Search

The introduction of Fredholm theory relative to general unital homomorphisms T : A → B between Banach algebras A and B, which involves the study of Fredholm, Weyl and Browder elements, was due to Harte (Math Z 179:431–436, 1982), after which this investigation was continued by several authors. Motivated by results of Alekhno (Positivity 11:375–386, 2007, Positivity 13:3–20, 2009), the definitions of upper Weyl and upper Browder elements in an ordered Banach algebra (OBA) were given in Benjamin and Mouton (Quaest Math 39:643–664, 2016), thereby initiating the study of the interplay between Fredholm theory and ordering. Inspired by an indication that these elements could be useful in the context of OBAs, certain variants of the invertible, Fredholm, Weyl and Browder elements (namely the r-invertible, r-Fredholm, r-Weyl and r-Browder elements) were investigated in Benjamin et al. (Glasgow Math J, 2018. 10.1017/S0017089518000393) in the context of general Banach algebras. Continuing the development of Fredholm theory in ordered Banach algebras, we now introduce the notions of upper r-Weyl, contractive upper r-Weyl, upper r-Browder and contractive upper r-Browder elements in an OBA and, motivated by previous results, investigate conditions under which the positive almost r-invertible r-Fredholm elements will be contractive upper r-Browder. This also leads to the recovering (and strengthening) of certain results in Benjamin and Mouton (Positivity 21:575–592, 2017) regarding the upper Browder spectrum property. [ABSTRACT FROM AUTHOR]

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14]. [ABSTRACT FROM AUTHOR]

In this paper we give an example of a Banach algebra A and a closed ideal I of A such that the multiplier algebra of I is equal to A but I does not have any bounded approximate identity. In the case that I has an approximate identity, we give a necessary condition on I for which A = M (I) , where M (I) denotes the multiplier algebra of I. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier–Stieltjes algebra. [ABSTRACT FROM AUTHOR]

We define a weighted extended g-Drazin inverse for Banach algebra elements, generalizing the concepts of extended g-Drazin inverse and weighted g-Drazin inverse. We characterize weighted extended g-Drazin invertible elements and present some representations of a weighted extended g-Drazin inverse. Applying these results, we introduce and investigate a weighted extended Drazin inverse. [ABSTRACT FROM AUTHOR]

The issue discussed in this paper is: how small can a sum of idempotents be? Here smallness is understood in terms of nilpotency or quasinilpotency. Thus the question is: given idempotents p 1 , … , p n in a complex algebra or Banach algebra, is it possible that their sum p 1 + ⋯ + p n is quasinilpotent or (even) nilpotent (of a certain order)? The motivation for considering this problem comes from earlier work by the authors on the generalization of the logarithmic residue theorem from complex function theory to higher (possibly infinite) dimensions. [ABSTRACT FROM AUTHOR]

We introduce and investigate a new generalized inverse as an extension of the g-Drazin inverse in a Banach algebra. This new inverse will be called an extended g-Drazin inverse. Using idempotents, we characterize this inverse and give some its representations. Also, we prove generalizations of Cline's formula for extended g-Drazin inverse. As a consequence of our results, we present the definition and characterizations of an extended Drazin inverse. [ABSTRACT FROM AUTHOR]

Let A be a complex commutative semisimple complete LMC algebra with respect to a topology τ 1 and a complete metrizable topological algebra with respect to a topology τ 2 . It is proved that every τ 1 -bounded set is a τ 2 -bounded set. This generalizes a result of R. L. Carpenter on uniqueness of Fréchet algebra topology for complex commutative semisimple algebras. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the existence of the generalized Drazin inverse of the sum a + b of two generalized Drazin invertible elements in a Banach algebra and present explicit expressions for the generalized Drazin inverse of this sum, under new conditions. [ABSTRACT FROM AUTHOR]

