scholarly journals The structure of a subclass of amenable banach algebras

2004 ◽  
Vol 2004 (55) ◽  
pp. 2963-2969 ◽  
Author(s):  
R. El Harti
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Sufficient Conditions ◽  
Full Matrix ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Maximal Left Ideal

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian*-algebras that are contractible.

Relative amenability and the non-amenability of B(l1)

2006 ◽  
Vol 80 (3) ◽  
pp. 317-333
Author(s):  
C. J. Read
Keyword(s):  
Banach Algebra ◽  
Free Group ◽  
Direct Proof ◽  
Group Algebras ◽  
Direct Sums ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Symmetric Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

AbstractIn this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

Spectral characterization of the socle in Jordan–Banach algebras

1995 ◽  
Vol 117 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Finite Rank ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
Bounded Operators ◽  
Finite Rank Operators ◽  
Finite Dimensional ◽  
Minimal Idempotent

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

scholarly journals Banach Algebras Which are a Direct Sum of Division Algebras

1988 ◽  
Vol 44 (2) ◽  
pp. 143-145
Author(s):  
Antonio Fernandez Lopez ◽  
Eulalia Garcia Rus
Keyword(s):  
Finite Number ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Division Algebras ◽  
Direct Sum

AbstractIn this note it is proved that a (real or complex) semiprime Banach algebra A satisfying xAx = x2Ax2 for every x ∈ A is a direct sum of a finite number of division Banach algebras.

scholarly journals Automatic continuity for Banach algebras with finite-dimensional radical

2006 ◽  
Vol 81 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Hung Le Pham
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity ◽  
Algebraic Condition ◽  
Finite Dimensional ◽  
Complete Algebra

AbstractThe paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

scholarly journals Some inequalities for norm unitaries in Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 17-23 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Unit Circle ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Image Position ◽  
Unital Banach Algebra

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

scholarly journals Supersets for the spectrum of elements in extended Banach algebras

1989 ◽  
Vol 12 (4) ◽  
pp. 823-824
Author(s):  
Morteza Seddighin
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Complex Numbers ◽  
Direct Sum

If A is a Banach Algebra with or without an identity, A can be always extended to a Banach algebraA¯with identity, whereA¯is simply the direct sum of A and C, the algebra of complex numbers. In this note we find supersets for the spectrum of elements ofA¯.

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 Strictly Convex Banach Algebras

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 221
Author(s):  
David Yost
Keyword(s):  
Banach Space ◽  
Banach Algebras ◽  
Infinite Dimensions ◽  
Strictly Convex ◽  
C Algebras ◽  
Finite Dimensional ◽  
And Algebra ◽  
Open Question

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C*-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there.

scholarly journals 2-локальные дифференцирования на алгебрах матрично-значных функций на компакте

2018 ◽  
Author(s):  
S.A. Ayupov ◽  
F.N. Arzikulov
Keyword(s):  
Hilbert Space ◽  
Banach Algebras ◽  
Algebraic Approach ◽  
Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Local Derivation

The present paper is devoted to 2-local derivations. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-lo\-cal derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and \mbox{2-local} derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every \mbox{2-local} derivation on a ∗-algebra C(Q,Mn(F)) or C(Q,Nn(F)), where Q is a compactum, Mn(F) is the ∗-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, Nn(F) is the ∗-subalgebra of Mn(F) defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.

scholarly journals Spectral properties of elements in different Banach algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 11-20 ◽  
Author(s):  
J. J. Grobler ◽  
H. Raubenheimer
Keyword(s):  
Banach Algebra ◽  
Spectral Properties ◽  
Banach Algebras ◽  
Sufficient Conditions

Let A be a Banach algebra with unit 1 and let B be a Banach algebra which is a subalgebra of A and which contains 1. In [5]Barnes gave sufficient conditions for B to be inverse closed in A. In this paper we consider single elements and study the question of how the spectrum relative to B of an element in B relates to the spectrum of the element relative to A.

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