normed algebra under a locally convex algebra topology

Contemp. Math. 427 (2007), 155–166.

On the completion of a C∗ -normed algebra under a locally convex algebra topology M. Fragoulopoulou, A. Inoue, and K.-D. K¨ ursten This paper is dedicated to the memory of Gerd Lassner Abstract. Structure and representations of a locally convex (quasi) ∗-algebra constructed as the completion of a C∗ -normed algebra under a locally convex algebra topology are discussed.

1. Introduction A mapping p of a ∗-subalgebra D(p) of a ∗-algebra A into R+ = [0, ∞) is said to be an unbounded C ∗ -(semi)norm if it is a C ∗ -(semi)norm on D(p). C∗ seminorms and unbounded C ∗ -seminorms on ∗-algebras are useful in many mathematical and physical subjects, such as Banach ∗-algebras and locally convex ∗algebras, differential structure of C∗ -algebras, moment problem, quantum field theory ([1, 2, 14, 16, 18, 29]). Since the investigation of unbounded C ∗ -seminorms seems to be far from being complete, we have tried to study them systematically and to apply those studies to locally convex ∗-algebras ([4, 5, 9, 10, 11, 12, 13, 22]). Suppose that A0 [∥ · ∥0 ] is a C∗ -normed algebra, i.e., A0 is a ∗ -algebra endowed with the C∗ -norm ∥ · ∥0 . Let τ be a topology on A0 such that A0 [τ ] is a locally convex ∗-algebra with topology τ . We assume that the norm topology and the topology τ are compatible in the sense that each net which is a Cauchy net in both of these topologies and converges in one of them, converges also in the other f0 [∥ · ∥0 ] and A f0 [τ ], one. We denote the completions of A0 [∥ · ∥0 ] and A0 [τ ] by A respectively. It is natural to consider the following cases: I. τ ≽ ∥ · ∥0 (the topology defined by the norm ∥ · ∥0 ). Then the identity map of A0 [τ ] onto A0 [∥ · ∥0 ] extends to a continuous ∗ -linear f0 [τ ] −→ A f0 [∥ · ∥0 ]. Since the topologies are assumed to be compatible, map: A f0 [τ ] into the this mapping is injective and we can regard it as an embedding of A ∗ ∗ f C -algebra A0 [∥ · ∥0 ]. The studies of C -normed algebras in this case are related to the differential structure of C∗ -algebras [13, 14]. II. τ ≼ ∥ · ∥0 . Then the identity map of A0 [∥·∥0 ] onto A0 [τ ] extends to an injective continuous f0 [∥ · ∥0 ] into A f0 [τ ] which will be considered as an ∗-linear map of the C∗ -algebra A f f f embedding A0 [∥ · ∥0 ] ⊂ A0 [τ ]. Hence, A0 [τ ] is the completion of the C∗ -algebra f0 [∥ · ∥0 ] with respect to the topology τ . A 2000 Mathematics Subject Classification. Primary 46K10; Secondary 47L60. Key words and phrases. GB∗ -algebra, unbounded C∗ -seminorm, quasi ∗-algebra, unbounded ∗-representation. The first author thankfully acknowledges partial support of this research by the Special Research Account: Grant Nr 70/4/5645, University of Athens. 1

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In this paper we shall consider Case II. In Section 2, we assume that the multiplication on A0 is jointly continuous with f0 [τ ] is a complete locally convex ∗respect to the topology τ , which implies that A ∗ f algebra containing the C -algebra A0 [∥ · ∥0 ] as a dense subalgebra. Upon replacing f0 [∥ · ∥0 ], we can assume that A0 [∥ · ∥0 ] itself is a C ∗ -algebra. It turns A0 [∥ · ∥0 ] by A f0 [τ ] is always a GB∗ -algebra. As a consequence, results out that in this case A f0 [τ ] is isomorphic to a ∗of Allan [2] show that whenever A0 is commutative, A ∗ algebra of C -valued continuous functions on a compact space, where C∗ stands for the extended complex plane C ∪ {∞} in its usual topology as the one-point compactification of C. Following [16], a set F of C∗ -valued continuous functions on a topological space Ω is said to be a ∗-algebra of C∗ -valued continuous functions if each f ∈ F takes the value ∞ on at most a nowhere dense subset of Ω and if F is a ∗-algebra under the operations αf, f + g, f g, f ∗ (= f ), consisting of defining the result of each operation pointwise on the dense subset of Ω where all the values involved are finite, and extending to obtain C∗ -valued continuous functions on Ω. f0 [τ ] is isomorIn the noncommutative case, results of Dixon [16] imply that A ∗ phic to an unbounded operator algebra, called extended C -algebra. When the multiplication on A0 is not jointly continuous with respect to τ , f0 [τ ] is not necessarily a locally convex ∗-algebra, but it has the structure of a A quasi ∗-algebra (cf. [3]). This quasi ∗-algebra will be investigated in Section 3. In f0 [τ ] will be characterized. particular, the existence of faithful ∗-representations of A 2. Completion with respect to a topology making the multiplication jointly continuous Throughout this section, A0 [∥ · ∥0 ] denotes a C∗ -algebra with identity 1 and τ denotes a topology on A0 , weaker than the norm topology, such that A0 [τ ] is a locally convex ∗-algebra with jointly continuous multiplication. Let us repeat the notion of a GB∗ -algebra introduced by G.R. Allan [2], which generalizes the concept of a C∗ -algebra. Given a locally convex ∗-algebra A[τ ] with identity 1 , let B∗ denote the collection of all closed, bounded, absolutely convex subsets B of A[τ ] satisfying 1 ∈ B, B∗ = B and B2 ⊂ B. For each B ∈ B∗ , the linear span A[B] of B is a normed ∗-algebra under the Minkowski functional ∥ · ∥B of B. If A[B] is complete for each B ∈ B∗ , then A[τ ] is said to be pseudo-complete. Note that every sequentially complete locally convex ∗-algebra with identity is pseudo-complete. A pseudo-complete locally convex ∗-algebra A[τ ] with identity, such that B∗ has a greatest member B0 and (1 + x∗ x)−1 ∈ A[B0 ] for each x ∈ A, is said to be a GB∗ -algebra over B0 . In this case, A[B0 ] is a C∗ -algebra and ∥ · ∥B0 is an unbounded C∗ -norm of A[τ ]. Let Bτ denote the τ -closure of the unit ball U(A0 ) ≡ {x ∈ A0 ; ∥x∥0 ≤ 1} of the C∗ -algebra A0 [∥ · ∥0 ]. The set Bτ plays an important role in this paper. Since the multiplication of A0 [τ ] is jointly continuous, Bτ ∈ B∗ . Furthermore, following f0 [τ ], the arguments of the proof of [4], Lemma 2.1, we conclude that for each x ∈ A the elements (2.1)

