µ = µ + µ . bx2,y x1,y x2,y x1+x2,y x1,y x2,y A similar calculation shows µx,y1+y2 = µx,y1 + µx,y2 , whenever x, y1, y2 ∈ Dc. ∗ Fix B ∈ B(S ) and define ΨB : Dc×Dc → C by (x, y) 7→ µx,y(B). Lemma 3.7 shows ΨB is sesquilinear. Given {x1, . . . , xn} ⊆ Dc and {c1, . . . , cn} ⊆ C define c ∗ P n ν ∈ M (S ) by ν := cjckµxj ,xk . Choose α ∈ A(S) with {xj}j=1 ⊆ Dα. j,k Using (3.10) it follows from the relevant calculation concerning ν in the proof of (α) Theorem 2 in [16] (performed now for U : S → L(Dα)) that ν > 0. Again the relevant calculation concerning ΨB in the proof of Theorem 2 in [16] shows ΨB is a positive definite kernel and

(3.11) |ΨB(x, y)| 6 kxk · kyk, (x, y) ∈ Dc × Dc.

Hence, ΨB has a unique extension to a sesquilinear form ΨB : H × H → C. So, there is a unique operator E(B) ∈ L(H) with kE(B)k 6 1 and hE(B)x, yi = ΨB(x, y), for (x, y) ∈ H × H. Of course, if (x, y) ∈ Dc × Dc, then hE(B)x, yi = µx,y(B). Moreover, (α) (3.12) E(B)x = E (B)x, if x ∈ Dα and α ∈ A(S). The claim is that E : B 7→ E(B), for B ∈ B(S∗), is the required spectral ∗ ∗ (α) measure. If x, y ∈ Dc, then hE(S )x, yi = µx,y(S ) = µx,y(e) = hUe x, yi = b ∗ hx, yi, for any α ∈ A(S) with x, y ∈ Dα. Since Dc is dense it follows E(S ) = I. The relevant calculation in the proof of Theorem 2 in [16] shows, for ﬁxed B ∈ ∗ ∗ B(S ), that hE(B) x, yi = hE(B)x, yi, for x, y ∈ Dc, and again density of Dc implies E(B)∗ = E(B). ∗ ∞ Fix B ∈ B(S ). Let x ∈ H and y ∈ Dc. Choose a sequence {xn}n=1 ⊆ Dc such that kx − xnk → 0 as n → ∞. Then it follows from (3.11) that

|µxn,y(B) − µxm,y(B)| = |ΨB(xn − xm, y)| 6 kxn − xmk · kyk, m, n ∈ N. ∗ Since B ∈ B(S ) is arbitrary, we have k|µxn,y − µxm,yk| 6 4kxn − xmk · kyk, ∗ for m, n ∈ N. By Lemma 3.1 there is a Radon measure µex,y : B(S ) → C such that k|µxn,y − µx,yk| → 0. By continuity of each E(B) ∈ L(H) and the iden- e ∗ tities µxn,y(B) = ΨB(xn, y) = hE(B)xn, yi, for B ∈ B(S ) and n ∈ N, we see hE(B)xn, yi → hE(B)x, yi. But, also µxn,y(B) → µx,y(B) and so hE(B)x, yi = ∗ e µx,y(B), for B ∈ B(S ). This shows hE(·)x, yi is σ-additive whenever x ∈ H and e ∞ y ∈ Dc. Now ﬁx x, y ∈ H. Choose {yn}n=1 ⊆ Dc such that kyn − yk → 0. For B ∈ B(S∗) we have |µ (B) − µ (B)| = |Ψ (x, y − y )| kxk · ky − y k, m, n ∈ . ex,ym ex,yn B m n 6 m n N An integral representation for semigroups 235

