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.

scholarly journals Strictly real Banach algebras

1993 ◽  
Vol 47 (3) ◽  
pp. 505-519 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Banach Algebra ◽  
Uniqueness Theorem ◽  
Banach Algebras ◽  
Complex Algebra ◽  
Conjugation Operator ◽  
Complex Banach Algebra

A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.

scholarly journals ON INTEGRAL REPRESENTATIONS OF THE DRAZIN INVERSE IN BANACH ALGEBRAS

2002 ◽  
Vol 45 (2) ◽  
pp. 327-331 ◽  
Author(s):  
N. Castro González ◽  
J. J. Koliha ◽  
Yimin Wei
Keyword(s):  
Integral Representation ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Integral Representations ◽  
Subject Classification ◽  
Drazin Inverse ◽  
General Situation ◽  
Primary 46H05 ◽  
C Algebra

AbstractThe purpose of this paper is to derive an integral representation of the Drazin inverse of an element of a Banach algebra in a more general situation than previously obtained by the second author, and to give an application to the Moore–Penrose inverse in a $C^*$-algebra.AMS 2000 Mathematics subject classification:Primary 46H05; 46L05

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.

scholarly journals An order-preserving representation theorem for complex Banach algebras and some examples

1973 ◽  
Vol 14 (2) ◽  
pp. 128-135 ◽  
Author(s):  
A. C. Thompson ◽  
M. S. Vijayakumar
Keyword(s):  
Banach Algebra ◽  
Representation Theorem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Order Structure ◽  
Commutative Case ◽  
Complex Matrices ◽  
Complex Banach Algebra

Let A be a complex Banach algebra with unit e of norm one. We show that A can be represented on a compact Hausdorff space ω which arises entirely out of the algebraic and norm structures of A. This space induces an order structure on A that is preserved by the representation. In the commutative case, ω is the spectrum of A, and we have a generalization of Gelfand's representation theorem for commutative complex Banach algebras with unit. Various aspects of this representation are illustrated by considering algebras of n × n complex matrices.

scholarly journals A generalization of Gelfand-Mazur theorem

1993 ◽  
Vol 16 (2) ◽  
pp. 401-402 ◽  
Author(s):  
Sung Guen Kim
Keyword(s):  
Banach Algebra ◽  
Complex Banach Algebra ◽  
Semisimple Complex

In this paper, we show that ifAis a unital semisimple complex Banach algebra with only the trivial idempotents and ifσA(x)is countable for eachx∈Fr(G(A)), thenA≅C, this generalizes the Gelfand-Mazur theorem.

Complemented Banach Algebras

1964 ◽  
Vol 16 ◽  
pp. 149-150
Author(s):  
A. Olubummo
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Complex Banach Algebra ◽  
Left Annihilator

Let A be a complex Banach algebra and Lr (Ll) be the lattice of all closed right (left) ideals in A. Following Tomiuk (5), we say that A is a right complemented algebra if there exists a mapping I —> IP of Lτ into Lr such that if I ∊ Lr, then I ∩ Ip = (0), (Ip)p = I, I ⴲ Ip = A and if I1, I2 ∊ Lr with I1 ⊆ I2 then .If in a Banach algebra A every proper closed right ideal has a non-zero left annihilator, then A is called a left annihilator algebra. If, in addition, the corresponding statement holds for every proper closed left ideal and r(A) = (0) = l(A), A is called an annihilator algebra (1).

scholarly journals On the strong radical of certain Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 81-85 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Complex Banach Algebra ◽  
Maximal Ideals

Let A be a complex Banach algebra. By an ideal in A we mean a two-sided idealunless otherwise specified. As in (7, p. 59) by the strong radical of A we mean theintersection of the modular maximal ideals of A (if there are no such ideals we set =A). Our aim is to discuss the nature of and the relation of to A for a specialclass of Banach algebras. Henceforth A will denote a semi-simple modular annihilatorBanach algebra (one for which the left (right) annihilator of each modular maximalright (left) ideal is not (0)). For the theory of such algebras see (2) and (9).

A theorem in Banach algebras and its applications

1979 ◽  
Vol 20 (2) ◽  
pp. 247-252 ◽  
Author(s):  
V.K. Srinivasan ◽  
Hu Shaing
Keyword(s):  
Banach Algebra ◽  
Principal Ideal ◽  
Banach Algebras ◽  
Negative Integer ◽  
Complex Field ◽  
Principal Ideal Domain ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Complex Banach Algebra ◽  
Divisor Of Zero

If A is a complex Banach algebra which is also a Bezout domain, it is shown that for any prime p and a non-negative integer n, pn is not a topological divisor of zero. Using the above result it is shown that a complex Banach algebra which is a principal ideal domain is isomorphic to the complex field.

scholarly journals Banach algebras, Samelson products, and the Wang differential

2014 ◽  
Vol 06 (02) ◽  
pp. 281-303
Author(s):  
Claude L. Schochet
Keyword(s):  
Banach Algebra ◽  
Principal Bundle ◽  
Banach Algebras ◽  
Homotopy Groups ◽  
Rational Homotopy ◽  
Connected Component ◽  
C Algebra ◽  
Close Analysis ◽  
K Theory ◽  
Compare And Contrast

Assume that given a principal G bundle ζ : P → Sk (with k ≥ 2) and a Banach algebra B upon which G acts continuously. Let [Formula: see text] denote the associated bundle and let [Formula: see text] denote the associated Banach algebra of sections. Then π* GL Aζ⊗B is determined by a mostly degenerate spectral sequence and by a Wang differential [Formula: see text] We show that if B is a C*-algebra then the differential is given explicitly in terms of an enhanced Samelson product with the clutching map of the principal bundle. Analogous results hold after localization and in the setting of topological K-theory. We illustrate our technique with a close analysis of the invariants associated to the C*-algebra of sections of the bundle [Formula: see text] constructed from the Hopf bundle ζ : S7 → S4 and by the conjugation action of S3 on M2 = M2(ℂ). We compare and contrast the information obtained from the homotopy groups π*( U ◦Aζ⊗M2), the rational homotopy groups π*( U ◦Aζ⊗M2) ⊗ ℚ and the topological K-theory groups K*(Aζ⊗M2), where U ◦B is the connected component of the unitary group of the C*-algebra B.

Spectra of approximate identities in Banach algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 271-278 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Unit Interval ◽  
Complex Banach Algebra ◽  
Approximate Identities

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 1].

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