Riesz Theory in Banach Algebras.
Schur Lemma and the Spectral Mapping Formula
Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula = generates a proper left idealUsing the Schur lemma and the Gelfand-Mazur theorem we prove that has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.
Semidirect products of locally convex algbras and the three-space-problem.
Semi-normes multiplicatives sur une algèbre de Banach ultramétrique
Spectral characterization of two-sided ideals in Banach algebras
Spectral characterization of two-sided ideals in Banach algebras
Sur la décomposition d'une algèbre topologique en produit cartésien d'idéaux
Sur la division et la composition dans des classes ultradifférentiables
We generalize to some classes of ultradifferentiable jets or functions the classical Łojasiewicz Division Theorem and Glaeser Composition Theorem. The proof uses the desingularization results by Hironaka, Bierstone and Milman.
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