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We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman's theorem can be derived from it. [ABSTRACT FROM AUTHOR]

AbstractLet K be a compact subgroup of a locally compact group G. In this work, we initiate an investigation of the Lebesgue-Fourier algebra S1 A(G/K) on the coset space G/K with pointwise multiplication; after some general results on the Banach algebra S1 A(G/K), as our main result, we give some necessary conditions for that S1 A(G/K) be approximately amenable in terms of algebraic and topological properties of G and K . [ABSTRACT FROM AUTHOR]

This paper presents a fairly general version of the well-known Gleason–Kahane– Z ˙ elazko (GKZ) theorem in the spirit of a GKZ type theorem obtained recently by Mashreghi and Ransford for Hardy spaces. In effect, we characterize a class of linear functionals as point evaluations on the vector space of all complex polynomials P . We do not make any topological assumptions on P . We then apply this characterization to present a version of the GKZ theorem for a vast class of topological spaces of complex-valued functions including the Hardy, Bergman, Dirichlet, and many more well-known function spaces. We obtain this result under the assumption of continuity of the linear functional, which we show, with the help of an example, to be a necessary condition for the desired conclusion. Lastly, we use the GKZ theorem for polynomials to obtain a version of the GKZ theorem for strictly cyclic weighted Hardy spaces. [ABSTRACT FROM AUTHOR]

This paper is an attempt to give an axiomatic approach to the investigation of various kinds of generalizations of Drazin invertibility in Banach algebras. We shall say that an element a of a Banach algebra $ \mathcal {A} $ A is generalized Drazin invertible relative to a regularity $ \mathcal {R} $ R if there is $ b\in \mathcal {A} $ b ∈ A such that $ ab=ba,\ bab=b $ ab = ba , bab = b and $ \sigma _{\mathcal {R}}(a-aba)\subset \{0\} $ σ R (a − aba) ⊂ { 0 }. The concept of Koliha-Drazin invertible elements, as well as some generalizations of this concept are described via the concept of generalized Drazin invertible elements relative to a regularity $ \mathcal {R} $ R which satisfies two properties: (D1) if $ a, b\in \mathcal {R} $ a , b ∈ R , p is an idempotent commuting with a and b, then $ ap+b(1-p)\in \mathcal {R} $ ap + b (1 − p) ∈ R ; (D2) if $ a\in \mathcal {R} $ a ∈ R , then a is almost invertible. If a regularity $ \mathcal {R} $ R satisfies the properties (D1) and (D2), we prove that $ a\in \mathcal {A} $ a ∈ A is generalized Drazin invertible relative to $ \mathcal {R} $ R if and only if 0 is not an accumulation point of $ \sigma _{\mathcal {R}}(a) $ σ R (a). In particular we define and characterize generalized Drazin-T-Riesz invertible elements relative to an arbitrary (not necessarily bounded) Banach algebra homomorphism T and so extend the concept of generalized Drazin-Riesz invertible operators introduced in [Živković-Zlatanović SČ, Cvetković MD. Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Linear Multilinear A. 2017;65(6):1171–1193]. Also we consider generalized Drazin invertibles relative to $ \mathcal {R} $ R in the case when $ \mathcal {R} $ R is the set of Drazin invertibles, as well as when $ \mathcal {R} $ R is the set of Koliha-Drazin invertibles. [ABSTRACT FROM AUTHOR]

For a Tychonoff space X, let C B (X) be the C ∗ -algebra of all bounded complex-valued continuous functions on X. In this paper, we mainly discuss Tychonoff one-point extensions of X arising from closed ideals of C B (X) . We show that every closed ideal H of C B (X) produces a Tychonoff one-point extension X (∞ H) of X. Moreover, every Tychonoff one-point extension of X can be obtained in this way. As an application, we study the partially ordered set of all Tychonoff one-point extensions of X. It is shown that the minimal unitization of a non-vanishing closed ideal H of C B (X) is isometrically ∗ -isomorphic with the C ∗ -algebra C B X (∞ H) . We provide a description for the Čech–Stone compactification of an arbitrary Tychonoff one-point extension of X as a quotient space of β X via a closed ideal of C B (X) . Then, we establish a characterization of closed ideals of C B (X) that have countable topological generators. Finally, an intrinsic characterization of the multiplier algebra of an arbitrary closed ideal of C B (X) is given. [ABSTRACT FROM AUTHOR]

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]

We establish some Lie–Trotter formulae for unital complex Jordan–Banach algebras, showing that for any elements a, b, c in a unital complex Jordan–Banach algebra A the identities lim n → ∞ e a n ∘ e b n n = e a + b , lim n → ∞ U e a n e b n n = e 2 a + b , and lim n → ∞ U e a n , e c n e b n n = e a + b + c hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in A . These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals f : A → C satisfying f (U x (y)) = U f (x) f (y) , for all x , y ∈ A . We prove that for any such a functional f, there exists a unique continuous (Jordan-) multiplicative linear functional ψ : A → C such that f (x) = ψ (x) , for every x in the connected component of the set of all invertible elements of A containing the unit element. If we additionally assume that A is a JB ∗ -algebra and f is continuous, then f is a linear multiplicative functional on A . The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Schulz, and Touré. [ABSTRACT FROM AUTHOR]