In this paper left, right and two-sided Mackey TQ -algebras are defined and main properties of these topological algebras are studied. Relations between Mackey TQ -algebras with other topological algebras are given. [ABSTRACT FROM AUTHOR]

In this note we investigate regularities (semiregularities) R and S in a Banach algebra A satisfying S ⊂ R and the corresponding spectra σS and σR Satisfying sup {|λ| : λ ∈ σR(a)} = sup{|λ| : λ ∈ σS (a)} for all a ∈ A. [ABSTRACT FROM AUTHOR]

In this note we investigate amenability properties in the class of the so‐called DF‐algebras. We also characterize, in terms of the defining sequence of weights, amenable Köthe co‐echelon algebras. [ABSTRACT FROM AUTHOR]

Under specific conditions, we develop new formulae for the generalized Drazin inverse of a block matrix in Banach algebras. Also, we present necessary and sufficient conditions for the existence and the expression of the group inverse of a block matrix. [ABSTRACT FROM AUTHOR]

Let L(X) be the Banach algebra of all bounded linear operators on a complex Banach space X. For an operator T∈L(X), let ιT(x) denote the inner local spectral radius of T at any vector x in X. We characterize maps ϕ (not necessarily linear nor surjective) on L(X) which satisfy ιT-S(x)=0ifandonlyifιϕ(T)-ϕ(S)(x)=0for every x∈X and T,S∈L(X). We also describe surjective linear maps ϕ on L(X) for which ϕ(I) is invertible and either ιT(x)=0⟹ιϕ(T)(x)=0for every x∈X and T∈L(X), or ιϕ(T)(x)=0⟹ιT(x)=0for every x∈X and T∈L(X). [ABSTRACT FROM AUTHOR]

One of the central problems investigated in Alekhno (Positivity 11(3):375-386, 2007, Positivity 13(1):3-20, 2009) is that of providing conditions under which the spectral radius of a positive operator T on a complex Banach lattice lies outside the lower Weyl spectrum of T given that it is not an element of its essential spectrum. In this paper the lower Weyl spectrum of an arbitrary positive ordered Banach algebra element is introduced and studied, and work done in the aforementioned papers are extended to general ordered Banach algebras. [ABSTRACT FROM AUTHOR]

Several new pre-orders are introduced and characterized on the set of all wg -Drazin invertible elements of a Banach algebra. We generalize recent results for rectangular matrices and bounded linear operators between Banach spaces, omitting some assumptions and adding new characterizations. [ABSTRACT FROM AUTHOR]

Under new conditions on Banach algebra elements a and b, we derive explicit expressions for the generalized Drazin inverse of the sum $$a+b$$ . As an application of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. [ABSTRACT FROM AUTHOR]

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $$*$$ -semigroup S when it is infinite non-discrete cancellative, $$M_a(S)^{**}$$ does not admit an involution, and $$M_a(S)^{**}$$ has a trivolution with range $$M_a(S)$$ if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A( G) if and only if G is finite. However, $$A(G)^{**}$$ always admits trivolution. [ABSTRACT FROM AUTHOR]

In this note we continue our study on the upper Browder spectrum initiated in Benjamin and Mouton (Quaest. Math. 39(5), 2016). Recall that, for an element a of an ordered Banach algebra A and w.r.t. a Banach algebra homomorphism $$T: A \rightarrow B,$$ we have inclusions where $$\sigma (Ta),$$ $$\beta _T(a),$$ $$ \beta _T^+(a)$$ and $$ \sigma (a)$$ denote the Fredholm, Browder, upper Browder and (usual) spectra of a, respectively (Benjamin and Mouton in Quaest. Math. 39(5), 2016). This paper concerns the following natural question: given that the spectral radius of a positive element is not in the Fredholm spectrum of the element, when will it be outside the upper Browder spectrum of that element? [ABSTRACT FROM AUTHOR]

We give a survey of the development of the spectral theory in ordered Banach algebras; from its roots in operator theory to the modern abstract context. [ABSTRACT FROM AUTHOR]

