scholarly journals Commutative Gelfand Theory for Real Banach Algebras: Representations as Sections of Bundles

1992 ◽  
Vol 44 (2) ◽  
pp. 342-356
Author(s):  
W. E. Pfaffenberger ◽  
J. Phillips
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Natural Classification ◽  
C Algebras ◽  
Gelfand Transform ◽  
Maximal Ideals ◽  
Almost Complex ◽  
Topological Obstruction ◽  
Algebra Homomorphisms

AbstractWe are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space.In § 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted ℳ, and the set of unital (real) algebra homomorphisms from B into C, denoted by ΦB. This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex.Finally, in § 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras “almost complex C*- algebras” and provide a natural classification scheme.

scholarly journals Estimations des solutions de l'équation de Bezout dans les algèbres de Beurling analytiques

2005 ◽  
Vol 96 (2) ◽  
pp. 307 ◽  
Author(s):  
O. El-Fallah ◽  
M. Zarrabi
Keyword(s):  
Banach Algebra ◽  
Existence Of Solutions ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Gelfand Transform ◽  
Maximal Ideals ◽  
Bezout Equation ◽  
Beurling Algebras

Let $A$ be a unitary commutative Banach algebra with unit $e$. For $f\in A$ we denote by $\hat f$ the Gelfand transform of $f$ defined on $\hat A$, the set of maximal ideals of $A$. Let $(f_1,\dots,f_n)\in A^n$ be such that $\sum_{i=1}^n\|f_i\|^2 \leq 1$. We study here the existence of solutions $(g_1,\dots,g_n)\in A^n$ to the Bezout equation $f_1g_1+\cdots+f_ng_n=e$, whose norm is controlled by a function of $n$ and $\delta=\inf_{\chi\in\hat A}(|\hat f_1(\chi)|^2+\cdots+|\hat f_n(\chi)|^2)^{1/2}$. We treat this problem for the analytic Beurling algebras and their quotient by closed ideals. The general Banach algebras with compact Gelfand transform are also considered.

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

2018 ◽  
Vol 17 (09) ◽  
pp. 1850169 ◽  
Author(s):  
Hossein Javanshiri ◽  
Mehdi Nemati
Keyword(s):  
Banach Algebra ◽  
Cohomology Group ◽  
Banach Algebras ◽  
Multiplier Algebra ◽  
Topological Center ◽  
Maximal Ideals ◽  
Open Questions ◽  
Arens Regularity ◽  
Amalgamated Duplication

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

scholarly journals Perturbation Theory for Quasinilpotents in Banach Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1163
Author(s):  
Xin Wang ◽  
Peng Cao
Keyword(s):  
Perturbation Theory ◽  
Banach Algebra ◽  
Perturbation Technique ◽  
Banach Algebras ◽  
C Algebras

In this paper, we prove the following result by perturbation technique. If q is a quasinilpotent element of a Banach algebra and spectrum of p + q for any other quasinilpotent p contains at most n values then q n = 0 . Applications to C* algebras are given.

scholarly journals Stable perturbation in Banach algebras

2007 ◽  
Vol 83 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Yifeng Xue
Keyword(s):  
Banach Algebra ◽  
Error Estimates ◽  
Generalized Inverse ◽  
Invertible Element ◽  
Banach Algebras ◽  
Gap Function ◽  
Drazin Inverse ◽  
C Algebras ◽  
Stable Perturbation ◽  
Unital Banach Algebra

AbstractLet be a unital Banach algebra. Assume that a has a generalized inverse a+. Then is said to be a stable perturbation of a if . In this paper we give various conditions for stable perturbation of a generalized invertible element and show that the equation is closely related to the gap function . These results will be applied to error estimates for perturbations of the Moore-Penrose inverse in C*–algebras and the Drazin inverse in Banach algebras.

Symmetric amenability and the nonexistence of Lie and Jordan derivations

1996 ◽  
Vol 120 (3) ◽  
pp. 455-473 ◽  
Author(s):  
B. E. Johnson
Keyword(s):  
Positive Result ◽  
Banach Algebra ◽  
Recent Work ◽  
Banach Algebras ◽  
Jordan Derivation ◽  
Symmetric Tensors ◽  
C Algebras ◽  
Semisimple Algebras ◽  
First Time ◽  
Semisimple Banach Algebra

A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

scholarly journals Some Inequalities arising from a Banach algebra norm

1983 ◽  
Vol 34 (2) ◽  
pp. 199-202 ◽  
Author(s):  
A. M. Russell
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Norm Inequality ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWe derive some specific inequalities involving absolutely continuous functions and relate them to a norm inequality arising from Banach algebras of functions having bounded k th variation.

Functionals on Real C(S)

1978 ◽  
Vol 30 (03) ◽  
pp. 490-498 ◽  
Author(s):  
Nicholas Farnum ◽  
Robert Whitley
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Codimension One ◽  
Maximal Ideals ◽  
Invertible Elements ◽  
Complex Valued

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property: (*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

scholarly journals REPRESENTATIONS OF REAL BANACH ALGEBRAS

2010 ◽  
Vol 88 (3) ◽  
pp. 289-300 ◽  
Author(s):  
F. ALBIAC ◽  
E. BRIEM
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Ideal Space ◽  
Gelfand Transform ◽  
Continuous Complex ◽  
Unital Banach Algebra ◽  
Complex Valued

AbstractA commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

scholarly journals Hermitian Operators and Isometries on Banach Algebras of Continuous Maps with Values in Unital Commutative C⁎-Algebras

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Osamu Hatori
Keyword(s):  
Banach Algebras ◽  
Continuous Functions ◽  
C Algebras ◽  
Lipschitz Maps ◽  
Spaces Of Continuous Functions ◽  
Continuous Maps

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.

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