We show that a free graded commutative Banach algebra over a (purely odd) Banach space $E$ is a Banach-Grassmann algebra in the sense of Jadczyk and Pilch if and only if $E$ is infinite-dimensional. Thus, a large amount of new examples of separable Banach-Grassmann algebras arise in addition to the only one example previously known due to A. Rogers.

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_d\ast g^{\ast d}+a_{d-1}\ast g^{\ast (d-1)}+\cdots +a₁\ast g+a₀=0$, where $a₀,...,a_d:\mathbb{N}\to ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form ${\sum }_{x\in X}f\left(x\right)e^{-sx}$ ($s\in {ℂ}^k$), where $X\subseteq {[0,\infty )}^k$ is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied,...

We study subalgebras of $C_b\left(X\right)$ equipped with topologies that generalize both the uniform and the strict topology. In particular, we study the Stone-Weierstrass property and describe the ideal structure of these algebras.

The generalized notion of weak amenability, namely $(\varphi ,\psi )$-weak amenability, where $\varphi ,\psi$ are continuous homomorphisms on a Banach algebra $𝒜$, was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(\varphi ,\psi )$-weak amenability on the measure algebra $M\left(G\right)$, the group algebra $L^1\left(G\right)$ and the Segal algebra $S^1\left(G\right)$, where $G$ is a locally compact group, are studied. As a typical example, the $(\varphi ,\psi )$-weak amenability of a special semigroup algebra is shown as well.

Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F\left(u\right){[F\left(u\right),u]}^n=0$ for all $u\in L$, where $n\ge 1$ is a fixed integer. Then one of the following holds: (1) ...

By considering arbitrary mappings $\omega$ from a Banach algebra $A$ into the set of all nonempty, compact subsets of the complex plane such that for all $a\in A$, the set $\omega \left(a\right)$ lies between the boundary and connected hull of the exponential spectrum of $a$, we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over ℂ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement...

Dans une algèbre de Banach et dans deux cas particuliers, nous montrons la continuité du centre du plus petit disque contenant le spectre. Pour a ∈ , on donne une condition nécessaire et suffisante pour avoir $R_K=d\left(a\right)$ où d(a) est la distance de a aux scalaires et $R_K$ le rayon du plus petit disque contenant K qui représente le spectre ou le domaine numérique algébrique de a. Dans un espace de Hilbert complexe, K peut représenter certains types de spectres ou de domaines numériques de a.

Let $A$ be a unital Banach algebra over $ℂ$, and suppose that the nonzero spectral values of $a$ and $b\in A$ are discrete sets which cluster at $0\in ℂ$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that $a$ and $b$ are quasinilpotent equivalent if...