(1 + x∗ x)−1 , x(1 + x∗ x)−1 , and (1 + x∗ x)−1 x exist and belong to Bτ .

Indeed: Let {xα } be a net in A0 such that τ -lim xα = x. Since A0 is a C ∗ -algebra, α

(1 + x∗α xα )−1 ∈ U(A0 ) for each index α. Moreover, for any τ -continuous seminorm

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p there exist γ > 0 and a τ -continuous seminorm q such that p((1 + x∗α xα )−1 − (1 + x∗β xβ )−1 ) = p((1 + x∗α xα )−1 (x∗β xβ − x∗α xα )(1 + x∗β xβ )−1 ) ≤ q((1 + x∗α xα )−1 )q((1 + x∗β xβ )−1 )q(x∗β xβ − x∗α xα ) ≤ γ∥(1 + x∗α xα )−1 ∥0 ∥(1 + x∗β xβ )−1 ∥0 q(x∗β xβ − x∗α xα ) ≤ γq(x∗β xβ − x∗α xα ). f0 [τ ] and y ≡ lim(1 + x∗α xα )−1 exists in Thus {(1 + x∗α xα )−1 } is a Cauchy net in A α

f0 [τ ] and belongs to Bτ . Taking the limit in A

1 = (1 + x∗α xα )(1 + x∗α xα )−1 = (1 + x∗α xα )−1 (1 + x∗α xα ), we obtain y = (1 + x∗ x)−1 . In a similar way it follows that x(1 + x∗ x)−1 ∈ Bτ and (1 + x∗ x)−1 x ∈ Bτ . Hence, A[Bτ ] is a symmetric Banach ∗-algebra. The following theorem states, in particular, that A[Bτ ] is a C∗ -algebra. Thus it solves some questions raised in [4]. f0 [τ ] is a GB∗ -algebra over Bτ . Theorem 2.1. A Proof. First of all, we show that A[Bτ ] is a C∗ -algebra. Let A be the C∗ algebra of all norm bounded nets {xα } with values in the C∗ -algebra A0 , where the index set is given as a basis of neighborhoods of zero for the topology τ and the norm is the supremum norm, i.e., ∥{xα }∥∞ ≡ supα ∥xα ∥0 . We put Ac = {{xα } ∈ A; {xα } is a τ -Cauchy net}, A0 = {{xα } ∈ A; τ - lim xα = 0}. α

∗

It follows that Ac is a C -subalgebra of the C∗ -algebra A[∥ · ∥∞ ] and that A0 is a closed ∗-ideal of Ac . Note that each a ∈ Bτ may be written as a = τ -limα xα for some {xα } ∈ Ac , which implies that the mapping Ac ∋ {xα } → τ -limα xα ∈ A[Bτ ] maps the unit ball of Ac onto Bτ . Since this mapping is also a ∗ -homomorphism with kernel A0 , it induces an isometric ∗ -isomorphism of the quotient C∗ -algebra Ac /A0 onto A[Bτ ]. Hence, A[Bτ ] is a C∗ -algebra. We show that Bτ is the greatest member in B∗ . Take an arbitrary B ∈ B∗ and ∗ f0 [τ ] containing h. h = h ∈ B. Let C be a maximal commutative ∗-subalgebra of A Then C is complete with respect to the topology induced by τ . We denote by B∗C the collection of all closed, bounded, absolutely convex subsets B1 of C satisfying 1 ∈ B1 , B∗1 = B1 and B21 ⊂ B1 . Then B∗C = {B2 ∩ C; B2 ∈ B∗ }. We show that B ∩ C ⊂ Bτ ∩ C. Since C is commutative and complete, it follows from [2], Theorem 2.10 that B∗C is directed, so there exists B1 ∈ B∗C such that (B∩C)∪(Bτ ∩C) ⊂ B1 . Furthermore, since the C∗ -algebra A[Bτ ∩C] = A[Bτ ]∩C is contained in the Banach ∗-algebra A[B1 ], it follows from [28], Proposition I.5.3 that ∥x∥Bτ = ∥x∥Bτ ∩C ≤ ∥x∥B1 ,

∀x ∈ A[Bτ ] ∩ C.

