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ContentsIntroduction .............................................................................................................................................................................. 51. Regular representations of algebras with approximate unit. A duality theorem..................................................... 92. Induced representations of algebras. The main duality............................................................................................. 203. Specialization;...

The class ω(A) of ideals consisting of topological zero divisors of a commutative Banach algebra A is studied. We prove that the maximal ideals of the class ω(A) are of codimension one.

Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula ${\sigma }_l\left(S\right)$ = ${\left(\lambda \left(s\right)\right)}_{s\in S}\in {ℂ}^S|{s-\lambda \left(s\right)}_{s\in S}$ generates a proper left ideal$.$Using the Schur lemma and the Gelfand-Mazur theorem we prove that ${\sigma }_l\left(S\right)$ 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.

We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $𝒜$ to have a non-void left joint spectrum ${\sigma }_l\left(S\right).$ In particular, for a Lie subalgebra $L\subset 𝒜$ we have ${\sigma }_l\left(L\right)\ne \varnothing$ if and only if $[L,L]$ generates in $𝒜$ a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

Agrafeuil and Zarrabi in [1] characterized all closed ideals with at most countable hull in a unital Banach algebra embedded in the classical disc algebra and satisfying certain conditions ((H1), (H2), (H3)), and the analytic Ditkin condition. We modify Ditkin’s condition and show that analogous result is true for a wider class of algebras. This is an extension of the result obtained in [1].

For a given unital Banach algebra A we describe joint spectra which satisfy the one-way spectral mapping property. Each spectrum of this class is uniquely determined by a family of linear subspaces of A called spectral subspaces. We introduce a topology in the space of all spectral subspaces of A and utilize it to the study of the properties of the spectra.

We study the relation between standard ideals of the convolution Sobolev algebra ${₊}^{\left(n\right)}\left(tⁿ\right)$ and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in ${₊}^{\left(n\right)}\left(tⁿ\right)$ with compact and countable hull are standard.

We define a new class of Banach algebras of holomorphic functions on the unit disc $𝔻$ which contains the algebras studied in [GMR2] and [GW]. To a function G of the class ${𝒜}^1\left(𝔻\right)$ nowhere vanishing in $𝔻$ we associate a Banach algebra ${𝒜}_G^n\left(𝔻\right)$ contained in the disc algebra $𝒜\left(𝔻\right)$. We prove that all closed ideals of ${𝒜}_G^n\left(𝔻\right)$ of at most countable hull are of the standard form.