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 Bundles of Banach algebras

1994 ◽  
Vol 17 (4) ◽  
pp. 671-680
Author(s):  
J. W. Kitchen ◽  
D. A. Robbins
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras

We study bundles of Banach algebrasπ:A→X, where each fiberAx=π−1({x})is a Banach algebra andXis a compact Hausdorff space. In the case where all fibers are commutative, we investigate how the Gelfand representation of the section space algebraΓ(π)relates to the Gelfand representation of the fibers. In the general case, we investigate how adjoining an identity to the bundleπ:A→Xrelates to the standard adjunction of identities to the fibers.

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).

Discontinuous homomorphisms from Banach *-algebras

1996 ◽  
Vol 120 (4) ◽  
pp. 703-708
Author(s):  
Volker Runde
Keyword(s):  
Banach Algebra ◽  
Open Problem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuum Hypothesis ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Commutative Banach Algebras ◽  
The Continuum

The long open problem raised by I. Kaplansky if, for an infinite compact Hausdorff space X, there is a discontinuous homomorphism from (X) into a Banach algebra was settled in the 1970s, independently, by H. G. Dales and J. Esterle. If the continuum hypothesis is assumed, then there is a discontinuous homomorphism from (X) (see [8] for a survey of both approaches and [9] for a unified exposition). The techniques developed by Dales and Esterle are powerful enough to yield discontinuous homomorphisms from commutative Banach algebras other than (X). In fact, every commutative Banach algebra with infinitely many characters is the domain of a discontinuous homomorphism ([7]).

scholarly journals BANACH ALGEBRAS OF VECTOR-VALUED FUNCTIONS

2013 ◽  
Vol 56 (2) ◽  
pp. 419-426 ◽  
Author(s):  
AZADEH NIKOU ◽  
ANTHONY G. O'FARRELL
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Function Algebra ◽  
Shilov Boundary ◽  
Function Algebras ◽  
Peak Points ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractWe introduce the concept of an E-valued function algebra, a type of Banach algebra that consists of continuous E-valued functions on some compact Hausdorff space, where E is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative E-valued function algebras. We give some specific examples.

HOMOMORPHISMS OF BANACH ALGEBRAS WITH RANGE IN C1[0, 1]

1994 ◽  
Vol 05 (02) ◽  
pp. 201-212 ◽  
Author(s):  
HERBERT KAMOWITZ ◽  
STEPHEN SCHEINBERG
Keyword(s):  
Analytic Functions ◽  
Compact Hausdorff Space ◽  
Unit Disc ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Locally Compact ◽  
Disc Algebra ◽  
Uniform Algebras ◽  
Closed Subalgebra ◽  
Uniformly Closed

Many commutative semisimple Banach algebras B including B = C (X), X compact, and B = L1 (G), G locally compact, have the property that every homomorphism from B into C1[0, 1] is compact. In this paper we consider this property for uniform algebras. Several examples of homomorphisms from somewhat complicated algebras of analytic functions to C1[0, 1] are shown to be compact. This, together with the fact that every homomorphism from the disc algebra and from the algebra H∞ (∆), ∆ = unit disc, to C1[0, 1] is compact, led to the conjecture that perhaps every homomorphism from a uniform algebra into C1[0, 1] is compact. The main result to which we devote the second half of this paper, is to construct a compact Hausdorff space X, a uniformly closed subalgebra [Formula: see text] of C (X), and an arc ϕ: [0, 1] → X such that the transformation T defined by Tf = f ◦ ϕ is a (bounded) homomorphism of [Formula: see text] into C1[0, 1] which is not compact.

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.

Bourgain Algebras of Douglas Algebras

1992 ◽  
Vol 44 (4) ◽  
pp. 797-804 ◽  
Author(s):  
Pamela Gorkin ◽  
Keiji Izuchi ◽  
Raymond Mortini
Keyword(s):  
Banach Algebra ◽  
Unit Circle ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Closed Subspace ◽  
Closed Subalgebra ◽  
Douglas Algebras

Let A be a Banach algebra and let B be a linear subspace of A. Recall that A has the Dunford Pettis property if whenever ƒn→ 0 weakly in A* and φn → 0 weakly in A* then φn(ƒn) → 0. Bourgain showed that H∞ has the Dunford Pettis property using the theory of ultraproducts. The Dunford Pettis property is related to the notion of Bourgain algebra, denoted Bb, introduced by [6] Cima and Timoney. The algebra Bb is the set of ƒ in A such that if ƒn → 0 weakly in B then dist(ƒƒn, B) —> 0. Bourgain showed [2] that a closed subspace X of C(L)y where L is a compact Hausdorff space, has the Dunford Pettis property if Xb — C(L). Cima and Timoney proved that Bb is a closed subalgebra of A and that if B is an algebra then B⊂Bb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.

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 Module bundles and module amenability

2021 ◽  
Vol 25 (1) ◽  
pp. 119-141
Author(s):  
Terje Hill ◽  
David A. Robbins
Keyword(s):  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Module Amenability

Let X be a compact Hausdorff space, and let {Ax : x ∈ X} and {Bx : x ∈ X} be collections of Banach algebras such that each Ax is a Bx-bimodule. Using the theory of bundles of Banach spaces as a tool, we investigate the module amenability of certain algebras of Ax-valued functions on X over algebras of Bx-valued functions on X.

scholarly journals A Banach algebra which is generated by idempotents

2015 ◽  
Vol 24 (1) ◽  
pp. 97-99
Author(s):  
A. ZIVARI-KAZEMPOUR ◽  
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Locally Compact ◽  
Locally Compact Hausdorff Space ◽  
Totally Disconnected

In this paper we show that the Banach algebra C0(X), where X is a locally compact Hausdorff space, is generated by idempotents if and only if X is totally disconnected.

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