On the other hand, Bτ ∩ C ⊂ B1 , therefore ∥x∥B1 ≤ ∥x∥Bτ ∩C = ∥x∥Bτ ,

∀x ∈ A[Bτ ] ∩ C.

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Thus we have ∥x∥B1 = ∥x∥Bτ ,

(2.2)

∀x ∈ A[Bτ ] ∩ C.

Next we show that Bτ ∩ C = B1 . Take an arbitrary x ∈ A[B1 ] and n ∈ N. By (2.1), 1 x(1 + x∗ x)−1 ∈ A[Bτ ]. n It is easily shown that {a, (1 + b∗ b)−1 ; a, b ∈ C} is commutative, so that by the maximality of C, {(1 + b∗ b)−1 ; b ∈ C} ⊂ C. Hence, x(1 +

1 ∗ −1 x x) ∈ A[Bτ ] ∩ C. n

The estimate 1 ∗ −1 1 1 x x) − x∥B1 = ∥xx∗ x(1 + x∗ x)−1 ∥B1 n n n 1 1 ∗ ≤ ∥xx x∥B1 ∥(1 + x∗ x)−1 ∥B1 n n 1 1 ∗ = ∥xx x∥B1 ∥(1 + x∗ x)−1 ∥Bτ n n 1 ∗ ≤ ∥xx x∥B1 n implies now that A[Bτ ]∩C is ∥·∥B1 -dense in A[B1 ]. Hence, A[Bτ ]∩C = A[C ∩Bτ ] = A[B1 ]. Thus from (2.2) we get that Bτ ∩ C = B1 . This implies that h ∈ B ∩ C ⊂ Bτ ∩ C. Thus we have shown that h ∈ Bτ for each h = h∗ ∈ B. Consequently, A[B] ⊂ A[Bτ ] and ∥x∥2Bτ = ∥x∗ x∥Bτ ≤ 1 for each x ∈ B. Hence B ⊂ Bτ and Bτ is the greatest member in B∗ . This completes the proof. ∥x(1 +

The C∗ -normed algebra A0 [∥ · ∥0 ] that determines the locally convex ∗-algebra Ae0 [τ ] is not unique. For this reason, we denote by C ∗ (A0 , τ ) the set of all C∗ -normed algebras A[∥ · ∥] such that A0 ⊂ A ⊂ Ae0 [τ ], τ ≼ ∥ · ∥ and ∥x∥ = ∥x∥0 , ∀x ∈ A0 . Then C ∗ (A0 , τ ) is an ordered set with the order: A1 [∥ · ∥1 ] ≼ A2 [∥ · ∥2 ] iff A1 ⊂ A2 and ∥x∥1 = ∥x∥2 , ∀ x ∈ A1 . Theorem 2.1 shows that A[Bτ ] is the largest element of C ∗ (A0 , τ ). It implies also the following result related to [8, 23, 24]. Corollary 2.2. The following statements are equivalent: (i) Ae0 [τ ] is a GB∗ -algebra over U(A0 ). (ii) U(A0 ) is τ -closed. We consider now the commutative case in more detail. Let A0 [∥ · ∥0 ] be a f0 [τ ] is a commutative commutative C∗ -algebra with identity 1 . By Theorem 2.1, A ∗ GB -algebra over Bτ , and so it is isomorphic to a ∗-algebra F(Ω) of C∗ -valued continuous functions on a compact space Ω. Here we briefly state some facts concerning this isomorphism and refer to [2] for more details. Let M (A[Bτ ]) denote the set of all non-zero multiplicative linear functionals on the C∗ -algebra A[Bτ ], in its usual weak∗ topology σ(M (A[Bτ ]), A[Bτ ]). Then

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f0 [τ ], we it is a non-empty compact Hausdorff space. For φ ∈ M (A[Bτ ]) and a ∈ A put { φ(a(1 +a∗ a)−1 ) if φ((1 + a∗ a)−1 ) ̸= 0 ∗ −1 ′ φ (a) = φ((1 +a a) ) ∞ if φ((1 + a∗ a)−1 ) = 0. f0 [τ ] such that Then φ′ is a C∗ -valued function on A (2.3)

f0 [τ ], {φ ∈ M (A[Bτ ]); φ′ (a) = ∞} is a nowhere dense for a ∈ A subset of M (A[Bτ ]);

(2.4)

φ′ ⊃ φ;

(2.5)

