On The Center Of Quasi-Central Banach Algebras With bounded Approximate Identity

1981 ◽  
Vol 33 (1) ◽  
pp. 68-90
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Central Complex ◽  
Double Centralizer ◽  
Completely Regular ◽  
Continuous Complex ◽  
Complex Valued ◽  
Canonical Image

Let A be a quasi-central complex Banach algebra with a bounded approximate identity and Prim A the structure space of A. In [15], we have shown that every central double centralizer T on A can be represented as a bounded continuous complex-valued function ΦT on Prim A such that Tx + P = ΦT(P)(x + P) for all x ∈ A and P ∈ Primal when the center Z(A) of A is completely regular. Here x + P for P ∈ Prim A denotes the canonical image of x in A/P. In particular, in the case of quasi-central C*-algebras, this result is equivalent to the Dixmier's representation theorem of central double centralizers on C*-algebras (see [3, Section 2] and [9, Theorem 5]).In this paper, it is shown that if Z(A) is completely regular then the space Prim A is locally quasi-compact and for each element z of Z(A), ΦLz vanishes at infinity, where Lz for z ∊ Z(i) is the central double centralizer on A defined by Lz(x) = zx for all x ∊ A.

Central Double Centralizers on Quasi-Central Banach Algebras with Bounded Approximate Identity

1983 ◽  
Vol 35 (2) ◽  
pp. 373-384
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Central Complex ◽  
Double Centralizer ◽  
Continuous Complex ◽  
Image Position ◽  
Centralizer Algebra ◽  
Complex Valued ◽  
Canonical Image

We assume throughout this paper that A is a semi-simple, quasi-central, complex Banach algebra with a bounded approximate identity {eα}. The author [6] has shown that every central double centralizer T on A can be, under suitable conditions, represented as a bounded continuous complex-valued function ΦT on Prim A, the structure space of A with the hull-kernel topology, such thatHere x + P for P ∊ Prim A denotes the canonical image of x in A/P. This map Φ is called Dixmier's representation of Z(M(A)), the central double centralizer algebra of A. We denote by τ the canonical isomorphism of A into the Banach algebra D(A) with the restricted Arens product as defined in [6]. Also denote by μ Davenport's representation of Z(M(A)). In fact, this map μ is given byfor each T ∊ Z(M(A)).

scholarly journals REPRESENTATIONS OF REAL BANACH ALGEBRAS

2010 ◽  
Vol 88 (3) ◽  
pp. 289-300 ◽  
Author(s):  
F. ALBIAC ◽  
E. BRIEM
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Ideal Space ◽  
Gelfand Transform ◽  
Continuous Complex ◽  
Unital Banach Algebra ◽  
Complex Valued

AbstractA commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

Cohen elements in Banach algebras

1979 ◽  
Vol 84 (1-2) ◽  
pp. 55-70 ◽  
Author(s):  
Allan M. Sinclair
Keyword(s):  
Banach Algebra ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Continuous Complex ◽  
Cohen Factorization ◽  
Definition Of ◽  
Invertible Elements ◽  
Unital Banach Algebra ◽  
Complex Valued ◽  
Locally Compact Hausdorff Space

SynopsisThe definition of Cohen elements in a commutative Banach algebra with a countable bounded approximate identity given by Esterle is modified slightly to be more analogous to the invertible elements in a unital Banach algebra. With the modified definition the n1-Cohen factorization results that were proved by Esterle are shown tohold in the semigroup of Cohen elements. If is the algebra of continuous complex valued functions vanishing at infinity on a σ-compact locally compact Hausdorff space X, then the Cohen elements in are identified and a natural quotient of a subsemigroup of Cohen elements is shown to be a group, isomorphic to the abstract index group of C(X∪{∞}).

Operator Algebras with Contractive Approximate Identities

1994 ◽  
Vol 46 (2) ◽  
pp. 397-414 ◽  
Author(s):  
Yiu-Tung Poon ◽  
Zhong-Jin Ruan
Keyword(s):  
Operator Algebra ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Approximate Identity ◽  
The Other ◽  
Double Centralizer ◽  
Approximate Identities ◽  
Centralizer Algebras ◽  
Matrix Norms ◽  
Centralizer Algebra

AbstractWe study operator algebras with contractive approximate identities and their double centralizer algebras. These operator algebras can be characterized as L∞- Banach algebras with contractive approximate identities. We provide two examples, which show that given a non-unital operator algebra A with a contractive approximate identity, its double centralizer algebra M(A) may admit different operator algebra matrix norms, with which M(A) contains A as an M-ideal. On the other hand, we show that there is a unique operator algebra matrix norm on the unitalization algebra A1 of A such that A1 contains A as an M-ideal.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

Amenability of Banach algebras

1989 ◽  
Vol 105 (2) ◽  
pp. 351-355 ◽  
Author(s):  
Frédéric Gourdeau
Keyword(s):  
Banach Algebra ◽  
Metric Space ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Commutative Banach Algebra ◽  
Cohomology Theory ◽  
Compact Metric Space ◽  
Amenable Banach Algebra ◽  
Second Dual

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

Weak Amenability of a Class of Banach Algebras

2001 ◽  
Vol 44 (4) ◽  
pp. 504-508 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Dual Algebra ◽  
Weak Amenability ◽  
Second Dual

AbstractWe show that, if a Banach algebra is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of implies the (2m+ 1)-weak amenability of for all m ≥ 1.

scholarly journals Factorization and bounded approximate identities for a class of convolution Banach algebras

1986 ◽  
Vol 28 (2) ◽  
pp. 211-214 ◽  
Author(s):  
S. I. Ouzomgi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Approximate Identities

An algebra A factors if, for each a ∈ A, there exist b, c ∈ A with a = bc. A bounded approximate identity for a Banach algebra A is a net (eα) ⊂ A such that aeα → a and eαa → a for each a ∈ A and such that sup ‖eα ‖ < ∞. It is well known [2, 11.10] that if A has a bounded approximate identity, then A factors. But a Banach algebra may factor even if it does not have a bounded approximate identity: an example which is non-commutative and separable, and an example which is commutative and nonseparable, are given in [3, §22]. However, we do not know an example of a commutative, separable Banach algebra which factors, but which does not have a bounded approximate identity. See 4 for related work.

The quasi-strict topology on the space of quasi-multipliers of a B*-algebra

1987 ◽  
Vol 101 (3) ◽  
pp. 555-566 ◽  
Author(s):  
M. S. Kassem ◽  
K. Rowlands
Keyword(s):  
General Theory ◽  
Banach Algebra ◽  
A Priori ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Strict Topology ◽  
General Object ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Special Case

The notion of a left (right, double) multiplier may be regarded as a generalization of the concept of a multiplier to a non-commutative Banach algebra. Each of these is a special case of a more general object, namely that of a quasi-multiplier. The idea of a quasi-multiplier was first introduced by Akemann and Pedersen in ([1], §4), where they consider the quasi-multipliers of a C*-algebra. One of the defects of quasi-multipliers is that, at least a priori, there does not appear to be a way of multiplying them together. The general theory of quasi-multipliers of a Banach algebra A with an approximate identity was developed by McKennon in [5], and in particular he showed that the quasi-multipliers of a considerable class of Banach algebras could be multiplied. McKennon also introduced a locally convex topology γ on the space QM(A) of quasi-multipliers of A and derived some of the elementary properties of (QM(A), γ).

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.

Sign in / Sign up

Export Citation Format

Share Document