Let X be an infinite-dimensional Banach space, and B (X) be the algebra of all bounded linear operators on X. A map Δ , from B (X) into a closed subsets of C is said to be ∂ -spectrum if ∂ (σ (T)) ⊆ Δ (T) ⊆ σ (T) for all T ∈ B (X) . Here, σ (T) is spectrum of T and ∂ (σ (T)) the boundary of σ (T) . In this paper, we determine the forms of all surjective maps ϕ from B (X) into itself that satisfy either Δ (ϕ (T) ϕ (S)) = Δ (T S) for all T , S ∈ B (X) or Δ (ϕ (T) ϕ (S) ϕ (T)) = Δ (T S T) for all T , S ∈ B (X) . [ABSTRACT FROM AUTHOR]

Let A be an algebra. Let N (A) be the set of all algebra norms on A . There is a natural equivalence relation ∼ on N (A) . Let N ~ (A) be the collection of equivalence classes in (N (A) , ∼) . Then there is a natural partial order ≤ on N ~ (A) . Using this partial relation, we study different types of algebra norms on A . We also study uniqueness and existence of norms with some specific properties. [ABSTRACT FROM AUTHOR]

Let B (X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. For A ∈ B (X) , σ 1 (A) , σ 2 (A) , and σ 3 (A) are the semi-Fredholm domain, the Fredholm domain, and the Weyl domain of A in the spectrum σ (A) , respectively. For an integer k ≥ 2 , let (i 1 , … , i m) be a finite sequence with terms chosen from { 1 , … , k } and assume that at least one of the terms in (i 1 , … , i m) appears exactly once. The generalized product of k operators A 1 , … , A k ∈ B (X) is defined by A 1 ∗ A 2 ∗ ⋯ ∗ A k = A i 1 A i 2 ⋯ A i m , and includes the usual product and the triple product. In this paper we characterize the form of the map ϕ from B (X) into itself which satisfies σ i (A 1 ∗ ⋯ ∗ A k) = σ i (ϕ (A 1) ∗ ⋯ ∗ ϕ (A k)) , for all A 1 , … , A k ∈ B (X) and i ∈ 1 , 2 , 3 . [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]

Let A be a Banach algebra and let J ⊂ A be a set that is contained in the set of all quasinilpotent elements of A . We say that a ∈ A is Drazin invertible relative to J if there exists b ∈ A commuting with a such that b = b a b and a - a b a ∈ J . Given a homomorphism T : A → B between Banach algebras, we give necessary and sufficient conditions for Ta to be Drazin invertible relative to J. What is more, several applications to the case of the Calkin homomorphism are indicated. In particular, we give a partial answer to a question related to the structure of polynomially compact operators on Banach spaces. [ABSTRACT FROM AUTHOR]

This paper is a contribution to understanding what properties should a topological algebra on a Stone space satisfy to be profinite. We reformulate and simplify proofs for some known properties using syntactic congruences. We also clarify the role of various alternative ways of describing syntactic congruences, namely by finite sets of terms and by compact sets of continuous self mappings of the algebra. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study and investigate the concept of weak topological BCK-algebras which is obtained by combining a certain topology to a BCK-algebra that makes the binary operation defined on the BCK-algebra compatible. Several properties of weak topological BCK-algebras are investigated. Continuity between weak topological BCK-algebra is studied and a new topology is constructed by using special subsets of a BCK-algebra and further results are obtained. [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]

In a Banach algebra A it is well known that the usual spectrum has the following property: σ (a b) \ { 0 } = σ (b a) \ { 0 } for elements a , b ∈ A . In this note we are interested in subsets of A that have the Jacobson Property, i.e. X ⊂ A such that for a , b ∈ A : 1 - a b ∈ X ⇒ 1 - b a ∈ X. [ABSTRACT FROM AUTHOR]

In this paper, we consider the notion of quasi-multipliers to the general topological algebra setting, not necessarily normed or locally convex. We discuss the quasi-uniform and quasi-strong operator topologies on the algebra QM(A) of all bilinear and jointly continuous quasi-multipliers on A and study their various properties. [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]

Let B (X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. For A ∈ B (X) , let F(A) be the set of all fixed points of A. For an integer k ≥ 2 , let (i 1 , ⋯ , i m) be a finite sequence with terms chosen from { 1 , ⋯ , k } and assume that at least one of the terms in (i 1 , ⋯ , i m) appears exactly once. The generalized product of k operators A 1 , ⋯ , A k ∈ B (X) is defined by A 1 ∗ A 2 ∗ ⋯ ∗ A k = A i 1 A i 2 ⋯ A i m and includes the usual product and the triple product. In this paper we characterize the form of surjective maps from B (X) into itself satisfying dim F (ϕ (A 1) ∗ ⋯ ∗ ϕ (A k)) = dim F (A 1 ∗ ⋯ ∗ A k) for all A 1 , ⋯ , A k ∈ B (X) . [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 define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$ -algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$ -Drazin invertible elements of a $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order. [ABSTRACT FROM AUTHOR]

The set E ′ (R) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of E ′ (R) is 2. [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]