The above argument can be repeated to deduce hE(·)x, yi is σ-additive. Hence, E : B(S∗) → L(H) is σ-additive for the weak (and so also strong) operator topology. Fix x ∈ Dc. Choose α ∈ A(S) with x ∈ Dα. By (3.12) and multiplicativity of E(α) we have E(A ∩ B)x = E(α)(A ∩ B)x = E(α)(A)E(α)(B)x = E(A)E(B)x, for all sets A, B ∈ B(S∗). The usual density argument yields E(A ∩ B) = E(A)E(B) in L(H). Accordingly, E is multiplicative. Hence, we have established that E is a selfadjoint spectral measure. It is Radon because Ex,x = µex,x is Radon, for x ∈ H. Moreover, if x ∈ Dc, then (α) Ex,x = Ex,x , for any α ∈ A(S) with x ∈ Dα, and so supp(Ex,x) = supp(Ex) = (α) supp(Ex,x ) is compact. Fix s ∈ S. Since sb is continuous it is Ex-integrable (still assuming x ∈ Dc) and Z Z (3.13) sbd(Ex), y = sbdEx,y, y ∈ H. S∗ S∗

But, if also y ∈ Dc, then it follows from Lemma 3.6 (after choosing another (α) α ∈ A(S), if necessary, such that both x, y ∈ Dα) and the identity Ex,y = Ex,y = µx,y that Z (α) (3.14) sbdEx,y = µbx,y(s) = hUs x, yi = hUsx, yi. S∗ R Since Dc is dense in H we conclude from (3.13) and (3.14) that Usx = sbd(Ex). S∗ So, (1.3) holds for all x ∈ Dc and s ∈ S. Suppose that F : B(S∗) → L(H) is another selfadjoint, Radon spectral measure such that supp(F x) is compact for each x ∈ Dc and Z 0 (1.3) Usx = sbdF x, x ∈ Dc, s ∈ S. S∗ 0 ∧ ∧ Then, for all x, y ∈ Dc, it follows from (1.3) and (1.3) that (Ex,y) = (Fx,y) . c ∗ ∗ Hence, Ex,y = Fx,y as elements of M (S ). So, for any A ∈ B(S ), we have hE(A)x, yi = hF (A)x, yi for all x, y ∈ Dc. By the usual density argument it follows that E(A) = F (A). Hence, we have established there exists a unique, selfadjoint Radon spectral ∗ ∗ measure E : B(S ) → L(H) such that supp(Ex) ⊆ S is compact, for x ∈ Dc, and Z (3.15) Usx = sbd(Ex), s ∈ S, x ∈ Dc. S∗ Lemma 3.8. Let U : S → N (H) be a ∗-representation and E : B(S∗) → L(H) be the unique selfadjoint, Radon spectral measure which satisﬁes (3.15). For each α ∈ A(S), let Qα ∈ L(H) be the orthogonal projection of H onto the closed subspace Dα. (i) lim Qα = I in the strong operator topology of L(H). a∈A(S) 236 P. Ressel and W.J. Ricker

(ii) For each α ∈ A(S) we have ∗ (3.16) QαE(A) = E(A)Qα,A ∈ B(S ). (iii) For each α ∈ A(S) and s ∈ S we have

(3.17) UsQαy = QαUsy, y ∈ D(Us).

Proof. (i) Fix x ∈ Dc. Choose α0 ∈ A(S) with x ∈ Dα0 , in which case Qα0 x = x. For α ∈ A(S) with α > α0 we have Dα0 ⊆ Dα and so Qαx = Qα0 x = x = Ix. This implies lim Qαx = Ix. Since Dc is dense and {Qα}α∈A(S) is α∈A(S) uniformly bounded it follows that lim Qα = I for the strong operator topology. α∈A(S) ∗ (ii) Fix α ∈ A(S) and A ∈ B(S ). By (3.12) we have E(A)Dα ⊆ Dα. So, given any y ∈ H we see that Qαy ∈ Dα and hence, E(A)Qαy ∈ Dα. Accordingly, QαE(A)Qαy = E(A)Qαy and, since y ∈ H is arbitrary, it follows that E(A)Qα = QαE(A)Qα. Taking adjoints gives QαE(A) = QαE(A)Qα and (3.16) follows. (iii) Fix α ∈ A(S) and s ∈ S. Suppose first that x ∈ Dc. Since Qαx ∈ Dα ⊆ Dc it follows from (3.15) and part (ii) that Z Z (3.18) UsQαx = sbd(E[Qαx]) = Qα sbd(Ex) = QαUsx, x ∈ Dc. S∗ S∗ ∗ Fix y ∈ D(Us). If x ∈ Dc, then hUsQαy, xi = hQαy, Us xi = hQαy, U(s−)xi; ∗ the first equality holds since Qαy ∈ Dα ⊆ D(Us) and x ∈ Dc ⊆ D(Us) = D(Us ), and the second holds since U is a ∗-representation. Moreover, hQαy, U(s−)xi = ∗ ∗ hy, QαU(s−)xi = hy, U(s−)Qαxi = hy, Us Qαxi; the first equality holds as Qα = Qα, the second holds by (3.18), and the third as U is a ∗-representation. But, ∗ hy, Us Qαxi = hUsy, Qαxi = hQαUsy, xi; the first equality holds by definition of ∗ adjoint operator (as y ∈ D(Us) and Qαx ∈ Dα ⊆ D(Us )), and the second holds ∗ since Qα = Qα. So, we have established hUsQαy, xi = hQαUsy, xi, for x ∈ Dc. Since Dc is dense in H, the identity (3.17) follows.

