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.

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.

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.

scholarly journals Continuity of Lie derivations on Banach algebras

1998 ◽  
Vol 41 (3) ◽  
pp. 625-630 ◽  
Author(s):  
M. I. Berenguer ◽  
A. R. Villena
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Lie Derivation ◽  
Separating Subspace ◽  
Lie Derivations ◽  
Semisimple Banach Algebra

The separating subspace of any Lie derivation on a semisimple Banach algebra A is contained in the centre of A.

scholarly journals 2-Local derivations of real AW*-algebras are derivation

Positivity ◽  
2021 ◽  
Author(s):  
R. A. Dadakhodjaev ◽  
A. A. Rakhimov
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Real Matrix ◽  
Inner Derivation ◽  
The Real ◽  
Matrix Algebras ◽  
Enveloping Algebra ◽  
Local Derivation

Abstract2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra $$M_{2^n}(A)$$ M 2 n ( A ) ($$n\ge 2$$ n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW$$^*$$ ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property.

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 Essentially defined derivations on semisimple Banach algebras

1997 ◽  
Vol 40 (1) ◽  
pp. 175-179 ◽  
Author(s):  
A. R. Villena
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Nonzero Ideal ◽  
C Algebra ◽  
Complex Banach Algebra ◽  
Essential Ideal ◽  
Semisimple Complex

We prove that every partially defined derivation on a semisimple complex Banach algebra whose domain is a (non necessarily closed) essential ideal is closable. In particular, we show that every derivation defined on any nonzero ideal of a prime C*-algebra is continuous.

ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

2021 ◽  
pp. 1-12
Author(s):  
PRAKASH A. DABHI ◽  
DARSHANA B. LIKHADA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Discrete Groups ◽  
Discrete Semigroups ◽  
Beurling Algebras

Abstract Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0\lt p\leq 1$ . We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$ . We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

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