φ′ is a partial ∗-homomorphism, in the following sense: • φ′ (λa) = λφ′ (a)

f0 [τ ]), where 0 · ∞ = 0, (λ ∈ C, a ∈ A

f0 [τ ]), • φ′ (a + b) = φ′ (a) + φ′ (b) (a, b ∈ A if either φ′ (a) ̸= ∞ or φ′ (b) ̸= ∞, f0 [τ ]), if φ′ (a)φ′ (b) ̸= 0 · ∞ or ∞ · 0, • φ′ (ab) = φ′ (a)φ′ (b) (a, b ∈ A f0 [τ ]), where ∞ = ∞. • φ′ (a∗ ) = φ′ (a) (a ∈ A Note that φ′ (a+b) is not determined by φ′ (a) and φ′ (b) in case φ′ (a) = φ′ (b) = ∞. Indeed, given a such that φ′ (a) = ∞, we can take b ≡ −a to get φ′ (a + b) = 0, but φ′ (a + a) = ∞. Similarly, φ′ (ab) is not determined by φ′ (a) and φ′ (b) in case φ′ (a) = ∞ and φ′ (b) = 0 (or φ′ (a) = 0 and φ′ (b) = ∞). Indeed, suppose φ′ (a) = ∞. Put ak = (1 + a∗ a)k , bl = (1 + a∗ a)−l (k, l ∈ N). Then φ′ (ak ) = ∞, φ′ (bl ) = 0, φ′ (a1 · b1 ) = 1, φ′ (a1 · b2 ) = 0, φ′ (a2 · b1 ) = ∞. f0 [τ ] we set For a ∈ A b a(φ) = φ′ (a),

φ ∈ M (A[Bτ ]).

∗

Then it turns out that b a is a C -valued continuous function on M (A[Bτ ]). f0 [τ ]} and collect some of the results We denote by F(M (A[Bτ ])) the set {b a; a ∈ A described before in a theorem. f0 [τ ] ∋ a → b Theorem 2.3. The mapping A a ∈ F(M (A[Bτ ])) is a ∗-isomorf0 [τ ] onto the ∗-algebra F(M (A[Bτ ])) of C∗ -valued continuous functions phism of A on M (A[Bτ ]). Next we consider the noncommutative case. Using Theorem 2.3, a functional f0 [τ ] is established (cf. [16], Theorem 4.12). calculus for any hermitian element of A ∗ f Suppose that h = h ∈ A0 [τ ]. Let C(h) denote a maximal commutative ∗f0 [τ ] containing h; set C ∗ (h) = A[Bτ ] ∩ C(h). Then the elements subalgebra of A 2 −1 (1 +h ) , h(1 +h2 )−1 belong to C ∗ (h) and C(h) is a commutative GB∗ -algebra over C ∗ (h)1 ≡ {x ∈ C ∗ (h); ∥x∥Bτ ≤ 1}. Hence, we can define the ∗-isomorphism a → b a of C(h) onto a ∗-algebra F(M (C ∗ (h))) of C∗ -valued continuous functions on the compact Hausdorff space M (C ∗ (h)). In this case, {b a; a ∈ C ∗ (h)} = C(M (C ∗ (h))). The spectrum σC ∗ (h) (a) of a ∈ C(h) is the subset of C∗ defined by the following two conditions: • for λ ∈ C, λ ∈ σC ∗ (h) (a) iff λ1 − a has no inverse belonging to C ∗ (h); • ∞ ∈ σC ∗ (h) (a) iff a ̸∈ C ∗ (h).

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Then (2.6)

σC ∗ (h) (a) = {b a(φ); φ ∈ M (C ∗ (h))},

∀

a ∈ C(h).

Denote by C0 (σC ∗ (h) (h)) the algebra of all continuous complex-valued functions on σC ∗ (h) (h) vanishing at ∞, and by C1 (σC ∗ (h) (h)) the set of all continuous complexvalued functions on σC ∗ (h) (h) ∩ C such that λ −→ f (λ)/(1 + |λ|2 )n belongs to C0 (σC ∗ (h) (h)) for some non-negative integer n. Clearly C1 (σC ∗ (h) (h)) is a ∗-algebra containing C0 (σC ∗ (h) (h)). Let f ∈ C0 (σC ∗ (h) (h)). By (2.6) f ◦ b h ∈ C(M (C ∗ (h))), and so by the well-known Gel’fand theorem there exists a unique element f (h) of C ∗ (h) such that fd (h)(φ) = f (b h(φ)) for each φ ∈ M (C ∗ (h)). Moreover, the mapping f 7→ f (h) is an isometric ∗-homomorphism of C0 (σC ∗ (h) (h)) into C ∗ (h). Now, given f ∈ C1 (σC ∗ (h) (h)) and n ∈ N0 such that gn (λ) ≡ f (λ)/(1 + |λ|2 )n ∈ C0 (σC ∗ (h) (h)), we set f (h) = gn (h)(1 + h∗ h)n . Then f (h) is independent of the choice of n such that gn ∈ C0 (σC ∗ (h) (h)), and the mapping f → f (h) is a ∗-isomorphism of C1 (σC ∗ (h) (h)) into C(h). Let us now collect some properties of this functional calculus. f0 [τ ] and that C(h) is a maximal Theorem 2.4. Suppose that h∗ = h ∈ A f commutative ∗-subalgebra of A0 [τ ] containing h. Then there exists a unique ∗isomorphism f 7→ f (h) of C1 (σC ∗ (h) (h)) into C(h) satisfying the following assertions: (i) if u0 (λ) ≡ 1 then u0 (h) = 1 ; (ii) if u1 (λ) ≡ λ then u1 (h) = h; (iii) for any f ∈ C1 (σC ∗ (h) (h)) and φ ∈ M (C ∗ (h)), fd (h)(φ) = f (b h(φ)); (iv) f −→ f (h) is an isometric ∗-isomorphism of the C ∗ -algebra C0 (σC ∗ (h) (h)) into C∗ (h) = A[Bτ ] ∩ C(h). f0 [τ ] and C(h) be a maximal commutative Corollary 2.5. Let h∗ = h ∈ A f0 [τ ] containing h. Then the following statements hold: ∗-subalgebra of A b (1) h(φ) ∈ R ∪ {∞} for each φ ∈ M (C ∗ (h)). (2) The following assertions (2)1 , (2)2 , (2)3 are equivalent to each other: (2)1 h = a∗ a for some a ∈ C(h). (2)2 h = k 2 for some hermitian k ∈ C(h). (2)3 For each φ ∈ M (C ∗ (h)), b h(φ) ≥ 0 or b h(φ) = ∞. If one of the three equivalent statements in (2) holds, then h is called positive. (3) If h is positive, there is a unique positive element k of C(h) such that k 2 = h. This k is said to be the square root of h and is denoted by h1/2 . (4) There exist unique positive elements h+ , h− in C(h) such that h = h+ − h− . f0 [τ ]. We begin with some Next, we describe a faithful ∗-representation of A basic terminology concerning unbounded ∗-representations. For more details we refer to [21, 27]. Let D be a dense subspace of a Hilbert space H. Denote by L(D) the algebra of all linear operators from D into D and define L† (D) = {X ∈ L(D); D(X ∗ ) ⊃ D and X ∗ D ⊂ D}.

