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 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.

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].

On Banach algebra elements of thin numerical range

1979 ◽  
Vol 86 (1) ◽  
pp. 71-83 ◽  
Author(s):  
R. R. Smith
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Natural Generalization ◽  
The Real ◽  
C Algebra ◽  
Image Position ◽  
Unital Banach Algebra ◽  
Real Subspace

Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W(h) ⊂ , where W(h) denotes the numerical range of h (7), or if ║eiλh║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the conditionis satisfied. These may be termed the elements of thin numerical range if δ is small.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

scholarly journals The polarization constant of finite dimensional complex spaces is one

2021 ◽  
pp. 1-19
Author(s):  
VERÓNICA DIMANT ◽  
DANIEL GALICER ◽  
JORGE TOMÁS RODRÍGUEZ
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Homogeneous Polynomial ◽  
Complex Space ◽  
Necessary Conditions ◽  
Best Constant ◽  
The Real ◽  
Complex Spaces ◽  
Finite Dimensional ◽  
Image Position

Abstract The polarization constant of a Banach space X is defined as \[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\] where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\] . We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\] . We derive some consequences of this fact regarding the convergence of analytic functions on such spaces. The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

scholarly journals Möbius disjointness along ergodic sequences for uniquely ergodic actions

2018 ◽  
Vol 39 (10) ◽  
pp. 2793-2826 ◽  
Author(s):  
JOANNA KUŁAGA-PRZYMUS ◽  
MARIUSZ LEMAŃCZYK
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Metric Space ◽  
Irrational Rotation ◽  
Compact Metric Space ◽  
Irrational Rotations ◽  
Ergodic Flow ◽  
Weakly Mixing ◽  
Image Position ◽  
Uniquely Ergodic

We show that there is an irrational rotation $Tx=x+\unicode[STIX]{x1D6FC}$ on the circle $\mathbb{T}$ and a continuous $\unicode[STIX]{x1D711}:\mathbb{T}\rightarrow \mathbb{R}$ such that for each (continuous) uniquely ergodic flow ${\mathcal{S}}=(S_{t})_{t\in \mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$ acting on $(X\times Y,\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D708})$ by the formula $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}(x,y)=(Tx,S_{\unicode[STIX]{x1D711}(x)}(y))$, where $\unicode[STIX]{x1D707}$ stands for the Lebesgue measure on $\mathbb{T}$ and $\unicode[STIX]{x1D708}$ denotes the unique ${\mathcal{S}}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak’s conjecture on the Möbius disjointness holds for all uniquely ergodic models of $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$. Moreover, we obtain a class of ‘random’ ergodic sequences $(c_{n})\subset \mathbb{Z}$ such that if $\boldsymbol{\unicode[STIX]{x1D707}}$ denotes the Möbius function, then $$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n\leq N}g(S_{c_{n}}y)\boldsymbol{\unicode[STIX]{x1D707}}(n)=0\end{eqnarray}$$ for all (continuous) uniquely ergodic flows ${\mathcal{S}}$, all $g\in C(Y)$ and $y\in Y$.

Minimal entropy and Mostow's rigidity theorems

1996 ◽  
Vol 16 (4) ◽  
pp. 623-649 ◽  
Author(s):  
Gérard Besson ◽  
Gilles Courtois ◽  
Sylvestre Gallot
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Metric Space ◽  
Universal Cover ◽  
Compact Metric Space ◽  
Entropy Functional ◽  
Volume Entropy ◽  
Image Position ◽  
A Minor ◽  
Special Metrics

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we definewhere B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define.

Embedding Smooth Dendroids in Hyperspaces

1979 ◽  
Vol 31 (1) ◽  
pp. 130-138 ◽  
Author(s):  
J. Grispolakis ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Function ◽  
Partial Order ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Vietoris Topology ◽  
Compact Metric Space ◽  
Partially Ordered ◽  
Arcwise Connected ◽  
Ordered Space ◽  
Image Position

A continuum will be a connected, compact, metric space. By a mapping we mean a continuous function. By a partially ordered space X we mean a continuum X together with a partial order which is closed when regarded as a subset of X × X. We let 2x (resp. C(X)) denote the hyperspace of closed subsets (resp. subcontinua) of X with the Vietoris topology which coincides with the topology induced by the Hausdorff metric. The hyperspaces 2X and C(X) are arcwise connected metric continua (see [3, Theorem 2.7]). If A ⊂ X we let C(A) denote the subspace of subcontinua of X which lie in A.If X is a partially ordered space we define two functions L, M : X → 2X by setting for each x ∊ X

ALMOST MULTIPLICATIVE MORPHISMS AND K-THEORY

2000 ◽  
Vol 11 (08) ◽  
pp. 983-1000 ◽  
Author(s):  
GUIHUA GONG ◽  
HUAXIN LIN
Keyword(s):  
Metric Space ◽  
Classification Theory ◽  
Compact Metric Space ◽  
Dimensional Image ◽  
C Algebras ◽  
Finite Dimensional ◽  
Finite Set ◽  
K Theory

Let X be a compact metric space and A=C(X). Suppose that ℬ is a class of unital C*-algebras satisfying certain conditions, we prove the following: For any ∊>0, finite set F⊂A, there is an integer l such that if ϕ, ψ:A→B(B∈ℬ) are sufficiently multiplicative morphisms (e.g. when both ϕ and ψ are *-homomorphisms) which induce same K-theoretical maps, then there are a unitary u∈Ml+1(B) and a homomorphism σ:A→Ml(B) with finite dimensional image such that [Formula: see text] for all f∈F. In particular, the integer l does not depend on B, ϕ and ψ. This feature has important applications to the classification theory of nuclear C*-algebras.

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:

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