On the semi-continuity of generalized inverses in Banach algebras

The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore-Penrose inverse in a C^∗-algebra and therefore, derive some new and well-known criteria in operator theory.     

Group inverses and Drazin inverses over Banach algebras

For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.     

Weakly almost periodic functionals on certain Banach algebras

For a Lau algebra A, we study the Banach space WAP(A) of all weakly almost periodic functionals on A to obtain some equivalent conditions for the existence of topological left invariant means on a topological left introverted subspace X of A contained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean on X and a common fixed point property.     

Radius preserving (semi)regularities in Banach algebras

Abstract

In this note we investigate regularities (semiregularities) R and S in a Banach algebra A satisfying S ? R and the corresponding spectra σ_S and σ_R Satisfying

sup {|λ| : λ ∈ σ_R(a)} = sup{|λ| : λ ∈ σ_S (a)}     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Semigroups and resolvents of bounded variation,imaginary powers andH ∞ functional calculus

Let −A be a linear, injective operator, on a Banach spaceX. We show that ∃ anH ^∞ functional calculus forA if and only if −A generates a bouned strongly continuous holomorphic semigroup of uniform weak bounded variation, if and only ifA(ζ+A) ⁻¹ is of uniform weak bounded variation. This provides a sufficient condition for the imaginary powers ofA, {A^−is} sεR, to extend to a strongly continuous group of bounded operators; we also give similar necessary conditions.     

Cohomological Characterizations of Biprojective and Biflat Banach Algebras

 Let A be a biprojective Banach algebra, and let A-mod-A be the category of Banach A-bimodules. In this paper, for every given -mod-A, we compute all the cohomology groups . Furthermore, we give some cohomological characterizations of biprojective Banach algebras. In particular, we show that the following properties of a Banach algebra A are equivalent to its biprojectivity: (i) for all -mod -A; (ii) for all -mod-A; (iii) for all -mod-A. (Here and are, respectively, the Banach A-bimodules of left, right and double multipliers of X.) Further, if A is a biflat Banach algebra and -mod-A, we compute all the cohomology groups , where is the Banach A-bimodule dual to X. Also, we give cohomological characterizations of biflat Banach algebras. We prove that a Banach algebra A is biflat if and only if any of the following conditions is valid: (i’) for all -mod-A; (ii’) for all -mod-A; (iii’) for all -mod-A. Received 16 June 1998     

ON INESSENTIAL IDEALS IN BANACH ALGEBRAS

Abstract

In [2], Aupetit studied the perturbation of elements of a Banach algebra A by elements of an inessential ideal I of A. The main result of his paper is based on a lemma ([2], theorem 1.1) obtained by the use of subharmonic methods and analytic multivalued functions. Our aim in this note is to prove Auptetit's perturbation theorem ([2], theorem 2.4) by making use of elementary methods.     

Polynomially Riesz elements

Abstract

A Banach algebra element a ∈ A is said to be “polynomially Riesz”, relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image T p(a) ∈ B is quasinilpotent.     

Sequences in the range of a vector measure and Banach spaces satisfying Π1(X,l 1) = Π2(X,l 1)

If X is a Banach space such that Π₁(X,l ₁) = Π₂ (X, l ₁), we prove the following results: 1) A bounded sequence (x _n ) lies inside the range of some X-valued measure if and only if the operator (α _n ) ? l ₁ → Σ _n α _n x _n ? X is 1-summing, and 2) If A is a bounded subset of X lying in the range of some X ^??-valued measure, then A is necessarily contained in the range of some X-valued measure.     

Advertibly complete locallym-convex two-sided algebras

Many results on two-sided Banach algebras remain valid form-convex normal ones; in particular, the commutativity modulo the Jacobson radical. Moreover advertible completeness appears to be sufficient.     

Commutatively ordered Banach algebras

Abstract

Spectral theory in ordered Banach algebras (OBAs) has been investigated and several authors have made contributions. However, the results are not applicable to non-commutative C^*-algebras, since a non-commutative C^*-algebra is not an OBA. In this paper we introduce a more general structure, called a commutatively ordered Banach algebra (COBA), which includes the class of OBAs. Every C^* - algebra is a COBA. We will give the basic properties of COBAs and show how known results in OBAs can be generalized to the COBA setting. We will then discuss two spectral problems regarding COBA elements. The results obtained, of course, hold true in an OBA as well. These results extend the theory of COBAs and OBAs.     

Spectral mapping theorems for the upper Weyl and upper Browder spectra

Abstract

We discuss the spectral mapping properties of the upper Weyl and upper Browder spectra of an arbitrary ordered Banach algebra element.     

Additive results for the generalized Drazin inverse in a Banach algebra

In this paper we investigate additive properties of the generalized Drazin inverse in a Banach algebra. We find some new conditions under which the generalized Drazin inverse of the sum a + b could be explicitly expressed in terms of a, a^d, b, b^d. Also, some recent results of Castro and Koliha [New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 1085-1097] are extended.     

The σ<Emphasis Type="Italic">g</Emphasis>-Drazin Inverse and the Generalized Mbekhta Decomposition

In this paper we define and study an extension of the g-Drazin for elements of a Banach algebra and for bounded linear operators based on an isolated spectral set rather than on an isolated spectral point. We investigate salient properties of the new inverse and its continuity, and illustrate its usefulness with an application to differential equations. Generalized Mbekhta subspaces are introduced and the corresponding extended Mbekhta decomposition gives a characterization of circularly isolated spectral sets.     

Fredholm theory in ordered Banach algebras

This paper illustrates some initial steps taken in the e?ort of unifying the theory of positivity in ordered Banach algebas (OBAs) with the general Fredholm theory in Banach algebras. We introduce here upper Weyl and upper Browder elements in an OBA relative to an arbitrary Banach algebra homomorphism and investigate the spectra corresponding to the sets of upper Weyl and upper Browder elements, which we shall refer to as the upper Weyl and upper Browder spectra, respectively.     

DENTABILITY IN LOCALLY CONVEX SPACES

Abstract

Let A be a non-empty bounded subset of a locally convex space E. We show that if all the separable subsets of A are weakly metrisable, then the weak*-compact subsets of E¹ satisfy geometrical conditions which are similar to the concept of “dentability” used to characterise the Radon-Nikodý Property in dual Banach spaces.     

COMPONENTS OF POSITIVE ELEMENTS IN RIESZ SPACES

Abstract

We find a necessary and sufficient condition on a Riesz space E such that a sub-Boolean algebra of components of a positive element generates the Boolean algebra of all components of the element. This condition yields a dual characterization of principal A-modules in the sense of D. Vuza. When applied to l-algebras, one finds improvements of results of J. Synnatzschke.