Automatic continuity and second order cohomology

The first main theorem of this paper asserts that any $$(\sigma , \tau )$$ -derivation d, under certain conditions, either is a $$\sigma $$ -derivation or is a scalar multiple of ( $$\sigma - \tau $$ ), i.e. $$d = \lambda (\sigma - \tau )$$ for some $$\lambda \in \mathbb {C} \backslash \{0\}$$ . By using this characterization, we achieve a result concerning the automatic continuity of $$(\sigma , \tau $$ )-derivations on Banach algebras which reads as follows. Let $$\mathcal {A}$$ be a unital, commutative, semi-simple Banach algebra, and let $$\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}$$ be two distinct endomorphisms such that $$\varphi \sigma (\mathbf e )$$ and $$\varphi \tau (\mathbf e )$$ are non-zero complex numbers for all $$\varphi \in \Phi _\mathcal {A}$$ . If $$d : \mathcal {A} \rightarrow \mathcal {A}$$ is a $$(\sigma , \tau )$$ -derivation such that $$\varphi d$$ is a non-zero linear functional for every $$\varphi \in \Phi _\mathcal {A}$$ , then d is automatically continuous. As another objective of this research, we prove that if $$\mathfrak {M}$$ is a commutative von Neumann algebra and $$\sigma :\mathfrak {M} \rightarrow \mathfrak {M}$$ is an endomorphism, then every Jordan $$\sigma $$ -derivation $$d:\mathfrak {M} \rightarrow \mathfrak {M}$$ is identically zero.

Automatic continuity of M-norms on [formula omitted]-algebras

On constructing multiplications on Banach spaces

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.

In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined byLet B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.

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.

Characterization of n-Jordan homomorphisms and their automatic continuity on banach algebras

Automatic Continuity of Lipschitz Algebras

This work has arisen from lecture courses given by the authors on important topics within functional analysis. The authors, who are all leading researchers, give introductions to their subjects at a level ideal for beginning graduate students, and others interested in the subject. The collection has been carefully edited so as to form a coherent and accessible introduction to current research topics. The first chapter by Professor Dales introduces the general theory of Banach algebras, which serves as a background to the remaining material. Dr Willis then studies a centrally important Banach algebra, the group algebra of a locally compact group. The remaining chapters are devoted to Banach algebras of operators on Banach spaces: Professor Eschmeier gives all the background for the exciting topic of invariant subspaces of operators, and discusses some key open problems; Dr Laursen and Professor Aiena discuss local spectral theory for operators, leading into Fredholm theory.

Let G be a locally compact abelian group and let \({\bf T}{\text{ = }}{\left\{ {T{\left( g \right)}} \right\}}_{{g \in G}} \) be a representation of G by means of isometries on a Banach space. We define WT as the closure with respect to the weak operator topology of the set \({\left\{ {\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)}:f \in L^{1} {\left( G \right)}} \right\}}, \) where \(\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)} = {\int\limits_G {f{\left( g \right)}T{\left( g \right)}dg} } \) is the Fourier transform of f ∈L1(G) with respect to the group T. Then WT is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that if the Arveson spectrum sp(T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra WT is semisimple.

Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may be found of the form C ( X ) C(X) for a compact space X X , ℓ 1 ( G ) \ell _1(G) for some torsion-free Abelian group, and a simple, unital AF C ∗ C^* -algebra. Extensions to higher densities are also presented.

Let A be a Banach algebra. For f ∈ A*, we inspect the weak sequential properties of the well-known map Tf : A → A*, Tf (a) = fa, where fa ∈ A* is defined by fa(x) = f (ax) for all x ∈ A. We provide equivalent conditions for when Tf is completely continuous for every f ∈ A*, and for when Tf maps weakly precompact sets onto L-sets for every f ∈ A*. Our results have applications to the algebra of compact operators K(X) on a Banach space X.

The main purpose of this paper is to present some new existence of hybrid best proximity point theorems involving the sum of two non-self operators in Banach spaces as well as the product of two non-self operators in Banach algebras. In this way, we extend and revisited the main conclusions of a recent paper by Kar and Veeramani [S. Kar, P. Veeramani, Best proximity version of Krasnoselskii?s fixed point theorem, Acta Sci. Math. (Szeged), 86 (2020), 265-271]. Illustrative examples are given to support our results.