Let Eb : S → N (H) be the canonical ∗-representation associated to E as in Section 2. Recall, in this case, that Z 2 (3.19) D(Eb(s)) = x ∈ H : |sb| dEx,x < ∞ , s ∈ S. S∗ To complete the proof of Theorem 1.2 it suﬃces to show that

(3.20) Us = Eb(s), s ∈ S.

Indeed, (3.15) then extends from Dc to D(Us), for s ∈ S (see (2.2)), and the claim in Theorem 1.2 that supp(Ex) is compact iff x ∈ Dc follows from Propo- sition 2.1 (ix). Since both Eb(s) and Us are normal, to establish (3.20) it suffices to show Us ⊆ Eb(s), for s ∈ S ([17], Theorem 13.32). So, let y ∈ D(Us). Since supp(Ex,x) is compact whenever x ∈ Dc, it is clear from (3.19) that {Qαy}α∈A(S) ⊆ Dc ⊆ D(Eb(s)). Accordingly, the definition of Eb (see Section 2) implies Eb(s)Qαy = R sbd(E[Qαy]) = UsQαy, where the last equality follows from (3.15) as Qαy ∈ Dc S∗ An integral representation for semigroups 237 for α ∈ A(S). Moreover, Lemma 3.8 (iii) shows UsQαy = QαUsy and hence, Eb(s)Qαy = QαUsy for α ∈ A(S). Since lim Qαy = y and lim QαUsy = Usy α∈A(S) α∈A(S)

(see Lemma 3.8 (i)) we see that lim (Qαy, Eb(s)Qαy) = (y, Usy) in the graph of α∈A(S)

Eb(s). Then the closedness of Eb(s) implies y ∈ D(Eb(s)) and Eb(s)y = Usy. This establishes (3.20) and thereby (ﬁnally!) completes the proof of Theorem 1.2.

We conclude with a criterion useful for verifying (v) of Deﬁnition 1.1.

Definition 3.9. Let S be a commutative, unital semigroup with an involution and U be a map from S into N (H). Then U is called orthogonally decomposable if there exists an orthogonal family of selfadjoint projections {Pβ}β∈J ⊆ L(H) such that: P (i) Pβ = I, where the series is strong operator convergent. β∈J S T (ii) PβH ⊆ D(Us). β∈J s∈S (iii) PβUs ⊆ UsPβ, s ∈ S, β ∈ J. T Remark 3.10. Conditions (i) and (ii) imply that D(Us) is dense in H. s∈S If H is separable, then (i) implies that J is a countable set.

Proposition 3.11. Let S be a commutative, unital semigroup with involution and U : S → N (H) be an orthogonally decomposable map satisfying conditions (i)–(iv) of Deﬁnition 1.1. Then Dc is dense in H. In particular, U is a ∗-representation.

