The compactum and finite dimensionality in Banach algebras.
The Gleason-Kahane-Zelazko theorem and function algebras
A theorem due to A. Gleason, J.-P. Kahane and W. Zelazko characterizes continuous characters within the space of all continuous linear forms of a locally multiplicatively convex, sequentially complete algebra. The present paper applies these results to investigate linear isometries of Banach algebras (with particular attention to normal uniform algebras) and of some locally multiplicatively convex algebras. The locally multiplicatively convex algebra of all holomorphic functions on a domain, will...
The index for Fredholm elements in a Banach algebra via a trace
We show that the existence of a trace on an ideal in a Banach algebra provides an elegant way to develop the abstract index theory of Fredholm elements in the algebra. We prove some results on the problem of the equality of the nonzero exponential spectra of elements ab and ba and use the index theory to formulate a condition guaranteeing this equality in a quotient algebra.
The second conjugate algebras of Banach algebras.
The socle and finite-dimensionality of a semiprime Banach algebra
The Spectral Radius Formula in Quotient Algebras.
The structure of L-ideals of measures algebras
The three-space-problem for locally-m-convex algebras.
We prove that a locally convex algebra A with jointly continuous multiplication is already locally-m-convex, if A contains a two-sided ideal I such that both I and the quotient algebra A/I are locally-m-convex. An application to the behaviour of the associated locally-m-convex topology on ideals is given.
Topological algebras of kernels.
Topological algebras of random elements
Let L₀(Ω;A) be the Fréchet space of Bochner-measurable random variables with values in a unital complex Banach algebra A. We study L₀(Ω;A) as a topological algebra, investigating the notion of spectrum in L₀(Ω;A), the Jacobson radical, ideals, hulls and kernels. Several results on automatic continuity of homomorphisms are developed, including versions of well-known theorems of C. Rickart and B. E. Johnson.
Topological algebras with maximal regular ideals closed
It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.
Topologie et traces sur les -algèbres
Topologies of compact families on the ideal space of a Banach algebra
Let be a family of compact sets in a Banach algebra A such that is stable with respect to finite unions and contains all finite sets. Then the sets , K ∈ define a topology τ() on the space Id(A) of closed two-sided ideals of A. is called normal if in (Id(A),τ()) and x ∈ A╲I imply . (1) If the family of finite subsets of A is normal then Id(A) is locally compact in the hull kernel topology and if moreover A is separable then Id(A) is second countable. (2) If the family of countable compact sets...
Topologies on the space of ideals of a Banach algebra
Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely , coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A)...
Totally convex algebras
By definition a totally convex algebra is a totally convex space equipped with an associative multiplication, i.eȧ morphism of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.