Charles John Read

Operator algebras with contractive approximate identities, II

We continue our study of operator algebras with contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain

Fréchet algebras of power series

C^*

Operator algebras with contractive approximate identities: Weak compactness and the spectrum

-algebraic results and theories to more general algebras. In the first part of this paper we do a more systematic development of this positivity and its associated ordering, proving many foundational facts. In much of this it is not necessary that the algebra have an approximate identity. In the second part we study interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series. Indeed the present work solves almost all of the open questions raised in these papers. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

We consider Fréchet algebras which are subalgebras of the algebra F = C [[X]] of formal power series in one variable and of Fn = C [[X1, . . . , Xn]] of formal power series in n variables, where n ∈ N. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, (F )-algebras, and other topological algebras, and recall some of their properties; we discuss Michael’s problem from 1952 on the continuity of characters on these algebras and some results on uniqueness of topology. A ‘test algebra’ U for Michael’s problem for commutative Fréchet algebras has been described by Clayton and by Dixon and Esterle. We prove that there is an embedding of U into F, and so there is a Fréchet algebra of power series which is a test case for Michael’s problem. We also discuss homomorphisms from Fréchet algebras into F. We prove that such a homomorphism is either continuous or a surjection, so answering a question of Dales and McClure from 1977. As corollaries, we note that a subalgebra A of F containing C[X] that is a Banach algebra is already a Banach algebra of power series, in the sense that the embedding of A into F is automatically continuous, and that each (F )-algebra of power series has a unique (F )-algebra 2010 Mathematics Subject Classification: Primary 46H40; Secondary 1325, 13J05, 46J99.

Order theory and interpolation in operator algebras

Abstract We continue our study of operator algebras with contractive approximate identities (cais) by presenting a couple of interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series, for example about whether, roughly speaking, ‘weak compactness’ of an operator algebra, or the lack of it, can be seen in the spectra of its elements.