Proof. Let Hβ = PβH, for β ∈ J. Fix s ∈ S. By (ii) of Definition 3.9 we have Hβ ⊆ D(Us). Let z ∈ Hβ. Then Pβz = z and so (iii) of Definition 3.9 implies Usz = UsPβz = PβUsz ∈ Hβ. This shows UsHβ ⊆ Hβ, for s ∈ S and β ∈ J, and β β so we may consider the restrictions Us = Us|Hβ from Hβ into itself. Then Us is a linear, everywhere defined operator on the closed subspace Hβ. Since Us ∈ N (H) β is closed, it follows Us is also closed and hence, by the closed graph theorem, we β ∗ β β ∗ deduce Us ∈ L(Hβ). Using Us− = Us it follows that U − = (Us ) , for s ∈ S and Ts β ∈ J. If z ∈ Hβ, then (ii) of Definition 3.9 shows z ∈ D(Ut) and hence, by (iii) t∈S of Definition 1.1, we have z ∈ D(UsUt) = D(Ust) ∩ D(Ut) and Ustz = UsUtz, for β β β s, t ∈ S. Then Ustz = Ustz = UsUtz = Us Ut z. Since z is arbitrary we conclude β β β Ust = Us Ut , for s, t ∈ S and β ∈ J. β For β ∈ J, define αβ : S → [0, ∞) by αβ(s) := kUs kL(Hβ ). Using the β properties of s 7→ Us established above it follows αβ ∈ A(S). Fix β ∈ J. If T β β 2 x ∈ Hβ ⊆ D(Us), then |hUsx, xi| = |hUs x, xi| 6 kUs kL(Hβ ) · kxk , for s ∈ S, s∈S S S which shows x ∈ Dαβ . Hence, Hβ ⊆ Dαβ and so Hβ ⊆ Dc. Since Hβ is β∈J β∈J dense (cf. (i) and (ii) of Definition 3.9) it follows Dc is also dense. 238 P. Ressel and W.J. Ricker

Remark 3.12. (a) The ∗-representation U of Example 2.7 is orthogonally decomposable. It suffices to take J = N and, for each n ∈ N, define Pn ∈ L(H) to be the projection Pn : h 7→ χ[n−1,n] h, for h ∈ H. (b) The ∗-representation U of Example 2.8 is also orthogonally decomposable. It suffices to take J = N 0 and, for each n ∈ N 0, define Pn ∈ L(H) to be the ∞ projection Pn : ξ → hξ, enien, for ξ ∈ H, with {en}n=0 the standard basis of 2 H = ` (N 0). Example 3.13. Let S be the family of all locally bounded Borel functions ϕ : [0, ∞) → C. With respect to pointwise multiplication S is a commutative semigroup with unit e = 11. As involution take complex conjugation ϕ 7→ ϕ 2 (defined pointwise). Let H = L ([0, ∞)). For ϕ ∈ S, let Uϕ be the operator with domain D(Uϕ) = {h ∈ H : ϕh ∈ H} given by Uϕ : h 7→ ϕh, for h ∈ H. Then U : S → N (H) so defined has properties (i)–(iv) of Definition 1.1. Note that Cc([0, ∞)) ⊆ DU and so DU is dense. It is routine to check the projections {Pn}n∈N ⊆ L(H) given in Remark 3.12 (a) form an orthogonal decomposition for U. By Proposition 3.11, U : S → N (H) is a ∗-representation.

4. THE SPECTRAL THEOREM

An examination of the proof of Theorem 1.2 shows it did not use the fact that a normal or selfadjoint operator has a spectral decomposition via its resolution of the identity. In fact, we now deduce the spectral theorem as a consequence of Theorem 1.2. Let T ∈ N (H) be selfadjoint. If S = N 0 is the semigroup of Lemma 2.6, then n it is routine to check that Un = T , for n ∈ N 0, satisfies (i)–(iv) of Definition 1.1. n Moreover, if αλ ∈ A(S) is defined by αλ(n) = λ , n ∈ N 0, for each λ > 0, then (1.2) implies ∞ \ n n n Dαλ = x ∈ D(T ): kT xk 6 λ kxk for all n ∈ N 0 . n=0 S Lemma 2.6 implies that Dc = Dαλ . Since Dαλ = F (T, λ), where F (T, λ) λ>0 is the space introduced in [13], p. 86, Lemma 4 of [13] shows that Dc is dense. Hence, U : N 0 → N (H) given by n 7→ Un, n ∈ N 0, is indeed a ∗-representation in our sense. By Theorem 1.2 there is a unique selfadjoint Radon spectral measure ∗ n R n E : B(S ) → L(H) such that Unx = T x = nb dEx, for n ∈ N 0 and x ∈ D(T ). S∗ But, S∗ can be identified with R via the isomorphism ξ 7→ ξe, where ξe(n) = ξn, n R n n ∈ N 0, for ξ ∈ R ([2], p. 115), and so T x = ξ dE(ξ)x, for n ∈ N 0 and R x ∈ D(T n). In particular, the choice n = 1 yields T x = R ξ dE(ξ)x, for x ∈ D(T ). R Remark 2.4 and (3.20), applied to U : N 0 → N (H), show that σ(T ) = supp(E) and so T x = R ξ dE(ξ)x, for x ∈ D(T ); this is the spectral theorem for T . σ(T ) Suppose T ∈ N (H) is normal. Let S = N 0 × N 0 be the commutative, unital semigroup with involution (m, n) 7→ (n, m), as defined in Lemma 2.6. Define An integral representation for semigroups 239

