The Brauer Group of a Compact Hausdorff Space and n-Homogeneous C ∗ -Algebras

1972 ◽  
Vol 34 (1) ◽  
pp. 209
Author(s):  
Roger Howe
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Brauer Group ◽  
C Algebras

Property T and Amenable Transformation Group C*-algebras

2015 ◽  
Vol 58 (1) ◽  
pp. 110-114 ◽  
Author(s):  
F. Kamalov
Keyword(s):  
Compact Hausdorff Space ◽  
Discrete Group ◽  
Hausdorff Space ◽  
Transformation Group ◽  
Amenable Group ◽  
Transformation Groups ◽  
C Algebra ◽  
C Algebras ◽  
Property T ◽  
Tracial States

AbstractIt is well known that a discrete group that is both amenable and has Kazhdan’s Property T must be finite. In this note we generalize this statement to the case of transformation groups. We show that if G is a discrete amenable group acting on a compact Hausdorff space X, then the transformation group C*-algebra C*(X; G) has Property T if and only if both X and G are finite. Our approach does not rely on the use of tracial states on C*(X; G).

Property T for general C*-algebras

2013 ◽  
Vol 156 (2) ◽  
pp. 229-239 ◽  
Author(s):  
CHI–KEUNG NG
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
C Algebras ◽  
Strong Property ◽  
Property T ◽  
Definition Of ◽  
Locally Compact Hausdorff Space

AbstractIn this paper, we extend the definition of property T and strong property T to general C*-algebras (not necessarily unital). We show that if an inclusion pair of locally compact groups (G,H) has property T, then (C*(G), C*(H)) has property T. As a partial converse, if T is abelian and C*(G) has property T, then T is compact. We also show that if Ω is a first countable locally compact Hausdorff space, then C0(Ω) has (strong) property T if and only if Ω is discrete. Furthermore, the non-unital C*-algebra $c_0(\mathbb{Z}^n)\rtimes SL_n(\mathbb{Z})$ has strong property T when n ≥ 3. We also give some equivalent forms of strong property T, which are new even in the unital case.

scholarly journals On the $KK$-theory of strongly self-absorbing $C^{*}$-algebras

2009 ◽  
Vol 104 (1) ◽  
pp. 95 ◽  
Author(s):  
Marius Dadarlat ◽  
Wilhelm Winter
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Unitary Equivalence ◽  
Norm Topology ◽  
Homotopy Classes ◽  
C Algebras ◽  
The Way

Let $\mathcal D$ and $A$ be unital and separable $C^{*}$-algebras; let $\mathcal D$ be strongly self-absorbing. It is known that any two unital ${}^*$-homomorphisms from $\mathcal D$ to $A \otimes \mathcal D$ are approximately unitarily equivalent. We show that, if $\mathcal D$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\mathcal D$ is asymptotically inner. Moreover, the space of automorphisms of $\mathcal D$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,(\mathrm{Aut}(\mathcal D)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\mathcal D$. As an application, we give a description of the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ is isomorphic to $K_0(A\otimes \mathcal D)$.

On the Bound of the C* Exponential Length

2014 ◽  
Vol 57 (4) ◽  
pp. 853-869
Author(s):  
Qingfei Pan ◽  
Kun Wang
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Inductive Limit ◽  
C Algebras

AbstractLet X be a compact Hausdorff space. In this paper, we give an example to show that there is u ∊⊗ C(X) Mn with det(u(x)) = 1 for all x ∊ X and u ~h 1 such that the C* exponential length of u (denoted by ~h cel(u)) cannot be controlled by π. Moreover, in simple inductive limit C*-algebras, similar examples also exist.

Positive Linear Mappings Between C*-Algebras

1995 ◽  
Vol 38 (2) ◽  
pp. 252-256
Author(s):  
Yong Zhong
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Von Neumann Algebra ◽  
Linear Mapping ◽  
Von Neumann ◽  
Jordan Homomorphism ◽  
C Algebras ◽  
Invertible Elements ◽  
Totally Disconnected

AbstractWe prove that a positive unital linear mapping from a von Neumann algebra to a unital C*-algebra is a Jordan homomorphism if it maps invertible selfadjoint elements to invertible elements, and that for any compact Hausdorff space X, all positive unital linear mappings from C(X) into a unital C*-algebra that preserve the invertibility for self-adjoint elements are *-homomorphisms if and only if X is totally disconnected.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

On the continuity of lattice isomorphisms on C(X, I)

2021 ◽  
Vol 71 (6) ◽  
pp. 1477-1486
Author(s):  
Vahid Ehsani ◽  
Fereshteh Sady
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Unit Interval ◽  
Lattice Isomorphism

Abstract We investigate topological conditions on a compact Hausdorff space Y, such that any lattice isomorphism φ : C(X, I) → C(Y, I), where X is a compact Hausdorff space and I is the unit interval [0, 1], is continuous. It is shown that in either of cases that the set of G δ points of Y has a dense pseudocompact subset or Y does not contain the Stone-Čech compactification of ℕ, such a lattice isomorphism is a homeomorphism.

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.

scholarly journals Recovering a compact Hausdorff space $X$ from the compatibility ordering on $C(X)$

2018 ◽  
Vol 242 (2) ◽  
pp. 187-205 ◽  
Author(s):  
Tomasz Kania ◽  
Martin Rmoutil
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space
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