scholarly journals ULTRAPOWERS OF BANACH ALGEBRAS AND MODULES

2008 ◽  
Vol 50 (3) ◽  
pp. 539-555 ◽  
Author(s):  
MATTHEW DAWS
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Main Tool ◽  
Bilinear Map ◽  
General Will ◽  
Arens Product ◽  
C Algebras ◽  
Local Reflexivity

AbstractThe Arens products are the standard way of extending the product from a Banach algebrato its bidual″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that ifis Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

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 Amenability and topological centres of the second duals of Banach algebras

2002 ◽  
Vol 65 (2) ◽  
pp. 191-197 ◽  
Author(s):  
F. Ghahramani ◽  
J. Laali
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Dual Algebra ◽  
Arens Product ◽  
Weak Amenability ◽  
Arens Regularity ◽  
Second Dual

Let  be a Banach algebra and let ** be the second dual algebra of  endowed with the first or the second Arens product. We investigate relations between amenability of ** and Arens regularity of  and the rôle topological centres in amenability of **. We also find conditions under which weak amenability of ** implies weak amenability of .

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.

The Second Conjugates of Certain Banach Algebras

1975 ◽  
Vol 27 (5) ◽  
pp. 1029-1035 ◽  
Author(s):  
Pak-Ken Wong
Keyword(s):  
Recent Result ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Arens Product ◽  
Conjugate Space ◽  
Natural Extensions ◽  
Semisimple Banach Algebra

Let A be a Banach algebra and A** its second conjugate space. Arens has denned two natural extensions of the product on A to A**. Under either Arens product, A** becomes a Banach algebra. Let A be a semisimple Banach algebra which is a dense two-sided ideal of a B*-algebra B and R** the radical of (A**, o). We show that A** = Q ⊕ R**, where Q is a closed two-sided ideal of A**, o). This was inspired by Alexander's recent result for simple dual A*-algebras (see [1, p. 573, Theorem 5]). We also obtain that if A is commutative, then A is Arens regular.

On amenability of the Banach algebras l∞(S, [Ufr ])

2002 ◽  
Vol 132 (2) ◽  
pp. 319-322
Author(s):  
FÉLIX CABELLO SÁNCHEZ ◽  
RICARDO GARCÍA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Short Note ◽  
Basic Information ◽  
Arens Product ◽  
Pointwise Multiplication ◽  
Weak Amenability ◽  
Usual Norm ◽  
Bounded Functions ◽  
Second Dual

Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ = sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write l∈([Ufr ]).In this short note, we exhibit examples of amenable (resp. weakly amenable) Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable), thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to [4, 9, 10] for background on amenability and weak amenability. For basic information about the Arens product in the second dual of a Banach algebra the reader can consult [5, 6].

Isomorphisms and multipliers on second dual algebras of Banach algebras

1992 ◽  
Vol 111 (1) ◽  
pp. 161-168 ◽  
Author(s):  
Fereidoun Ghahramani ◽  
Anthony To-Ming Lau
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Weakly Compact ◽  
Dual Algebra ◽  
Arens Product ◽  
Second Dual

Suppose that A is a Banach algebra and let A be the second dual algebra of A equipped with the first Arens product 3. In this paper we characterize compact and weakly compact multipliers of A, when A possesses a bounded approximate identity and is a two sided ideal in A. We use this to study the isomorphisms between second duals of various classes of Banach algebras satisfying the above properties.

scholarly journals ZERO JORDAN PRODUCT DETERMINED BANACH ALGEBRAS

2020 ◽  
pp. 1-14
Author(s):  
J. ALAMINOS ◽  
M. BREŠAR ◽  
J. EXTREMERA ◽  
A. R. VILLENA
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Group Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Continuous Linear ◽  
Bilinear Map ◽  
Locally Compact ◽  
Jordan Product

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.

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.

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