Noninvertibility preservers on Banach algebras

It is proved that a linear surjection Ф: A → B, which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective.     

Homotopical stable ranks for Banach algebras

The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.     

Nilpotent ideals in a class of Banach algebras

We introduce the concepts of approximately complemented subspaces of normed spaces and approximately biprojective algebras. We prove that any approximately biprojective Banach algebra with left and right approximate identities does not have a nontrivial nilpotent ideal whose closure is approximately complemented.

     

Star ring homomorphisms between commutative Banach algebras

We consider a -ring homomorphism from a commutative Banach algebra with an involution to a commutative Banach algebra with a symmetric involution. We give the Gelfand transform of the -ring homomorphism image.

     

An example on ordered Banach algebras

Let be a complex unital Banach algebra. We consider the Banach algebra ordered by the algebra cone , and investigate the connection between results on ordered Banach algebras and the right bound of the numerical range of elements in .

     

The Socle and finite dimensionality of some Banach algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L ¹(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal.     

Algebraic properties of isometries between groups of invertible elements in Banach algebras

We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then (T(eA))⁻¹T is extended to an isometric real algebra isomorphism from A onto B. In particular, A−1 is isometric as a metric space to B−1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of ?elazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.     

Asymptotic approximations for second-order linear difference equations in Banach algebras, I

An asymptotic approximation theory is developed for some classes of linear second-order difference equations in Banach algebras, subject to “finite moments perturbations.” The special case of linear matrix difference equations (or, equivalently, of second-order systems) is included. Rigorous and explicitly computable bounds for the error terms are obtained.     

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Let K be an ultrametric complete field and let E be an ultrametric space. Let A be the Banach K-algebra of bounded continuous functions from E to K and let B be the Banach K-algebra of bounded uniformly continuous functions from E to K. Maximal ideals and continuous multiplicative semi-norms on A (resp. on B) are studied by defining relations of stickiness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of A (resp. of B) is in bijection with the set of equivalence classes with respect to stickiness (resp. to contiguousness). Every prime ideal of A or B is included in a unique maximal ideal and every prime closed ideal of A (resp. of B) is a maximal ideal, hence every continuous multiplicative semi-norms on A (resp. on B) has a kernel that is a maximal ideal. If K is locally compact, every maximal ideal of A (resp. of B) is of codimension 1. Every maximal ideal of A or B is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, on A as on B the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of A, Max(A) and the Banaschewski compactification of E which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of A (resp. B) is equal to the whole set of continuous multiplicative semi-norms.