This is a list of open problems posed at the workshop on Ordered Banach Algebras held at the Lorentz Center, Leiden, during the week 21-25 July, 2014. [ABSTRACT FROM AUTHOR]

In this article, the ( b, c)-inverse will be studied. Several equivalent conditions for the existence of the ( b, c)-inverse in rings will be given. In particular, the conditions ensuring the existence of the ( b, c)-inverse, of the annihilator ( b, c)-inverse and of the hybrid ( b, c)-inverse, will be proved to be equivalent, provided b and c are regular elements in a unitary ring $$\mathcal {R}$$ . In addition, the set of all ( b, c)-invertible elements will be characterized and the reverse order law will be also studied. Moreover, the relationship between the ( b, c)-inverse and the Bott-Duffin ( p, q)-inverse will be considered. In the context of Banach algebras, integral, series and limit representations will be given. Finally, the continuity of the ( b, c)-inverse will be characterized. [ABSTRACT FROM AUTHOR]

In a Banach algebra, the spectrum satisfies σ ( a b ) ∖ { 0 } = σ ( b a ) ∖ { 0 } for each pair of elements a , b . We show that this is no longer true for the exponential spectrum, thereby solving a problem open since 1992. Our proof depends on the fact that the homotopy group π 4 ( G L 2 ( C ) ) is non-trivial. [ABSTRACT FROM AUTHOR]

We study a finite dimensional invariant subspace property similar to Fan's Theorem on semigroups for arbitrary Banach algebras A in terms of amenability of X ( A , ϕ ) , the closed subalgebra of A generated by the set of all maximal elements in A with respect to a character ϕ . As a consequence, we offer some applications to the measure algebra M ( G ) and the generalized Fourier algebra A p ( G ) of a locally compact group G . [ABSTRACT FROM AUTHOR]

For a Lau algebraA, we study the Banach space WAP(A) of all weakly almost periodic functionals onAto obtain some equivalent conditions for the existence of topological left invariant means on a topological left introverted subspaceXofAcontained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean onXand a common fixed point property. [ABSTRACT FROM PUBLISHER]

Let $${\mathcal{A}}$$ and $${\mathcal{B}}$$ be unital semisimple complex Banach algebras, and let $${\varphi_1}$$ and $${\varphi_2}$$ be maps from $${\mathcal{A}}$$ onto $${\mathcal{B}}$$ . We show that if the socle of $${\mathcal{A}}$$ is an essential ideal of $${\mathcal{A}}$$ , and $${\varphi_1}$$ and $${\varphi_2}$$ satisfy for all $${a,b\in \mathcal{A}}$$ , then $${\varphi_1\varphi_2(1)}$$ and $${\varphi_1(1)\varphi_2}$$ coincide and are Jordan isomorphisms. We also show that a map $${\varphi}$$ from $${\mathcal{A}}$$ onto $${\mathcal{B}}$$ satisfies for all $${a,b\in \mathcal{A}}$$ if and only if $${\varphi(1)}$$ is a central invertible element of $${\mathcal{B}}$$ for which $${\varphi(1)^3=1}$$ and $${\varphi(1)^2\varphi}$$ is a Jordan isomorphism. [ABSTRACT FROM AUTHOR]

Let A be a semisimple Banach algebra with non-trivial, and possibly infinite-dimensional socle. Addressing a problem raised in [5, p. 1399] , we first define a characteristic polynomial for elements belonging to the socle, and we then show that a Generalized Cayley–Hamilton Theorem holds for the associated polynomial. The key arguments leading to the main result follow from the observation that a purely spectral approach to the theory of the socle carries alongside it an efficient method of dealing with relativistic problems associated with infinite-dimensional socles. [ABSTRACT FROM AUTHOR]

