Theory Of Banach Algebras Research Articles

Equivalence of some properties in the theory of Banach algebras and applications

In this work, we establish a sequential characterization of the notion of relatively weak compactness of Banach algebras introduced recently by J. Banaś and L. Olszowy. Moreover, we show that this structure is one of the most important properties which could be lifted from a Banach algebra X to C(K,X) and L1(μ,X). In addition, fixed point theorems for the product of two nonlinear operators acting on RWC-Banach algebra are proved by the help of the De Blasi measure of weak noncompactness.

On the equivalence of some concepts in the theory of Banach algebras

Our principal aim in this article is to show the equivalence of two concepts used recently in the theory of Banach algebras. The result we present here solves an open problem raised by Jeribi and Krichen in their 2015 book.

The bidual of a Banach algebra associated with a bilinear map

We associate a Banach algebra \(\mathbf A _\Phi \) to each bounded bilinear map \(\Phi : \mathrm X\times \mathrm Y\rightarrow \mathrm {Z}\) on Banach spaces, and we find out that this Banach algebra can be useful for many purposes in the theory of Banach algebras. For example, the construction of \(\mathbf A _\Phi \) enables us to provide many simple examples of Banach algebras with different topological centers, which are neither Arens regular nor either left or right strongly Arens irregular. It also gives examples of Banach algebras which are not n-weakly amenable for each natural number n. We also find out that the dual of \(\mathbf A _\Phi \) does not enjoy the factorization property of any level.

The order spectrum of convolution operators in discrete Cesàro spaces

The spectrum of convolution operators acting in the discrete Cesàro spaces ces(p),1<p<∞, is determined in Ricker [1]. The method of proof is based partially on Banach algebra techniques and partially on an analysis of the shift operator acting in the dual Banach spaces (ces(p))∗,1<p<∞. Unfortunately, the spaces (ces(p))∗ are rather unwieldy. One aim of this note is to provide an alternative proof for the formula of the spectrum of a convolution operator; it is based entirely on the theory of Banach algebras and makes no use of the dual spaces (ces(p))∗. An advantage is that this approach can also be used to show that the spectrum of a convolution operator coincides with its order spectrum when it is considered as an element of the Banach algebra of all regular operators acting in the complex Banach lattice ces(p).

When is multiplication in a Banach algebra open?

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ1(N0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ1(Z) nor in ℓ1(Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.

COMMUTATORS AND SQUARE-ZERO ELEMENTS IN BANACH ALGEBRAS

Our initial result states that, in a certain class of Banach algebras which includes |$C^*$|-algebras and group algebras of locally compact groups, every commutator lies in the closed linear span of square-zero elements. The proof relies on the theory of Banach algebras with property |${\mathbb {B}}$|⁠. We then study several variations and extensions of this result. For instance, we show that in a von Neumann algebra every commutator is actually a finite sum of square-zero elements. We also consider the commutator ideal and the existence of some special square-zero elements.

The ergodic theorems for Markov chains with an arbitrary phase space

The ergodic theorems are formulated both is the case of the uniform ergodicity and in the absence of this latter under more general conditions than before. The conditions laid down on the transition function in the case of nonuniform ergodicity are less restrictive than these by Athreya-Ney and Nummelin. In contrast to these latter the restrictions are set not on the initial transition function, but on that of the imbedded Markov chain which is constructed by the instants of the recurrence to some fixed set. The proof is based on the spectral theory of Banach algebras.

Topological Methods in Analysis

Topological methods have played a seminal role in functional analysis since its birth in the early twentieth century. The Baire category theorem, for example, is the bedrock on which rest such basic principles of functional analysis as the open mapping theorem and the principle of uniform boundedness. Initial (weak) topologies, compactness, and the Tikhonov theorem drive such classical results of duality theory as the Banach-Alaoglu and Krein-Milman theorems. Topological methods also play a crucial role in Banach algebra theory (Gelfand topology), harmonic analysis (locally compact groups and function spaces), differential equations (fixed point theorems and Ascoli-Arzela theorem), and nonlinear analysis (fixed point existence theorems and topological degree theory) to mention just a few. The lofty goal of this special issue is to present some new outstanding contributions in this research area. They include results on fixed point theorems, metric spaces, C-algebras, modular spaces, algebraic hypersurfaces, and topological groups. Our hope is that this wide spectrum of topics will be of interest to many researchers in the area.