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L† (D) is a ∗-algebra under the usual algebraic operations and the involution X → X † ≡ X ∗ ⌈D. Furthermore, L† (D) is a locally convex ∗-algebra with respect to the topology τs∗ generated by the family of seminorms {p†ξ (·); ξ ∈ D}, where

p†ξ (X) = ∥Xξ∥ + ∥X † ξ∥, X ∈ L† (D). A ∗-subalgebra of L† (D) containing the identity operator I of D as an element is said to be an O∗ -algebra on D. Let A be a ∗-algebra. A ∗-homomorphism π : A → L† (D) is called (unbounded) ∗representation of A on the Hilbert space H, with domain D. If A has an identity 1 , we suppose that π(1 ) = I. Usually, the domain of π and the corresponding Hilbert space are denoted by D(π) and Hπ , respectively. A ∗-representation π of A is said to be faithful if π(a) = 0, implies a = 0. A ∗-representation π of a locally convex ∗-algebra A[τ ] is said to be (τ − τs∗ ) continuous if it is a continuous mapping from A[τ ] into L† (D(π))[τs∗ ]. f0 [τ ]+ the set of all positive elements of A f0 [τ ]. By Corollary 2.5 We denote by A f and [16], Theorem 5.5, A0 [τ ]+ is a positive cone, that is, f0 [τ ]+ whenever a, b ∈ A f0 [τ ]+ and λ ≥ 0, • wedge: a + b, λa ∈ A f f f f0 [τ ]) ≡ {a∗ a; a ∈ A f0 [τ ]}. • A0 [τ ]+ ∩ (−A0 [τ ]+ ) = {0} and A0 [τ ]+ = P(A f Moreover, [16], Theorem 6.5 may be applied to conclude that A0 [τ ]+ is τ -closed, since the property [A] used in that theorem is weaker than the joint continuity of the product. Thus an application of the Hahn Banach Theorem shows that the set f0 [τ ] is separating. F of all τ -continuous positive linear functionals on A Let (πf , λf , Hf ) be the GNS-construction for f ∈ F. This means, in particular, f0 [τ ] and that λf is a mapping of A f0 [τ ] onto that πf is a ∗-representation of A D(πf ) ⊂ Hf = Hπf . The direct orthogonal sum π of the representations πf , f ∈ F , is defined now by the following settings:  ⊕ ∑  Hπ ≡ Hf = {(xf )f ∈F ; xf ∈ Hf for all f ∈ F and ∥xf ∥2 < ∞}.     f ∈F f ∈F  ⊕   Hf ; xf ∈ A0 for all f ∈ F and the set D(π) = {(λf (xf ))f ∈F ∈ f ∈F     {f ∈ F; xf ̸= 0} is finite}.    π(a)((λ (x )) f f f ∈F ) = (πf (a)(λf (xf )))f ∈F = (λf (axf ))f ∈F . f0 [τ ], and the faithfulness of π Then π is a (τ -τs∗ )-continuous ∗-representation of A follows from the fact that F is separating. Furthermore, the mapping a ∈ A[Bτ ] → π(a) ∈ B(H) is isometric, because it is a faithful ∗ -representation of a C∗ -algebra. f0 [τ ], π(a) ∈ B(H) if and only if π(1 + a∗ a) ∈ B(H). But this is Given now a ∈ A the case if and only if π((1 + a∗ a)−1 ) and (1 + a∗ a)−1 are invertible in π(A[Bτ ]) f0 [τ ]; π(a) ∈ B(H)} = A[Bτ ]. and A[Bτ ], respectively. Thus {a ∈ A ∗ Note that an O -algebra M on D in H is said to be an extended C∗ -algebra if Mb ≡ {X; X ∈ M and X ∈ B(H)} is a C∗ -algebra on H and (I + X ∗ X)−1 ∈ Mb for each X ∈ M ([17, 20]). Thus the properties of π may be summarized as follows. Theorem 2.6. The representation π is a (τ -τs∗ )-continuous ∗-isomorphism of f0 [τ ] onto an extended C∗ -algebra. A

