Let be a Banach algebra. is called ideally amenable if for every closed ideal of , the first cohomology group of with coefficients in is zero, i.e. . Some examples show that ideal amenability is different from weak amenability and amenability. Also for , is called -ideally amenable if for every closed ideal of , . 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...
For commuting elements x, y of a unital Banach algebra ℬ it is clear that . On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form for all , where and c, s are constants.
Semigroups S for which the Banach algebra is injective are investigated and an application to the work of O. Yu. Aristov is described.
A concept of amenability for an arbitrary Lau algebra called inner amenability is introduced and studied. The inner amenability of a discrete semigroup is characterized by the inner amenability of its convolution semigroup algebra. Also, inner amenable Lau algebras are characterized by several equivalent statements which are similar analogues of properties characterizing left amenable Lau algebras.
The paper gives new integral representations of the -Drazin inverse of an element of a -algebra that require no restriction on the spectrum of . The representations involve powers of and of its adjoint.