Double multipliers on topological algebras

Abstract

Abstract. The main purpose of this paper is to investigate some topological properties of the double multiplier algebra on a topological algebra. Let M d (A) be the double multiplier algebra on a topological algebra A, and let u and s be the uniform and strong operator topologies on M d (A), respectively. It is shown, under some additional hypotheses on A, In view of the applications of (nonnormed) topological algebras in other fields such as quantum mechanics and quantum statistics (see, e.g., Lassner [10,11]) and recent developments in the theory of topological algebras (see, for instance, the book of Mallios [13]), it is important to consider operators on more general classes of topological algebras. More recently, Phillips [14,15] has studied inner and approximately inner derivations on pro-C * -algebras (inverse limits of C * -algebras, also called LMC * -algebras) using multipliers, while Van Daele [20] has considered multipliers on Hopf algebras which provide a natural framework to study quantum groups. Therefore, it is important to develop the theory of multipliers for general topological algebras and, in particular, for metrizable topological algebras.In this paper, we are mainly concerned with the linear topological properties of

Topological modules of continuous homomorphisms

“…The following results generalize some results of [5,15,19,31] to modules of continuous homomorphisms. …”

“…If E = X = A, then Hom A (A, A) is the usual multiplier algebra of A, and is denoted by M(A). In fact, there is a vast literature dealing with the notions of left multiplier, right multiplier, multiplier and double multiplier (see, e.g., [5,10,12,13,15,18,29]). Proof.…”

Quasimultipliers on F‐Algebras

“…Theorem 3.7. a Suppose that A is a strongly factorable F-normed algebra. If T ∈ M A (resp., M r A ), then T is linear and continuous (see [32,33]). b Suppose that A is a faithful F-algebra.…”

“…b Suppose that A is a faithful F-algebra. If S, T ∈ M d A , then S and T are linear and continuous; in particular each T ∈ M A is linear and continuous (see [31,33]). Convention 1.…”

Semisimple Segal Fréchet algebras

“…In addition, in [11], Lawson and Read introduced and studied the notions of approximate amenability and approximate contractibility of Fréchet algebras. We also refer to [10,16] and [17] as papers that are closely related to the present work.…”