Functional continuity of topological algebras with orthonormal bases

(i) Every complex [Formula: see text]-algebra with an identity and with an orthonormal basis is functionally continuous; (ii) Every complex complete LMC algebra with an orthogonal basis is functionally continuous; (iii) Every complex sequentially complete locally convex algebra with an unconditional orthonormal basis and with an element [Formula: see text] for which [Formula: see text]th coefficient functional value tends to infinity as [Formula: see text] tends to infinity is functionally continuous. These results are proved and an example is provided for non-extendability of these results. A representation for positive linear functionals on a sequentially complete locally convex algebra with an unconditional orthonormal basis, with an identity, and with an element [Formula: see text] mentioned in (iii) is obtained. All results are obtained only for commutative algebras.

Extending Characters of Fixed Point Algebras

A dynamical system is a triple ( A , G , α ) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism α : G → Aut ( A ) that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units A × is open in A and the inversion map ι : A × → A × , a ↦ a − 1 is continuous at 1 A . Given a dynamical system ( A , G , α ) with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A.

On the Arens product and approximate identity in locally convex algebras

Let A' and A'' be the dual and bidual spaces of a locally convex algebra A with dual and weak* topology, respectively. In this paper, we show that A has a bounded right (left) approximate identity if and only if A'' has a right (left) unit with respect to the first (second) Arens product.

GEOMETRIC CHARACTERIZATION OF HERMITIAN ALGEBRAS WITH CONTINUOUS INVERSION

A hermitian algebra is a unital associative ℂ-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ℝ. In the case of an algebra 𝒜 endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra 𝒜 is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups Un(𝒜) on each flag manifold is transitive.

DENSE m-CONVEX FRÉCHET SUBALGEBRAS OF OPERATOR ALGEBRA CROSSED PRODUCTS BY LIE GROUPS

Let A be a dense Fréchet *-subalgebra of a C*-algebra B. (We do not require Fréchet algebras to be m-convex.) Let G be a Lie group, not necessarily connected, which acts on both A and B by *-automorphisms, and let σ be a sub-polynomial function from G to the nonnegative real numbers. If σ and the action of G on A satisfy certain simple properties, we define a dense Fréchet *-subalgebra G ⋊σ A of the crossed product L1 (G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in σ. We give conditions on σ and the action of G on A which imply the m-convexity of the dense subalgebra G ⋊σ A. A locally convex algebra is said to be m-convex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Fréchet algebra, and is useful in modern operator theory. If G acts as a transformation group on a locally compact space M, we develop a class of dense subalgebras for the crossed product L1 (G, C0 (M)), where C0 (M) denotes the continuous functions on M vanishing at infinity with the sup norm topology. We define Schwartz functions S (M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense Fréchet *-subalgebra G ⋊σ S (M) of rapidly vanishing, G-differentiable functions from G to S (M). If the reciprocal of σ is in Lp (G) for some p, we prove that our group algebras Sσ (G) are nuclear Fréchet spaces, and that G ⋊σ A is the projective completion [Formula: see text].

A characterisation of C*-algebras

It is of some interest to the theory of locally convex *-algebras to know under what conditions such an algebra A is a pre-C*-algebra (the topology of A can be described by a submultiplicative norm such that ‖x*x‖ = ‖x‖2, ∀x∈A). We recall that a locally convex *-algebra is a complex *-algebra A with the structure of a Hausdorff locally convex topological vector space such that the multiplication is separately continuous, and the involution is continuous.

Quotient bounded elements in locally convex algebras

The quotient bounded and the universally bounded elements in a calibrated locally convex algebra are defined and studied. In the case of a generalized B*-algebra A, they are shown to form respectively b* and B*-algebras, both dense in A. An internal spatial characterization of generalized B*-algebras is obtained. The concepts are illustrated with the help of examples of algebras of measurable functions and of continuous linear operators on a locally convex space.

Infrasequential Topological Algebras

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

Multiplicative functionals and a class of topological algebras

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.