On smooth modules

In this note we introduce the notion of smooth module to extend results from homology theory of Banach algebras to the locally convex category. A complete locally convex module over an m-convex algebra is shown to be smooth if and only if it is topologically isomorphic to a reduced inverse limit of Banach modules over Banach algebras. Stability properties for smoothness are discussed and conditions under which an arbitrary locally convex module is rendered smooth are given.

Reducibility in Aℝ(K), Cℝ(K), and A(K)

AbstractLet K denote a compact real symmetric subset of ℂ and let Aℝ(K) denote the real Banach algebra of all real symmetric continuous functions on K that are analytic in the interior K◦ of K, endowed with the supremum norm. We characterize all unimodular pairs ( f , g) in Aℝ(K)2 which are reducible. In addition, for an arbitrary compact K in ℂ, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of A(K) is 1. Finally, we also characterize all compact real symmetric sets K such that Aℝ(K), respectively Cℝ(K), has Bass stable rank 1.

Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices

The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted lp-spaces. In particular, the stability of the finite section method on l2 implies its stability on weighted lp-spaces. Our approach uses recent results from the theory of Banach algebras of matrices with off-diagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of unstructured non-hermitian matrices as well as to least squares problems.

Constructive theory of Banach algebras

We present a way to organize a constructive development of the theory of Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We illustrate this by giving elementary proofs of Wiener's result on the inverse of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we show how this can be used in a localic, or point-free, description of the spectrum of a Banach algebra.

The spectral topology in rings

The spectral topology of a ring is easily defined, has familiar applications in elementary Banach algebra theory, and appears relevant to abstract Fredholm and stable range theory. 0. Introduction. The spectral topology of a ring is motivated by the simple Banach algebra observation that, for x ∈ A, (0.1) ‖x‖ < 1 ⇒ 1− x ∈ A−1. This leads to simple observations about the (norm) closure of the invertible group and its intersection with the one-sided invertibles, relatively regular and Fredholm elements. In a topological algebra the relationship between the spectral and the original topology is able to decide whether or not the invertible group is open, and the spectral topology offers another version of the idea of “stable rank”. Precisely what is the spectral closure of the invertibles in the familiar disc algebra seems to be a delicate question. In this writing each definition and theorem ushers in a new section. We define the spectral closure in §1, prove it gives a topology in §2, with jointly continuous addition and multiplication; in §3 we check some “spectral permanence” and work out some obvious examples. In §4 we make the observation that the spectral topology is the weakest ring topology giving an open group of invertibles, and in §5 we suggest that it offers a possible extension of the “quasinilpotent” concept to general rings. In §6 we chart the interaction between the spectral topology and the “regular” elements of the ring. In §7 we find that onto homomorphisms T : A → B are automatically continuous, that Gelfand homomorphisms are relatively open, and in §8 that if the homomorphism has inverse closed range then the nearly invertible “Fredholm” elements are “Weyl”. In §9 we make an extension of the spectral topology to the space An of n-tuples, and use it to show, in §10, 2010 Mathematics Subject Classification: Primary 46L05.

The Gelfand-Naimark theorem for C*-algebras over a ring of measurable functions

One of the most important results in the theory of Banach algebras is the Gelfand–Naimark theorem. It describes a commutative C∗-algebra U over the field of complex numbers C as an algebra of continuous complex-valued functions defined on the set of pure states of U endowed with an ∗-weak topology. The development of the general theory of Banach–Kantorovich C∗-algebras over a ring of measurable functions naturally leads to the question about an analog of the Gelfand–Naimark theorem for such C∗-modules. The structural theory of C∗-modules originates from the work of I. Kaplansky [1], who used these objects in the algebraic approach to the theory of W ∗-algebras. Considering C∗-algebras, AW ∗-algebras, andW ∗-algebras as modules over their centers, one can describe some properties of the mentioned classes of ∗-algebras with the help of Boolean analysis methods (e.g., [2–5]). In particular, in [4] for C∗-algebras, representing modules over the Stone algebra, one obtains vector analogs of Gelfand–Mazur and Gleason–Zhelyazko–Kahan theorems. C∗-modules serve as useful examples of Banach–Kantorovich modules, whose theory is being actively developed now (e.g., [5, 6]). An efficient tool for the study of these Banach–Kantorovich modules, together with the Boolean-valued analysis, is the theory of continuous and measurable Banach bundles [6]. In particular, this enables one to represent a C∗-module over a ring of measurable functions as a measurable bundle of classical C∗-algebras [7]. This, in turn, allows one to establish properties ofC∗-modules, “gluing” the corresponding properties of C∗-algebras over the field C. In this paper we apply this approach, proving one version of the Gelfand– Naimark theorem for C∗-modules. We use the terminology and the notation of the theory of Banach– Kantorovich spaces [5] and the theory of measurable bundles stated in [6].

