scholarly journals On a problem of Barnes and Duncan

1991 ◽  
Vol 34 (2) ◽  
pp. 321-323
Author(s):  
R. G. McLean
Keyword(s):  
Banach Algebra ◽  
Free Monoid ◽  
Finite Dimensional ◽  
Image Position ◽  
Funct Anal

Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:Let R have its natural involution * in which each element of P is Hermitian. We show that the Banach *-algebra ℓ1(R) has a separating family of finite dimensional *-representations and consequently is *-semisimple. This generalizes a result of B. A. Barnes and J. Duncan (J. Funct. Anal.18 (1975), 96–113.) dealing with the case where P has two elements.

scholarly journals Spaces of sections of Banach algebra bundles

2012 ◽  
Vol 10 (2) ◽  
pp. 279-298 ◽  
Author(s):  
Emmanuel Dror Farjoun ◽  
Claude L. Schochet
Keyword(s):  
Spectral Sequence ◽  
Banach Algebra ◽  
Metric Space ◽  
Compact Metric Space ◽  
The Real ◽  
Finite Dimensional ◽  
Image Position

AbstractSuppose thatBis a G-Banach algebra over= ℝ or ℂXis a finite dimensional compact metric space, ζ :P → Xis a standard principalG-bundle, andAζ= Γ(X,P×GB) is the associated algebra of sections. We produce a spectral sequence which converges to π*(GLoAζ) withA related spectral sequence converging toK*+1(Aζ) (the real or complex topologicalK-theory) allows us to conclude that ifBis Bott-stable, (i.e., if π*(GLoB) →K*+1(B) is an isomorphism for all * > 0) then so isAζ.

scholarly journals A Note on Finite-Dimensional Differentiable Mappings

1969 ◽  
Vol 9 (3-4) ◽  
pp. 405-408 ◽  
Author(s):  
K. J. Palmer ◽  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Continuous Linear ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Image Position ◽  
Differentiable Mappings

Let E be a real infinite-dimensional Banach space. Let ℒ be the Banach algebra of all continuous linear mappings of E into itself with topology defined by the norm:

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals A note on liftings of hermitian elements and unitaries

1980 ◽  
Vol 21 (1) ◽  
pp. 183-185
Author(s):  
C. K. Fong
Keyword(s):  
Banach Algebra ◽  
Present Note ◽  
Partial Answer ◽  
General Question ◽  
Quotient Mapping ◽  
Unitary Element ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
Special Cases

Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u−1∥ = 1. An element h in A is said to be hermitian if ∥exp(ith)∥ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

Derivations Tangential to Compact Group Actions: Spectral Conditions in the Weak Closure

1985 ◽  
Vol 37 (1) ◽  
pp. 160-192 ◽  
Author(s):  
Ola Bratteli ◽  
Frederick M. Goodman
Keyword(s):  
Finite Elements ◽  
Common Core ◽  
Dimensional Space ◽  
List Type ◽  
Finite Dimensional Space ◽  
Type Number ◽  
Compact Lie Group ◽  
Parameter Group ◽  
Finite Dimensional ◽  
Image Position

Let G be a compact Lie group and a an action of G on a C*-algebra as *-automorphisms. Let denote the set of G-finite elements for this action, i.e., the set of those such that the orbit {αg(x):g ∊ G} spans a finite dimensional space. is a common core for all the *-derivations generating one-parameter subgroups of the action α. Now let δ be a *-derivation with domain such that Let us pose the following two problems:Is δ closable, and is the closure of δ the generator of a strongly continuous one-parameter group of *-automorphisms?If is simple or prime, under what conditions does δ have a decompositionwhere is the generator of a one-parameter subgroup of α(G) and is a bounded, or approximately bounded derivation?

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

scholarly journals On Kadison's condition for extreme points of the unit ball in a B*-algebra

1969 ◽  
Vol 16 (3) ◽  
pp. 245-250 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Unit Ball ◽  
Banach Algebra ◽  
Extreme Points ◽  
Complex Banach Algebra ◽  
Image Position

Let B be a complex Banach algebra with an identity 1 and an involution x→x*. Kadison (1) has shown that, if B is a B*-algebra, [the set of extreme points of its unit ball coincides with the set of elements x of B for which

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.

scholarly journals Boundary problems for Riccati and Lyapunov equations

1986 ◽  
Vol 29 (1) ◽  
pp. 15-21 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Boundary Condition ◽  
Boundary Problem ◽  
Transport Theory ◽  
Adjoint Operator ◽  
Linear Operators ◽  
Dimensional Case ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Image Position

The resolution problem of the systemwhere U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problemwhere t ∈[0,b].In this paper we consider the more general boundary problemwhere all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).

scholarly journals Automatic continuity for Banach algebras with finite-dimensional radical

2006 ◽  
Vol 81 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Hung Le Pham
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity ◽  
Algebraic Condition ◽  
Finite Dimensional ◽  
Complete Algebra

AbstractThe paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

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