Let $𝔐$ be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$,   $a,b\in 𝔐$(ii) $\parallel a^2\parallel =\parallel a{\parallel }^2$(iii) $\parallel a^2\parallel \le \parallel a^2+b^2\parallel$.$𝔐$ is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set ${𝔐}^{+}$ of squares in $𝔐$ is a closed convex cone. $(𝔐,{𝔐}^{+},\mathbf{1})$ is a complete ordered vector space with $\mathbf{1}$ as a order unit. In addition, we assume $𝔐$ to be monotone complete (i.e. $𝔐$ coincides with the bidual ${𝔐}^{**}$), and that there exists a finite normal faithful trace $\phi$ on $𝔐$.Then the completion $\{{𝔐}^{+}{\}}_{\phi }$ of ${𝔐}^{+}$ with respect to the Hilbert structure...

We give a survey of our recent results on homological properties of Köthe algebras, with an emphasis on biprojectivity, biflatness, and homological dimension. Some new results on the approximate contractibility of Köthe algebras are also presented.

Let S be a Rees semigroup, and let ℓ¹(S) be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of ℓ¹(S) are isomorphic to those of the underlying discrete group algebra.

For every closed subset C in the dual space ${Ĥ}_n$ of the Heisenberg group $H_n$ we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra $S\left(H_n\right)$ and we show that in general for two closed subsets $C_1,C_2$ of ${Ĥ}_n$ the product of $j\left(C_1\right)$ and $j\left(C_2\right)$ is different from $j\left(C_1\cap C_2\right)$.

Let $𝒜$ be a Banach algebra. $𝒜$ is called ideally amenable if for every closed ideal $I$ of $𝒜$, the first cohomology group of $𝒜$ with coefficients in $I^{*}$ is zero, i.e. $H^1(𝒜,I^{*})=\left\{0\right\}$. Some examples show that ideal amenability is different from weak amenability and amenability. Also for $n\in N$, $𝒜$ is called $n$-ideally amenable if for every closed ideal $I$ of $𝒜$, $H^1(𝒜,I^{\left(n\right)})=\left\{0\right\}$. In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.

This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach-algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (or HSA's, which are in some sense a generalization of ideals). Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which...

The structure of closed ideals of a regular algebra containing the classical A∞ is considered. Several division and approximation results are proved and a characterization of those ideals whose intersection with A∞ is not {0} is obtained. A complete description of the ideals with countable hull is given, with applications to synthesis of hyperfunctions.

Using the holomorphic functional calculus we give a characterization of idempotent elements commuting with a given element in a Banach algebra.

In this paper, we begin the study of the phenomenon of the “invisible spectrum” for commutative Banach algebras. Function algebras, formal power series and operator algebras will be considered. A quantitative treatment of the famous Wiener-Pitt-Sreider phenomenon for measure algebras on locally compact abelian (LCA) groups is given. Also, our approach includes efficient sharp estimates for resolvents and solutions of higher Bezout equations in terms of their spectral bounds. The smallest “spectral...