Characterizations of $C\sp{\ast} $-algebras. II

p-Adic Numbers, Ultrametric Analysis and Applications, 2018

The main goal of this work is to study the Gelfand spaces of some commutative Banach algebras with unit within the space of bounded linear operators. We will also show, under special condition, that this algebra is isometrically isomorphic to some space of continuous functions defined over a compact. Such isometries preserve idempotent elements. This fact will allow us to define the respective associated measure which is known as spectral measure. Let us also notice that this measure is obtained by restriction of the reciprocal of the Gelfand transform to the set of characteristic functions of clopen subsets of the spectrum of above algebra. We will finish this work showing that each element of such algebras described above can be represented as an integral of some continuous function, where the integral has been defined through the spectral measure.

2004

Using the notion of a Banach operator, we have obtained a decompositional property of a Hilbert space, and the equality of two invertible bounded linear multiplicative operators on a normed algebra with identity.

International Journal of Nonlinear Analysis and Applications, 2021

In this paper, we study some maps from a $C^*$-algebra into a Banach algebra or a Banach module. Under some conditions, by extending on unitization of Banach algebras, we prove that the maps defined into Banach algebras are homomorphisms and the others defined into Banach modules are derivations. Applications of our results in the context of $C^*$-algebras are also provided.

Journal of Function Spaces, 2018

We study isometries on algebras of the Lipschitz maps and the continuously differentiable maps with the values in a commutative unital C⁎-algebra. A precise proof of a theorem of Jarosz concerning isometries on spaces of continuous functions is exhibited.

In this thesis, we investigate some results on multipliers and bounded approximate identities in Banach algebras. In doing this, we first review the general theory of Banach algebras. We define some basic and important concepts relating to Banach algebras, we give some vital examples, mention and prove some important results, and form new Banach algebras from known ones. We then define the notion of bounded approximate identities in Banach algebras, study and establish some basic results, and then consider bounded approximate identities in group algebras. Finally, we define and give examples of multipliers on Banach algebras without order (faithful) and then investigate some interesting results.

Bulletin of the London Mathematical Society, 2009

For a complex Banach algebra A with unit 1, we give several characterizations of the scalars, that is, multiples of the identity. To a large extent, this work is a continuation and generalization of the work done on characterizations of the radical in Banach algebras. In particular it is shown that if a ∈ A has the property that the number of elements in the spectrum of ax is less than or equal to the number of elements in the spectrum of x for all x in an arbitrary neighbourhood of 1, then a is a scalar. Moreover, as a consequence of some of the results, new spectral characterizations of commutative Banach algebras are obtained. In particular, A is commutative if and only if it has the property that the number of elements in the spectrum remains invariant under all permutations of three elements in some neighbourhood of the identity.

2014

In this thesis we discuss amenability properties of the Banach algebra-valued continuous functions on a compact Hausdorff space X . Let A be a Banach algebra. The space of Avalued continuous functions on X , denoted by C(X,A), form a new Banach algebra. We show that C(X,A) has a bounded approximate diagonal (i.e. it is amenable) if and only if A has a bounded approximate diagonal. We also show that if A has a compactly central approximate diagonal then C(X,A) has a compact approximate diagonal. We note that, unlike C(X), in general C(X,A) is not a C∗-algebra, and is no longer commutative if A is not so. Our method is inspired by a work of M. Abtahi and Y. Zhang. In addition to the above investigation, we directly construct a bounded approximate diagonal for C0(X), the Banach algebra of the closure of compactly supported continuous functions on a locally compact Hausdorff space X .

Rocky Mountain Journal of Mathematics, 2016

Let X be a completely regular Hausdorff space. We denote by C(X, A) the algebra of all continuous functions on X with values in a complex commutative unital Banach algebra A. Let C b (X, A) be its subalgebra consisting of all bounded continuous functions and endowed with the uniform norm. In this paper, we give conditions equivalent to the density of a natural continuous image of X × M(A) in the maximal ideal space of C b (X, A).

Bulletin of the London Mathematical Society, 2003

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies u = u −1 = 1. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A . If X is a complex Hilbert space, then the algebra BL(X) of all bounded linear operators on X is unitary by the Russo-Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) BL(X) is unitary and, for Y equal to X, X * or X * * , there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y .

Author please supply abstract 1. Introduction and preliminaries Let L be a subset of a ring A. A functional identity on L is an identity which involves maps from L into A and holds for all elements in L. Much work has been done on functional identities satisfied by derivations, automorphisms and some other special additive maps (see [3]). Posner [18] proved that if A is prime and δ is a non-zero commuting derivation on A, that is, [δ(a), a] = 0 for each a ∈ A, then A has to be commutative. Making use of this, Mathieu and Murphy [15] generalized the result of Singer and Wermer and showed that any bounded commuting derivation on a Banach algebra maps it into the radical. The boundedness assumption was later dropped in [16]. Bresar [5] initiated an intensive study of commuting maps T from L into A: