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Let $\mathcal{A}$ and $\mathcal{B}$ be commutative Banach algebras. Then a multiplicative linear functional $\theta$ on $\mathcal{B}$ induces a multiplication on the Cartesian product space $\mathcal{A}\times\mathcal{B}$ given by $(a,b)(c,d)=(ac+\theta(d)a+\theta(b)c,bd)$ for all $(a,b),(c,d)\in\mathcal{A}\times\mathcal{B}$ . We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn–Banach property, and the unique semisimple norm property under certain conditions on $\theta$ .

The purpose of this paper is to present some old and recent results for the class of $F$-algebras which include most classes of Banach algebras that are important in abstract harmonic analysis. We also introduce a subclass of the class of $F$-algebras, called normal $F$-algebras, that captures better the measure algebras and the (reduced) Fourier-Stieltjes algebras, and use this to give new characterisations the reduced Fourier-Stieltjes algebras of discrete groups.

Let $T$ be a unital, continuous linear functional defined on complex Banach algebra $A$. First, we prove an approximate version of the Gleason-Kahane-\.Zelazko theorem: given $\epsilon >0$, there exists an $M>0$ such that, if $$ T(\exp x)\neq 0,\quad x\in A,\ \|x\|\leq M, $$ then $T$ is $\epsilon $-almost multiplicative. Then, we show that this result remains true if the exponential function is replaced by a nonsurjective entire function~$F$ with $F'(0)\neq 0$.

A necessary and sufficient condition for a Lau type binary operation defined by two mappings to be an algebra-operation is given in terms of multipliers. Also a characterization of multipliers of a Lau algebra constructed by semisimple commutative Banach algebras is given in terms of multipliers of original Banach algebras.

Suppose that $H$ is a complex Hilbert space and that $\mathcal{B}(H)$ denotes the bounded linear operators on $H$ . We show that every abelian, amenable operator algebra is similar to a $C^{*}$ -algebra. We do this by showing that if $\mathcal{A}\subseteq\mathcal{B}(H)$ is an abelian algebra with the property that given any bounded representation $\varrho:\mathcal{A}\to\mathcal{B}(H_{\varrho})$ of $\mathcal{A}$ on a Hilbert space $H_{\varrho}$ , every invariant subspace of $\varrho(\mathcal{A})$ is topologically complemented by another invariant subspace of $\varrho(\mathcal{A})$ , then $\mathcal{A}$ is similar to an abelian $C^{*}$ -algebra.

This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C*-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

Let \zeta be an n-dimensional complex matrix bundle over a compact metric space X and let A_\zeta denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_\zeta, the group of unitaries of A_\zeta. The answer turns out to be independent of the bundle \zeta and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X., Comment: Final version. To appear in J. of Topology and Analysis. Garbled text in abstract removed

We examine the uniqueness of the bounded structure of semisimple and Mackey complete uniformly $A$-convex algebras. We also consider the particular locally $C^{\ast}$-case and the uniform one.

We show that for every Banach space $X, \dim X>1$, every maximal commutative subalgebra of $L(X)$ has uncountably many copies between maximal commutative subalgebras of $L(X)$. Answering to a question of Aleksander Pe{\l}czy\'nski, we show also that for an arbitrary infinite dimensional Banach space $X$ there are at least countably many multiplications making of $X$ a commutative unital Banach algebra.

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over $\mathbb C$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb R)$ with $\det M>0$ is a product of two real exponential matrices., 9 pages

For semisimple, commutative Banach algebras $\mathcal A$ and $\mathcal B$, some Banach algebra properties of the Cartesin product $\mathcal A \times \mathcal B$ are characterized in terms of $\mathcal A$ and $\mathcal B$. A couple of results are also proved for non-commutative Banach algebras.