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¨ M. FRAGOULOPOULOU, A. INOUE, AND K.-D. KURSTEN

3. Completion with respect to a topology making the multiplication separately continuous Since the completion of a C∗ -normed algebra with respect to a locally convex topology with separately continuous multiplication is a partial ∗-algebra, but need not be an algebra, we recall some basic definitions and properties of partial ∗algebras and quasi ∗-algebras (for more details, see [3]). A partial ∗-algebra is a vector space A, endowed with a vector space involution x → x∗ and a partial multiplication defined on a set Γ ⊂ A×A (a binary relation) such that the following properties are satisfied: (i) (x, y) ∈ Γ implies (y ∗ , x∗ ) ∈ Γ; (ii) (x, y1 ), (x, y2 ) ∈ Γ and λ, µ ∈ C imply (x, λy1 + µy2 ) ∈ Γ; (iii) for any (x, y) ∈ Γ, a product xy ∈ A is defined, such that xy depends linearly on y and satisfies the relation (xy)∗ = y ∗ x∗ . Whenever (x, y) ∈ Γ, we say that x is a left multiplier of y and y is a right multiplier of x, and we write x ∈ L(y) respectively y ∈ R(x). Next we introduce partial O∗ -algebras and (locally convex) quasi ∗-algebras as important examples of partial ∗-algebras. Given a dense linear subspace D of a Hilbert space H, let L† (D, H) denote the set of all linear operators X from D to H such that D(X ∗ ) ⊃ D. Then L† (D, H) is a partial ∗-algebra with respect to the usual sum, scalar multiplication, the involution defined by X † = X ∗ ⌈D , and the (weak) partial multiplication X Y = X †∗ Y , defined whenever X is a left multiplier of Y (X ∈ L(Y )), that is, iff Y D ⊂ D(X †∗ ) and X † D ⊂ D(Y ∗ ). A partial ∗-subalgebra of the partial ∗-algebra L† (D, H) is said to be a partial O∗ -algebra on D. Let A be a vector space and let A0 be a subspace of A, which is also a ∗-algebra. A is said to be a quasi ∗-algebra with distinguished ∗-algebra A0 (or, simply, a quasi ∗-algebra over A0 ) if • the left multiplication ax and the right multiplication xa of an element a of A with an element x of A0 , that extend the multiplication of A0 , are always defined and are bilinear; • x1 (x2 a) = (x1 x2 )a, (ax1 )x2 = a(x1 x2 ) and x1 (ax2 ) = (x1 a)x2 for all x1 , x2 ∈ A0 and a ∈ A; • an involution ∗ that extends the involution of A0 is defined on A and satisfies the properties: (ax)∗ = x∗ a∗ and (xa)∗ = a∗ x∗ for all x ∈ A0 and a ∈ A. f0 [τ ] of A0 [τ ] is a Let A0 [τ ] be a locally convex ∗-algebra. Then the completion A quasi ∗-algebra over A0 with respect to the following left and right multiplications: ax := lim xα x and xa := lim xxα α

α

f0 [τ ]), (x ∈ A0 and a ∈ A

where {xα } is a net in A0 converging to a with respect to the topology τ . The f0 [τ ] is the continuous extension of the involution given on A0 . A involution of A f0 [τ ] containing A0 is said to be a quasi-∗-subalgebra ∗-invariant subspace A of A f of A0 [τ ] if ax and xa belong to A for all x ∈ A0 and a ∈ A. Then, it is readily shown that A is a quasi ∗-algebra over A0 . Moreover, A[τ ] is a locally convex space containing A0 as a dense subspace and the maps A[τ ] ∋ a → ax ∈ A[τ ] and A[τ ] ∋ a → xa ∈ A[τ ] are continuous for each fixed x ∈ A0 . An algebra of this kind is said to be a locally convex quasi ∗-algebra over A0 .

