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We present new results for the generalized Drazin inverse of 2 × 2 anti-triangular matrices with commuting entries over a Banach algebra. As an application, the g-Drazin invertibility of block-operator matrices is obtained under new wider conditions. [ABSTRACT FROM AUTHOR]

Let A be an ordered Banach algebra and a, b ∈ A such that 0 ≤ a ≤ b. The domination problem involves finding natural conditions under which properties of b will be inherited by a. In [12] this problem was investigated for ergodic elements, where a ∈ A is ergodic if the sequence converges in A. In this paper we consider the more general problem where the assumption 0 ≤ a ≤ b is replaced by the weaker condition ±a ≤ b. [ABSTRACT FROM AUTHOR]

Let A and B be complex unital Banach algebras, and let σ (x) and σ ^ (x) denote the spectrum of x and its polynomially convex hull, respectively. A spectrally additive map ϕ : A → B is a surjective function which satisfies σ (x + y) = σ (ϕ (x) + ϕ (y)) for each x , y ∈ A . By establishing a new additive characterization of rank one elements in a Banach algebra, we prove that any spectrally additive map acting on a semisimple domain preserves rank one elements in both directions. This settles an open question raised in [1], which ultimately then classifies spectrally additive maps on a large class of Banach algebras as Jordan-isomorphisms. By refining the techniques in [1] even further, we are able to prove the following more general result: If A is semisimple and either A or B has an essential socle, then any surjective map ϕ : A → B with the property that σ ^ (x + y) = σ ^ (ϕ (x) + ϕ (y)) for each x , y ∈ A is a continuous Jordan-isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, the concept of left ϕ-approximate Connes amenability for a dual Banach algebra 풜 is studied, where ϕ is a weak * -continuous multiplicative linear functional on 풜 . Some characterizations for module extension dual Banach algebras and matrix algebras are given. Moreover, the notion of approximate left ϕ-Connes biprojectivity for dual Banach algebras is introduced and some concrete examples regarding this new notion are indicated. [ABSTRACT FROM AUTHOR]

For a Banach algebra A and a norm one character ϕ on A , we characterize ϕ -amenability and ϕ -contractibility of A in terms of metrically homological properties of some Banach A -modules. Also, for any Banach A -module, metrically injectivity and projectivity of submodules and quotient modules are investigated. Finally, we give some results on metrically injectivity of A ∗ and its submodules. [ABSTRACT FROM AUTHOR]

We introduce the e–core–EP inverse and the core–EP inverse for elements of Banach algebras. Several characterizations and representations for these inverses are presented using idempotents and Drazin inverses. Then, we define the e–core inverse of elements in Banach algebras and give its characterizations, generalizing the related results of the core inverse. [ABSTRACT FROM AUTHOR]

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$ , every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$ , $b\in A$ are such that $ab+ba=0$ , is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$. We show that all $C^{\ast }$ -algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications. [ABSTRACT FROM AUTHOR]

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms. [ABSTRACT FROM AUTHOR]

Let ${\mathcal{A}}$ be a complex unital Banach algebra, let $a$ be an element in it and let $0. In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the $\unicode[STIX]{x1D716}$ -level set of a condition spectrum at a given $(\unicode[STIX]{x1D716},a)$ plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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 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]

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]

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 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]

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]

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 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]

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]

Let A and B be Banach algebras and T: B → A be a continuous homomorphism. n-weak amenability of the Banach algebra A × T B (defined in Bade, W. G., Curtis, P. C., Dales, H. G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc., 55(2), 359-377 (1987)) is studied. The new version of a Banach algebra defined with a continuous homomorphism is introduced and Arens regularity and various notions of amenability of this algebra are studied. [ABSTRACT FROM AUTHOR]

Under some conditions, the reverse order laws for the generalized Drazin inverse are investigated in Banach algebras. Similar results related to the Drazin inverse in a ring are presented too. [ABSTRACT FROM AUTHOR]

A Banach algebra elementa∈Ais said to be “polynomially Riesz”, relative to the homomorphismT:A→B, if there exists a nonzero complex polynomialp(z) such that the imageT p(a) ∈Bis quasinilpotent. [ABSTRACT FROM PUBLISHER]

We establish some triple reverse order laws for the generalized Drazin inverse and then obtain from one of them various expressions of the generalized Drazin inverse of a block matrix in a Banach algebra. [ABSTRACT FROM AUTHOR]

As a follow-up to a paper of D. Petz and J. Zemánek [1] , a number of equivalent conditions which characterize the trace among linear functionals on matrix algebras, finite rank operators and the socle elements of semisimple Banach algebras in general are given. Moreover, the converse problem is also addressed; that is, given the equivalence of certain conditions which characterize the trace, what can be said about the structure of the socle? In particular, we characterize those socles isomorphic to matrix algebras in this manner, as well as those socles which are minimal two-sided ideals. [ABSTRACT FROM AUTHOR]

The objective of this paper is to derive formulae for the generalized Drazin inverse of a block matrix in a Banach algebra $$\mathcal{A}$$ under different conditions. Let $$x=\left[ \begin{array}{c@{\quad }c} a&{}b\\ c&{}d\end{array}\right] \in \mathcal{A}$$ relative to the idempotent $$p\in \mathcal{A}$$ and $$a\in p\mathcal{A}p$$ be generalized Drazin invertible. The formulae for the generalized Drazin inverse are obtained under the more general case that the generalized Schur complement $$s=d-ca^db$$ is generalized Drazin invertible, which covers the cases that $$s$$ is Drazin invertible, $$s$$ is group invertible, or $$s$$ is equal to zero. Thus, recent results on the Drazin inverse of block matrices and block-operator matrices are extended to a more general setting. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish explicit representations of the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra, which covers the case when the generalized Schur complement is equal to zero. [ABSTRACT FROM AUTHOR]

We define and characterize inner generalized inverses with prescribed idempotents. These classes of generalized inverses are natural algebraic extension of generalized inverses of linear operators with prescribed range and kernel. We consider the reverse order rule for inner generalized inverses of elements of a ring, some perturbation bounds, and we construct an iterative method for computing a (p, q)-inner inverse in Banach algebras. [ABSTRACT FROM AUTHOR]