m ∗ n 0 U : S → N (H) by U(m,n) = T (T ) , where T := I. It is routine to verify U T satisfies (i)–(iv) of Definition 1.1. Let DU = D(U(m,n)); using the definition (m,n) of the domain of products (and powers) of unbounded operators it follows DU = ∞ T D((T ∗T )n). Define n=1 m ∗ n m+n 2 Da = {x ∈ DU : |hT (T ) x, xi| 6 a kxk for all (m, n) ∈ S}, for each a > 0 and, for each b > 0, define ∗ k k 2 Cb = {x ∈ DU : |h(T T ) x, xi| 6 b kxk for all k ∈ N 0}.

Lemma 4.1. Cb = Db1/2 , for each b > 0.

Proof. Fix x ∈ Cb. Suppose (m, n) ∈ S and m + n is even. Using the property kRxk = kR∗xk for x ∈ D(R) = D(R∗), whenever R ∈ N (H), it follows that m ∗ n ∗ (m+n)/2 (m+n)/2 1/2 m+n (4.1) kT (T ) xk = k(T T ) xk 6 b kxk = (b ) kxk. m ∗ n Deﬁne the positive deﬁnite function ϕx : S → C by ϕx(m, n) = hT (T ) x, xi, for (m, n) ∈ S. By the Cauchy-Schwarz inequality and (4.1)

1/2 m+n 2 (4.2) |ϕx(m, n)| 6 (b ) kxk , m + n even. − 2 − − Since ϕx is positive deﬁnite, |ϕx(s + t )| 6 ϕx(s + s )ϕx(t + t ), for s, t ∈ S ([2], Chapter 4). Hence, with (m, n) = s and (0, 0) = t, we have 2 2 |ϕx(m, n)| = |ϕx((m, n) + (0, 0))| 6 ϕx((m, n) + (n, m))ϕx(0, 0) 2 1/2 (2m+2n) 4 = ϕx(m + n, m + n)kxk 6 (b ) kxk ; the last inequality follows from (4.2). Since s = (m, n) ∈ S is arbitrary we see (via Proposition 3.2 (iv)) that x ∈ Db1/2 . The reverse containment Db1/2 ⊆ Cb is obvious. S S S Lemma 4.1 implies Cb = Da = Dc, where Dc = Da follows from b>0 a>0 a>0 Lemma 2.6 (ii). But, the above consideration for selfadjoint operators implies S ∗ Cb is dense (as T T is selfadjoint) and hence, Dc is dense. Accordingly, U is b>0 a ∗-representation. By Theorem 1.2 there is a unique selfadjoint Radon spectral measure E : B(S∗) → L(H) with T m(T ∗)nx = R (m, n)∧ dEx, for (m, n) ∈ S and S∗ x ∈ D(T m(T ∗)n). But, S∗ can be identiﬁed with C via the isomorphism z 7→ z, for m n m ∗ n R m n e z ∈ C, where ze(m, n) = z z , (m, n) ∈ S, and so T (T ) x = z z dE(z)x for C (m, n) ∈ S and x ∈ D(T m(T ∗)n). In particular, the choice (m, n) = (1, 0) yields T x = R z dE(z)x, for x ∈ D(T ). Again Remark 2.4 and (3.20), applied this time C R to U : N 0 × N 0 → N (H), show σ(T ) = supp(E) and so T x = z dE(z)x, for σ(T ) x ∈ D(T ); this is the spectral theorem for T . 240 P. Ressel and W.J. Ricker