A LOCALLY CONVEX ∗-ALGEBRA STRUCTURE OF A C∗ -NORMED ALGEBRA

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Topological quasi ∗-algebras were introduced by Lassner [25, 26] with the purpose of dealing with the thermodynamical limit of local observables of certain quantum statistical models that did not fit into the set-up of the algebraic formulation of quantum theories developed by Haag and Kastler [19]. Furthermore, quasi ∗-algebras appeared in [3, 5, 6, 7]. Thus it is important to investigate the structure and the representation theory of locally convex quasi ∗-algebras. Let D be a dense linear subspace of a Hilbert space H. The algebraic conjugate dual D† of D (i.e., the set of all conjugate linear functionals on D), becomes a vector space in a natural way. Denote by L(D, D† ) the set of all linear maps from D to D† . Then, L(D, D† ) is a ∗-invariant vector space under the usual operations and the involution T → T † such that < T † ξ, η >:= < T η, ξ >, ξ, η ∈ D, where < f, η >≡ f (η) for f ∈ D† . Any linear operator X : D → H is regarded as an element of L(D, D† ) satisfying < Xξ, η >= (Xξ|η), ξ, η ∈ D. Thus L† (D, H) is regarded as a ∗-subspace of L(D, D† ). For any X ∈ L† (D) and T ∈ L(D, D† ) we define the multiplications X ◦ T and T ◦ X by < (X ◦ T )ξ, η >:=< T ξ, X † η > and < (T ◦ X)ξ, η >:=< T Xξ, η > . Under these multiplications, L(D, D† ) is a quasi ∗-algebra over L† (D). The weak topology τw on L(D, D† ) is defined to be the locally convex topology generated by the family of seminorms {pξ,η (·) : ξ, η ∈ D} given by pξ,η (T ) := | < T ξ, η > |, T ∈ L(D, D† ). It is not difficult to show that L(D, D† ), the set of all sesquilinear forms on D × D, and the completion of L† (D)[τw ] coincide with each other. Furthermore, L(D, D† )[τw ] is a locally convex quasi ∗-algebra over L† (D). More generally, for f w ] is a locally convex quasi ∗-algebra over M. any O∗ -algebra M on D, M[τ A ∗-representation of a partial ∗-algebra A is a ∗-homomorphism π of A into a partial O∗ -algebra L† (D, H), in the sense of [3], Definition 2.1.6, satisfying π(1 ) = I, whenever 1 ∈ A. A quasi ∗-representation of a quasi ∗-algebra A over A0 is naturally defined as a linear mapping π of A into a quasi ∗-algebra L(D, D† ) over L† (D) such that: • π is a ∗-representation of the ∗-algebra A0 ; • π(a)† = π(a∗ ), ∀ a ∈ A; • π(ax) = π(a) ◦ π(x) and π(xa) = π(x) ◦ π(a), ∀ a ∈ A, ∀ x ∈ A0 . From now on, let A0 [∥ · ∥0 ] be a C ∗ -algebra with 1 and let τ be a topology on A0 such that τ ≼ ∥ · ∥0 and A0 [τ ] is a locally convex ∗-algebra whose multiplication f0 [τ ] is a locally convex quasi ∗-algebra over A0 , is not jointly continuous. Then A but not a ∗-algebra, in general. For this reason, it is natural to use the theory of e ]=A f0 [τ ] as locally quasi *-algebras. We remark that for any A ∈ C ∗ (A0 , τ ), A[τ e f convex spaces, but A[τ ] is different from A0 [τ ] as a quasi ∗-algebra. Moreover, the f0 [τ ]+ of the quasi*-algebra A f0 [τ ] over A0 , defined now as the τ -closure of wedge A e ]+ of the the positive cone (A0 )+ , does not necessarily coincide with the wedge A[τ e quasi *-algebra A[τ ] over A, in contrast to the situation discussed in Section 2. f0 [τ ], such that f (x) ≥ 0, for each x ∈ A f0 [τ ]+ , is said A linear functional f on A f0 [τ ] over A0 . The to be a strongly positive linear functional on the quasi ∗-algebra A f next results concern the representation theory of A0 [τ ]. Theorem 3.1. The following statements are equivalent:

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¨ M. FRAGOULOPOULOU, A. INOUE, AND K.-D. KURSTEN

f0 [τ ]+ ∩ (−A f0 [τ ]+ ) = {0}. (i) A (ii) There exists a faithful (τ − τw )-continuous quasi ∗-representation of the f0 [τ ] over A0 . quasi ∗-algebra A Proof. Since it is easy to see that (ii) implies (i), we assume that (i) is satisfied and construct a faithful (τ −τw )-continuous quasi ∗-representation. Let F be the set f0 [τ ] of all τ -continuous strongly positive linear functionals on the quasi ∗-algebra A over A0 . For any f ∈ F we denote by (πf , λf , Hf ) the GNS-construction for f ⌈A0 . ff (a) of λf (A0 )† = (D(πf ))† f0 [τ ] we define an element λ Let f ∈ F. For any a ∈ A by ff (a), λf (x) >= f (x∗ a), x ∈ A0 . = ff (af ), λf (xf ) >= < (λ = f (a) ̸= 0, this implies that πf (a) ̸= 0, f0 [τ ] we obtain π(a) ̸= 0 by writing and so π(a) ̸= 0. For an arbitrary 0 ̸= a ∈ A f a as a = a1 + ia2 with a1 , a2 ∈ A0 [τ ]h . We refer to [4], Theorem 3.3 for more details.  It is natural to investigate the following question: When there exists a faithful f0 [τ ] over A0 (into L† (D(π), Hπ ))? For ∗-representation π of the quasi ∗-algebra A that, we define the following notion: A subset G of F is said to be separating if f0 [τ ] and f (a) = 0 for all f ∈ G imply a = 0. For example, if F is separating a∈A and G is dense in F with respect to the weak∗ -topology, then G is separating. The following result is found in [4], as Proposition 3.4. Theorem 3.2. The following statements are equivalent: (i) There exists a faithful (τ − τw )-continuous ∗-representation π of the quasi f0 [τ ] over A0 (into L† (D(π), Hπ )). ∗-algebra A f0 [τ ]+ ∩ (−A f0 [τ ]+ ) = {0} and Fb is separating, where (ii) A f0 [τ ], Fb = {f ∈ F ; ∀ a ∈ A

∃

γa > 0 such that |f (a∗ x)|2 ≤ γa f (x∗ x), ∀ x ∈ A0 }.