Given a (not necessarily continuous) homomorphism between Banach algebras $${{\mathcal{T} : \mathcal{A} \to \mathcal{B}}}$$ , an element $${a \in {\mathcal{A}}}$$ will be said to be B-Fredholm (respectively, generalized B-Fredholm) relative to $${\mathcal{T}}$$ , if $${{\mathcal{T}(a) \in \mathcal{B}}}$$ is Drazin invertible (respectively, Koliha-Drazin invertible). In this article, the aforementioned elements will be characterized and their main properties will be studied. In addition, perturbation properties will be also considered. [ABSTRACT FROM AUTHOR]

This paper illustrates some initial steps taken in the effort 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. [ABSTRACT FROM AUTHOR]

Various kinds of the reverse order laws for the strong Drazin inverse are characterized in a ring. Then we define the generalized strong Drazin inverse in a Banach algebra and present similar results related to reverse order laws for this new inverse. [ABSTRACT FROM AUTHOR]

For a space X denote by $$\mathrm {C}_0^\sigma (X)$$ the set of all bounded continuous $$f:X\rightarrow \mathbb {R}$$ such that $$|f|^{-1}([\epsilon ,\infty ))$$ is $$\sigma $$ -compact for all $$\epsilon >0$$ . We give sufficient (and in certain cases, necessary) conditions for a Banach algebra to be of the form $$\mathrm {C}_0^\sigma (X)$$ for some locally compact space X. [ABSTRACT FROM AUTHOR]

This paper is devoted to the generalized Drazin inverse of a block matrixin a Banach algebraA, under specific conditions. We focus on deriving formulae for the generalized Drazin inverse ofxin terms of the generalized Drazin inverses of the elementsa,aπ bc,a2ad+aad bcadand the generalized Schur complements=d−cad b. [ABSTRACT FROM PUBLISHER]

We show that the index defined via a trace for Fredholm elements in a Banach algebra has the property that an index zero Fredholm element can be decomposed as the sum of an invertible element and an element in the socle. We identify the set of index zero Fredholm elements as an upper semiregularity with the Jacobson property. The Weyl spectrum is then characterized in terms of the index. [ABSTRACT FROM AUTHOR]

LetAbe a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map intoA, we derive a generalization of Gelfand’s famous power boundedness theorem. Elaborating on these ideas, with the help of a Phragm´en-Lindel¨of device for subharmonic functions, it is then shown, as the main result, that two normal elements of aC∗-algebra are equal if and only if they are quasinilpotent equivalent. [ABSTRACT FROM PUBLISHER]

We continue the work begun in [12] on noncommutativeC∗-Segal algebras, with an emphasis on the order structure. We extend and improve the main results of [12], and characterize theC∗-Segal algebras with an order unitization. [ABSTRACT FROM AUTHOR]

We give an example of a positive element a in some ordered Banach algebra A such that its spectrum is equal to {1} and it is not greater than or equal to the unit element of A. [ABSTRACT FROM AUTHOR]

A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence(xn) in A, (xn) is Cauchy whenever the sequence bn(xn − xn-1) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra,then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theoremfor a large class of F-algebras, containing both locally bounded algebras and locally convex algebras withbounded elements.

As a follow-up to work done in [7] , some new insights to the structure of the socle of a semisimple Banach algebra are obtained. In particular, it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals. Remarkably, the finite-dimensional case here reduces to the classical Wedderburn–Artin Theorem, and this approach does not use any continuous irreducible representations of the algebra in question. Furthermore, the structure of the socles for which the classical Shoda's Theorem for matrices can be extended, is characterized exactly as those socles which are minimal two-sided ideals. It is then shown that the set of commutators in the socle (i.e. { x y − y x : x , y ∈ Soc A } ) is a vector subspace. Finally, we characterize those socles which belong to the center of a Banach algebra and obtain results which suggest that the dimension of certain subalgebras of the socle in fact provides a measure, to some extent, of commutativity. [ABSTRACT FROM AUTHOR]