On Frobenius, Mazur, and Gelfand-Mazur theorems on division algebras

The main purpose of this paper is to introduce the reader to the theory of normed division algebras. It is written with a graduate student of mathematics in mind, who only had exposure to Advanced Calculus and to an introduction to Functional Analysis. We establish the equivalence of the Mazur and Gelfand-Mazur theorems using only elementary properties of normed algebras. Mazur's theorem [19] states that every normed division algebra over the field of real numbers is isomorphic to either the field R of real numbers, the field C of complex numbers, or the non-commutative algebra Q of quaternions. Gelfand [15] proved that every normed division algebra over the field C is isomorphic to C. He named this theorem, which is fundamental for the development of the theory of Banach Algebras, the Gelfand-Mazur theorem. An analysis of Mazur's proof (see [27], p. 18–23) shows that he proved that theorem firstly for algebras over C (using the theory of Analytic Functions) and then generalized the proof to the case of algebras over R (using the theory of Harmonic Functions.) We prove the theorems that are essential for the development in detail. These include the construction of the algebra of quaternions and the theorems of Frobenius [14], Arens [1], and Stone [26].

Ideals in constructive Banach algebra theory

We introduce the notion of a ‘q-ideal in a commutative Banach algebra, and investigate the relation between q-ideals and maximal ideals constructively.

ON DERIVATIONS IN NONCOMMUTATIVE SEMIPRIME RINGS AND BANACH ALGEBRAS

Let R be a noncommutative semi prime ring. Suppose that there exists a derivation d : R <TEX>$\to$</TEX> R such that for all x <TEX>$\in$</TEX> R, either [[d(x),x], d(x)] = 0 or <TEX>$\langle$</TEX><TEX>$\langle(x),\;x\rangle,\;d(x)\rangle$</TEX> = 0. In this case [d(x), x] is nilpotent for all x <TEX>$\in$</TEX> R. We also apply the above results to a Banach algebra theory.

BARRY EDWARD JOHNSON 1937–2002

Barry Johnson made major contributions to the theory of Banach algebras, by stimulating research on automatic continuity and cohomology in these algebras. His research on the continuous Hochschild cohomology of Banach and operator algebras led to major developments in these areas, and to a recognition of ‘amenability’ as more than simply a group-theoretic idea, but also one that is widely applicable in modern analysis.

Inductive limits of locally m-convex algebras

In this short note we prove that the locally convex inductive limit of commutative locally m-convex algebras is already locally m-convex whenever multiplication is continuous. A topological algebra is an (associative) algebra endowed with a vector space topology such that multiplication is jointly continuous. If one considers inductive limits (in the category of topological vector spaces) of topological algebras multiplication may fail to be continuous since it is a bilinear map on the product which is certainly not linear, and forming inductive limits only respects the linear structure (although this is quite clear, the dierence between linearity and bilinearity in this context has been overlooked several times in the literature). The most important category of topological algebras is that of locally m-convex (l.m.c.) algebras having a 0-neighbourhood basis consisting of m-convex sets (i.e. absolutely convex sets B which are multiplicative in the sense that B 2 B). This class was introduced by Michael [9] and Arens [2] since it allows generalizations and applications of the theory of Banach algebras. Since their work a number of authors studied the question whether the inductive limit (in the category of locally convex spaces) of l.m.c. algebras is again l.m.c., see e.g. [1, 3, 4, 5, 7, 8, 10]. This is indeed true for countable inductive limits of normed algebras [1, 4]. On the other hand, Warner [10] gave an example of an inductive limit of metrizable l.m.c. algebras which is not l.m.c., and in [4] it is shown that [

Barry Edward Johnson. 1 August 1937 – 5 May 2002

Barry Johnson made major contributions to the theory of Banach algebras, stimulating research on automatic continuity and cohomology in these algebras. His research on the continuous Hochschild cohomology of Banach and operator algebras led to major developments in these areas, and to ‘amenability’ moving from a group theoretic idea to one that is widely applicable in modern analysis.