We investigate some homological notions of Banach algebras. In particular, for a locally compact group G we characterize the most important properties of G in terms of some homological properties of certain Banach algebras related to this group. Finally, we use these results to study generalized biflatness and biprojectivity of certain products of Segal algebras on G.

We find some relations between module biprojectivity and module biflatness of Banach algebras $𝒜$ and $ℬ$ and their projective tensor product $𝒜\widehat{\otimes }ℬ$. For some semigroups $S$, we study module biprojectivity and module biflatness of semigroup algebras $l^1\left(S\right)$.

For two Banach algebras and ℬ, an interesting product ${\times }_{\theta }ℬ$, called the θ-Lau product, was recently introduced and studied for some nonzero characters θ on ℬ. Here, we characterize some notions of amenability as approximate amenability, essential amenability, n-weak amenability and cyclic amenability between and ℬ and their θ-Lau product.

Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^p\left(I\right)$, 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$ and their preduals.

We show that every continuous derivation of a countably dominated Fréchet GB*-algebra A is spatial whenever A is additionally an AO*-algebra.

We study general Hilbert modules over the disc algebra and exhibit necessary spectral conditions for the vanishing of certain associated extension groups. In particular, this sheds some light on the problem of identifying the projective Hilbert modules. Part of our work also addresses the classical derivation problem.

Let M be a Beurling-type submodule of $H²{(}_d)$, the Hardy space over the unit ball ${}_d$ of ${ℂ}^d$, and let $N=H²{(}_d)/M$ be the associated quotient module. We completely describe the spectrum and essential spectrum of N, and related index theory.

We revisit the old result that biflat Banach algebras have the same cyclic cohomology as C, and obtain a quantitative variant (which is needed in separate, joint work of the author on the simplicial and cyclic cohomology of band semigroup algebras). Our approach does not rely on the Connes-Tsygan exact sequence, but is motivated strongly by its construction as found in [2] and [5].

We study the structure of certain classes of homologically trivial locally C*-algebras. These include algebras with projective irreducible Hermitian A-modules, biprojective algebras, and superbiprojective algebras. We prove that, if A is a locally C*-algebra, then all irreducible Hermitian A-modules are projective if and only if A is a direct topological sum of elementary C*-algebras. This is also equivalent to A being an annihilator (dual, complemented, left quasi-complemented, or topologically...

Un $C^{*}$-modulo hilbertiano destro $X$ su una $C^{*}$-algebra $\mathcal{A}$ dotato di uno ${}^{*}$-omomorfismo isometrico $\varphi :\mathcal{A}\to {\mathcal{L}}_{\mathcal{A}}\left(X\right)$ viene qui considerato come un oggetto $X_{\mathcal{A}}$ della $C^{*}$-categoria degli $\mathcal{A}$-moduli Hilbertiani destri. Come in [11], associamo ad esso una $C^{*}$-algebra ${\mathcal{O}}_{X_{\mathcal{A}}}$ contenente $X$ come un «$\mathcal{A}$-bimodulo hilbertiano in ${\mathcal{O}}_{X_{\mathcal{A}}}$». Se $X$ è pieno e proiettivo finito ${\mathcal{O}}_{X_{\mathcal{A}}}$ è la $C^{*}$-algebra $C^{*}\left(X\right)$, la generalizzazione delle algebre di Cuntz-Krieger introdotta da Pimsner [27] (e in un caso particolare da Katayama [31]). Più in generale, $C^{*}\left(X\right)$ è canonicamente immersa...

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.