A LOCALLY CONVEX ∗-ALGEBRA STRUCTURE OF A C∗ -NORMED ALGEBRA

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Thus we have obtained some results for representations of the locally convex f0 [τ ], whose structure is rather complicated, so that it needs further quasi ∗-algebra A investigation. For further studies in this direction, we shall begin with generalizing and investigating the notion of a GB∗ -algebra in the context of quasi ∗-algebras. References [1] J. Alcantara and J. Yngvason, Algebraic quantum field theory and non commutative moment problem, Part II, Ann. Inst. Henri Poincar´ e 48 (1988), 161-173. [2] G.R. Allan, On a class of locally convex algebras, Proc. London Math. Soc. (3) 17 (1967), 91-114. [3] J.P. Antoine, A. Inoue and C. Trapani, Partial ∗-Algebras and their Operator Realizations, Math. Appl. 553, Kluwer Academic, Dordrecht, 2003. [4] F. Bagarello, M. Fragoulopoulou, A. Inoue and C. Trapani, The completion of a C∗ -algebra with a locally convex topology, J. Operator Theory (2) 56 (2006), 101-120. [5] F. Bagarello, A. Inoue and C. Trapani, Unbounded C ∗ -seminorms and ∗-representations of partial ∗-algebras, Z. Anal. Anwend. 20 (2001), 1-20. [6] F. Bagarello and C. Trapani, States and representations of CQ∗ -algebras, Ann. Inst. H. Poincar´ e 61 (1994), 103-133. [7] F. Bagarello and C. Trapani, CQ∗ -algebras: Structure properties, Publ. RIMS, Kyoto Univ. 32 (1996), 85-116. [8] S.J. Bhatt, A note on generalized B∗ -algebras I, II, J. Indian Math. Soc. 43 (1979), 253-257; J. Indian Math. Soc. 44 (1980), 285-290. [9] S.J. Bhatt, M. Fragoulopoulou and A. Inoue, Existence of well-behaved ∗-representations of locally convex ∗-algebras, Math. Nachr. 279 (2006), 86-100. [10] S.J. Bhatt, M. Fragoulopoulou and A. Inoue, Existence of spectral well-behaved ∗-representations, J. Math. Anal. Appl. 317 (2006), 475-495. [11] S.J. Bhatt, A. Inoue and K.-D. K¨ ursten, Well-behaved unbounded operator representations and unbounded C∗ -seminorms, J. Math. Soc. Japan 56 (2004), 417-445. [12] S.J. Bhatt, A. Inoue and H. Ogi, Unbounded C ∗ -seminorms and unbounded C ∗ -spectral algebras, J. Operator Theory 45 (2001), 53-80. [13] S.J. Bhatt, A. Inoue and H. Ogi, Spectral invariance, K-theory isomorphism and an application to the differential structure of C*-algebras, J. Operator theory, 49 (2003), 389-405. [14] B. Blackadar and J. Cuntz, Differential Banach algebra norms and smooth subalgebras of C*- algebras, J. Operator Theory 26 (1991), 255-282. [15] N. Bourbaki, Espaces Vectoriels Topologiques, Hermann, Paris, 1966. [16] P.G. Dixon, Generalized B ∗ -algebras, Proc. London Math. Soc. 21 (1970), 693-715. [17] P.G. Dixon, Unbounded operator algebras, Proc. London Math. Soc. 23 (1971), 53-69. [18] M. Dubois-Violette, A generalization of the classical moment problem on ∗-algebras with applications to relativistic quantum theory, Parts I and II, Commum. Math. Phys. 43 (1975), 225-254 and 54 (1977), 151-172. [19] R. Haag and D. Kastler, An algebraic approach to quantum field theorem, J. Math. Phys. 5 (1964), 848-861. [20] A. Inoue, On a class of unbounded operator algebras, Pacific J. Math. 65 (1976), 77-95. [21] A. Inoue, Tomita-Takesaki Theory in Algebras of Unbounded Operators, Lecture Notes Math. No 1699, Springer-Verlag, 1998. [22] A. Inoue and K.-D. K¨ ursten, On C ∗ -like locally convex ∗-algebras, Math. Nachr. 235 (2002), 51-58. [23] W. Kunze, Zur algebraischen Struktur der GC∗ -Algebren, Math. Nachr. 88 (1979), 7-11. [24] W. Kunze, Halbordnung und Topologie in GC∗ -Algebren, Wiss. Z. KMU Leipzig, Math. Naturwiss. R. 31 (1982), 55-62. [25] G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. KMULeipzig, Math. Naturwiss. R. 30 (1981), 572-595. [26] G. Lassner, Algebras of unbounded operators and quantum dynamics, Physica A 124 (1984), 471-480. udgen, Unbounded Operator Algebras and Representation Theory, Birkh¨ auser-Verlag, [27] K. Schm¨ Basel, 1990.

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[28] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York-Heidelberg-Berlin, 1979. [29] J. Yngvason, Algebraic quantum field theory and non-commutative moment problem, Part I, Ann. Inst. Henri Poincar´ e 48 (1988), 147-159. Department of Mathematics, University of Athens, Athens 15784 Greece E-mail address: [email protected] Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan E-mail address: [email protected] Mathematisches Institut, Augustusplatz 10, D-04109 Leipzig, Germany E-mail address: [email protected]