scholarly journals Finite dimensionality in scole of Banach algebras

1984 ◽  
Vol 7 (3) ◽  
pp. 519-522 ◽  
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Semisimple Banach Algebra

It is shown that if the soclesoc(A)of a semisimple Banach algebraAis norm-closed, thensoc(A)is already finite dimensional. The proof makes use of the Al-Moajil theorem. However it is remarked that our main theorem is an extension of the Al-Moajil's.

scholarly journals Finite dimensionality, nilpotents and quasinilpotents in Banach algebras

1974 ◽  
Vol 19 (1) ◽  
pp. 45-49 ◽  
Author(s):  
J. Duncan ◽  
A. W. Tullo
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Finite Dimensional ◽  
Finite Dimensionality

In this note we present some rather loosely connected results on Banach algebras together with some illustrative examples. We consider various conditions on a Banach algebra which imply that it is finite dimensional. We also consider conditions which imply the existence of non-zero nilpotents, and hence the existence of finite dimensional subalgebras. In the setting of Banach algebras quasinilpotents figure more prominently than nilpotents. We give an example of a non-commutative Banach algebra in which 0 is the only quasinilpotent; this resolves a problem of Hirschfeld and Zelazko (4).

scholarly journals Conditions on Banach algebras which imply finite dimensionality

1976 ◽  
Vol 20 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Alan W. Tullo
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Complex Numbers ◽  
Finite Dimensional ◽  
Finite Dimensionality

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. We shall assume throughout that we are dealing with Banach algebras over the field of complex numbers,C.

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.

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.

2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

2020 ◽  
pp. 1-12
Author(s):  
WENBO HUANG ◽  
JIANKUI LI
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Subspace Lattice ◽  
Local Derivation ◽  
Essential Ideal ◽  
Semisimple Banach Algebra

Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$ . We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$ , then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$ -subspace lattice algebras, is a derivation.

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.

Banach algebras satisfying certain chain conditions on closed ideals

2020 ◽  
Vol 57 (3) ◽  
pp. 290-297
Author(s):  
Abdullah Alahmari ◽  
Falih A. Aldosray ◽  
Mohamed Mabrouk
Keyword(s):  
Banach Algebra ◽  
Jacobson Radical ◽  
Banach Algebras ◽  
Chain Condition ◽  
Chain Conditions ◽  
Finite Dimensional ◽  
Descending Chain Condition ◽  
Unital Banach Algebra ◽  
Ascending Chain Condition

AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

scholarly journals The compactum and finite dimensionality in Banach algebras

1982 ◽  
Vol 5 (2) ◽  
pp. 275-280 ◽  
Author(s):  
Abdullah H. Al-Moajil
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
General Properties ◽  
Finite Dimensionality

Given a Banach algebraA, the compactum ofAis defined to be the set of elementsx∈Asuch that the operatora→xaxis compact. General properties of the compactum and its relation to the socle ofAare discussed. Characterizations of finite dimensionality of a semi-simple Banach algebra are given in terms of the compactum and the socle ofA.

scholarly journals On one-sided primitivity of Banach algebras

2010 ◽  
Vol 53 (1) ◽  
pp. 111-123 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Semigroup Algebra ◽  
Alternative Proof ◽  
Infinite Dimensional ◽  
Finite Dimensional

AbstractLet S be the semigroup with identity, generated by x and y, subject to y being invertible and yx = xy2. We study two Banach algebra completions of the semigroup algebra ℂS. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that ℂS is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for ℂS is finite dimensional and hence that ℂS has a separating family of such modules.

scholarly journals A note on annihilator Banach algebras

1988 ◽  
Vol 38 (1) ◽  
pp. 77-81
Author(s):  
Pak-Ken Wong
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Dense Subalgebra ◽  
Semisimple Banach Algebra

LetAbe a semisimple Banach algebra with ‖ · ‖, which is a dense subalgebra of a semisimple Banach algebraBwith | · | such that ‖ · ‖ majorises | · | onA. The purpose of this paper is to investigate the annihilator property between the algebrasAandB.

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