5. PERFECT SEMIGROUPS

A commutative semigroup S (still unital and with an involution) is called perfect if every positive deﬁnite function on it is the generalized Laplace transform of a unique positive Radon measure on S∗, ([2], p. 203). Examples are ﬁnite semigroups, idempotent semigroups, abelian groups with the involution s− = s−1, but also (and somewhat surprising) the semigroups (Q+, +) and (Q, +), with the identity function as involution! Finite products, countable direct sums, and homo- morphic images of perfect semigroups are again perfect ([2], Chapter 6, Section 5). We shall see that for perfect semigroups S an analogue of Theorem 1.2 holds under considerably weaker assumptions. Let D ⊆ H be any subspace of a Hilbert space H, and denote by L(D) the set of all linear operators T : D → D, without any continuity assumptions. We can and do assume that D = H in the sequel. Let T ∈ L(D). If, for every y ∈ D there exists Ry ∈ D such that hT x, yi = hx, Ryi, for x ∈ D, then Ry is unique with this property and R ∈ L(D). In this case we write R = T #. If T # = T , the operator T is called symmetric. A map U : S → L(D) satisfying Ue = I will be called a ∗-homomorphism if # # Us exists for each s ∈ S and Us = Us− , and if Ust = UsUt for all s, t ∈ S (in the usual sense of composing maps). Theorem 5.1. Let D be a dense linear subspace of a Hilbert space H, let S be a perfect, commutative, unital semigroup with involution and let U : S → L(D) be a ∗-homomorphism. Then there exists a unique selfadjoint Radon spectral measure E : B(S∗) → L(H) such that Z Usx = sbdEx, x ∈ D, s ∈ S. S∗

Proof. Again we use the functions ϕx(s) := hUsx, xi, for s ∈ S and x ∈ D, n n 2 which are positive definite since P c c ϕ (s s−) = P c U x . By perfect- j k x j k j sj j,k=1 j=1 ∗ 1 ness of S there is a unique Radon measure µx > 0 on S such that sb ∈ L (µx), for s ∈ S, and ϕx = µx ([2], Definition 6.5.1). Repeating partly the proof of Theorem 2 b 1 in [16] we define complex Radon measures µx,y := 4 (µx+y −µx−y +iµx+iy −iµx−iy), leading to µx,y(s) = hUsx, yi for x, y ∈ D and s ∈ S (note µx,x = µx). The kernel b ∗ from D × D to C given by (x, y) 7→ µx,y(B), for each fixed B ∈ B(S ), turns out again to be positive (semi-)definite, and is a bounded sesquilinear form since

1/2 ∗ ∗ 1/2 |µx,y(B)| 6 [µx,x(B)µy,y(B)] 6 [µx,x(S )µy,y(S )] 1/2 = [ϕx(0)ϕy(0)] = kxk · kyk.

So, there is a unique operator E(B) ∈ L(H) such that hE(B)x, yi = µex,y(B), for x, y ∈ H, where each µex,y is a Radon measure and µex,y = µx,y whenever x, y ∈ D. The same calculations and arguments used in the proof of Theo- rem 2 of [16] lead to the fact that E : B(S∗) → L(H) is actually a selfadjoint, Radon spectral measure; here we make crucial use of the injectivity of the generalized Laplace transformation µ 7→ µb on the complex linear space generated by An integral representation for semigroups 241

