A note on annihilator Banach algebras

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.

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

Journal of the Australian Mathematical Society ◽

10.1017/s144678872000021x ◽

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.

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The Second Conjugates of Certain Banach Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-108-9 ◽

1975 ◽

Vol 27 (5) ◽

pp. 1029-1035 ◽

Cited By ~ 3

Author(s):

Pak-Ken Wong

Keyword(s):

Continuity of Lie derivations on Banach algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500019933 ◽

1998 ◽

Vol 41 (3) ◽

pp. 625-630 ◽

Cited By ~ 14

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.

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Finite dimensionality in scole of Banach algebras

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171284000570 ◽

1984 ◽

Vol 7 (3) ◽

pp. 519-522 ◽

Cited By ~ 2

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.

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Annihilator and complemented banach*-algebras

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010594 ◽

1972 ◽

Vol 13 (1) ◽

pp. 47-66 ◽

Cited By ~ 4

Author(s):

B. J. Tomiuk ◽

Pak-Ken Wong

Keyword(s):

Banach Algebra ◽

Structure Theorem ◽

Banach Algebras ◽

Direct Sum ◽

Dense Subalgebra

The study of complemented Banach*-algebras taken up in [1] was confined mainly toB*-algebras. In the present paper we extend this study to (right) complemented Banach*-algebras in whichx*x= 0 impliesx= 0. We show that ifAis such an algebra then every closed two-sided ideal ofAis a *-ideal. Using this fact we obtain a structure theorem forAwhich states that ifAis semi-simple thenAcan be expressed as a topological direct sum of minimal closed two sided ideals each of which is a complemented Banach*-algebra. It follows thatAis an A*-algebra and is a dense subalgebra of a dual B*-algebraU, which is determined uniquely up to *-isomorphism.

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ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720001392 ◽

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.

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

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500213 ◽

2018 ◽

Vol 11 (02) ◽

pp. 1850021 ◽

Cited By ~ 3

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.

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Compact linear operators from an algebraic standpoint

Glasgow Mathematical Journal ◽

10.1017/s0017089500000069 ◽

1967 ◽

Vol 8 (1) ◽

pp. 41-49 ◽

Cited By ~ 4

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

Journal of Algebra and Its Applications ◽

10.1142/s0219498818501694 ◽

2018 ◽

Vol 17 (09) ◽

pp. 1850169 ◽

Cited By ~ 3

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.

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