B ∗ -Algebras and their Representation

1971/01/01 by Constantin Apostol · 5 citations
Mathematics · Computer Science · #Advanced Banach Space Theory #Advanced Algebra and Logic #Advanced Operator Algebra Research

paper · doi:10.1112/jlms/s2-3.1.30

Abstract

Let A be a complete lmc-∗-algebra with unit whose topology is given by a family P of submultiplicative pseudonorms p such that p(x∗ x) = p(x)2, x ∈ A (i.e. A is a b∗-algebra). We study the general properties of A (§2) and especially its representation in the commutative case. The description of the form of submultiplicative pseudonorms p such that p(x∗x) = p(x)2, x ∈ A (Lemma 3.2) enables us to obtain some structure theorems (§4). Thus we give an abstract characterisation of the algebra of all continuous complex functions on a locally compact Hausdorff space, endowed with the compact-open topology [Th. 4.1, Th. 4.2].

Cited by

• Isometric representations of calibrated ordered spaces on C(X)
• Some inverse limits of Cuntz algebras
• Representing Locally Hilbert Spaces and Functional Models for Locally Normal Operators
• Structure theory of homologically trivial and annihilator locally C*-algebras
• On Intrinsic Characterization of Real Locally C*- and Locally JB-Algebras

Related

• Functoriality of group algebras acting on L^p-spaces
• C^*-algebras isomorphically representable on l^p
• Extending representations of Banach algebras to their biduals
• A modern look at algebras of operators on L^p-spaces
• Symmetrized pseudofunction algebras from L^p-representations and amenability of locally compact groups