∗ 1 µ ∈ M+(S ): s ∈ L (µ) for all s ∈ S ([2], Proposition 6.5.2). Now Ex,y = µx,y b R R whenever x, y ∈ D and so hUsx, yi = µbx,y(s) = sbdµx,y = sbdEx,y, for x, y ∈ D S∗ S∗ R and s ∈ S. On the other hand, the vector integral sbdEx exists in H since S∗ 2 − 1 2 2 |s| = (ss ) ∈ L (µx), i.e. s ∈ L (µx) = L (Ex,x) for x ∈ D. Therefore Rb bR b R sbdEx,y = h sbdEx, yi for x, y ∈ D and so indeed Usx = sbdEx for x ∈ D S∗ S∗ S∗ and s ∈ S. Remark 5.2. The above proof shows D ⊆ D(Eb(s)) for s ∈ S, where Eb : S → N (H) is the canonical ∗-representation associated to E, and Eb(s)|D = Us. Hence, D is contained in T D(Eb(s)) and invariant under Eb(s), for each s ∈ S. s∈S As mentioned in the introduction, the conclusion of Theorem 5.1 does not hold, in general, for non-perfect semigroups. In [4], Proposition 2 a result with some similarity to Theorem 5.1 is shown for “operator semiperfect” semigroups, with L(D) replaced by the set of sesquilinear forms on some vector space. The relation between semiperfect (a weakened version of “perfect”) and operator semiperfect semigroups is only partly clarified; for finitely generated semigroups the two notions coincide ([4], Theorem 1). As further related work we mention [18] where the author studies “∗-representations” U : A → L(D) defined on an algebra A with involution, rather than on the more general notion of a semigroup. Let us now consider the particular semigroup S = (Q+, +) of non-negative rational numbers which is perfect ([2], Proposition 6.5.6), but not finitely generated. ∗ st Its dual S can be identified with ([−∞, ∞), +) via ρt(s) = e and ρ−∞ = χ{0} for t ∈ R, s ∈ Q+. On Q+ there is no involution other than the identity. Hence, a ∗-homomorphism U : Q+ → L(D) takes its values in the symmetric operators on D. Corollary 5.3. (i) Let D be a dense linear subspace of a Hilbert space H, and let U : Q+ → L(D) be a ∗-homomorphism. Then there is a unique selfadjoint Radon spectral measure E : B([−∞, ∞)) → L(H) such that Z st Usx = χ{0} (s) · E({−∞})x + e dE(t)x, s ∈ Q+, x ∈ D.

(ii) Weak continuity of U, i.e. continuity of s 7→ hUsx, yi, s ∈ Q+, for each x, y ∈ D, is equivalent with E(R) = I (i.e. E({−∞}) = 0). (iii) Let U : R+ → L(D) be a weakly continuous ∗-homomorphism. Then there is a unique selfadjoint Radon spectral measure E : B(R) → L(H) such that Z st Usx = e dE(t)x, s ∈ R+, x ∈ D.

R Proof. Part (i) is an immediate consequence of Theorem 5.1. From Z 2 st hUsx, xi = χ{0} (s) · kE({−∞})xk + e dEx,x(t), s ∈ Q+, x ∈ D,

R 242 P. Ressel and W.J. Ricker one easily deduces (ii). To see the last part we note that the restriction of U to R st Q+ has (by (i)) the representation Usx = e dE(t)x, for s ∈ Q+ and x ∈ D, R R st implying hUsx, yi = e dEx,y(t), for s ∈ Q+ and x, y ∈ D. Since both sides of R this equation are continuous functions of s on R+ (for x, y ∈ D) and agree on Q+, they must agree on R+. The semigroup (Q, +) of all rational numbers (with the identity function as ∗ ∼ involution!) is also perfect ([2], Proposition 6.5.10), and Q = (R, +) via ρt(s) = est. In particular, all characters on (Q, +) are continuous (a rare exception), and so every ∗-homomorphism U :(Q, +) → L(D) has a unique representation Z st Usx = e dE(t)x, s ∈ Q, x ∈ D,

R as does every weakly continuous ∗-homomorphism U :(R, +) → L(D). Corre- k k k k sponding results hold likewise for the semigroups Q+, Q , R+ and R since, as already mentioned, ﬁnite products of perfect semigroups are again perfect ([2], Theorem 6.5.4).

Acknowledgements. The support of the German Academic Exchange Scheme (DAAD) is gratefully acknowledged.

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P. RESSEL W.J. RICKER Math.-Geogr. Fakultät School of Mathematics Katholische UniversitätEichstätt University of New South Wales D–85071 Eichstätt Sydney, N.S.W. 2052 GERMANY AUSTRALIA

Current address:

Math.-Geogr. Fakultät Katholische UniversitätEichstätt D–85071 Eichstätt GERMANY

Received August 18, 1999; revised May 24, 2000.