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]).

ONE FUNCTIONAL OPERATOR INVERSION FORMULA

2001 ◽  
Vol 6 (1) ◽  
pp. 138-146 ◽  
Author(s):  
P. Plaschinsky
Keyword(s):  
Banach Algebra ◽  
Explicit Form ◽  
Maximal Ideal ◽  
Inversion Formula ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Functional Operator ◽  
Inversion Operator ◽  
Commutative Banach Algebras ◽  
Algebra Theory

Some results about inversion formula of functional operator with generalized dilation are given. By means of commutative Banach algebra theory the explicit form of inversion operator is expressed. Some commutative Banach algebras with countable generator systems are constructed, their maximal ideal spaces are investigated.

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).

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.

Commutative Banach Algebras with Idempotent Maximal Ideals

1969 ◽  
Vol 9 (3-4) ◽  
pp. 275-286 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Complex Field ◽  
Linear Combinations ◽  
Maximal Ideals ◽  
Commutative Banach Algebras ◽  
Finite Linear

Letbe a commutative Banach algebra over the complex fieldC,Man ideal of. Denote byM2the set of all finite linear combinations of products of elements fromM.Mwill be termed idempotent ifM2=M. The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.

scholarly journals Higher point derivations on commutative Banach algebras, III

1981 ◽  
Vol 24 (1) ◽  
pp. 31-40 ◽  
Author(s):  
H. G. Dales ◽  
J. P. McClure
Keyword(s):  
Banach Algebra ◽  
Infinite Order ◽  
Infinite Sequence ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Complex Field ◽  
Linear Functionals ◽  
Point Derivation ◽  
Commutative Banach Algebras ◽  
Image Position

Let A be a commutative Banach algebra with identity 1 over the complex field C, and let d0 be a character on A. We recall that a (higher) point derivation of order q on A at d0 is a sequence d1, …, dq of linear functionals on A such that the identitieshold for each choice of f and g in A and k in {1, …, q}. A point derivation of infinite order is an infinite sequence {dk} of linear functionals such that (1.1) holds for all k. A point derivation is continuous if each dk is continuous, totally discontinuous if dk is discontinuous for each k≧1, and degenerate if d1 = 0.

DITKIN CONDITIONS

2017 ◽  
Vol 60 (1) ◽  
pp. 153-163
Author(s):  
AZADEH NIKOU ◽  
ANTHONY G. O'FARRELL
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Commutative Banach Algebra ◽  
Algebraic Properties ◽  
The Relationship

AbstractThis paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra E with unit and the associated commutative Banach algebra C(X, E) of all continuous functions from a compact Hausdorff space X into E. The properties concern Ditkin's condition and bounded relative units. We show that these properties are shared by E and C(X, E). We also consider the relationship between these properties in the algebras E, B and $\~{B}$ that appear in the so-called admissible quadruples (X, E, B, $\~{B}$).

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.

scholarly journals Spatial numerical ranges of elements of subalgebras ofC0(X)

2000 ◽  
Vol 23 (12) ◽  
pp. 827-831
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Numerical Range ◽  
Commutative Banach Algebra ◽  
Locally Compact ◽  
Continuous Complex ◽  
Complex Valued ◽  
Locally Compact Hausdorff Space

WhenAis a subalgebra of the commutative Banach algebraC0(X)of all continuous complex-valued functions on a locally compact Hausdorff spaceX, the spatial numerical range of element ofAcan be described in terms of positive measures.

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.

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