CAPACITIES ON C*-ALGEBRAS

2003 ◽  
Vol 06 (03) ◽  
pp. 373-388
Author(s):  
HENRI COMMAN
Keyword(s):  
Hausdorff Space ◽  
Large Deviation ◽  
Radon Measures ◽  
C Algebra ◽  
C Algebras ◽  
Upper Semicontinuous ◽  
Large Deviations Theory ◽  
Noncommutative Version ◽  
Compactness Theorems ◽  
Locally Compact Hausdorff Space

We distinguish three classes of capacities on a C*-algebra: subadditive, additive and maxitive. A tightness notion for capacities, the vague and narrow topologies on the set of capacities are introduced. The vague space of additive capacities which are finite on compact projections is a noncommutative version of the usual vague space of Radon measures on a locally compact Hausdorff space X. We give criterions of vague and narrow relative compactness in various classes of capacities. This allows us to extend most classical compactness theorems for Radon measures. The set of bounded (resp. tight) maxitive capacities is in bijection with the set of positive q-upper semicontinuous (resp. strongly q-upper semicontinuous) operators. This allows us to define a vague (resp. narrow) large deviation principle for a net of capacities as a vague (resp. narrow) convergence of this net towards a maxitive capacity, generalizing the classical notion for Radon probability measures on X. Next, we apply compactness theorems in order to extend some results in large deviations theory.

scholarly journals The Weak Ideal Property and Topological Dimension Zero

2017 ◽  
Vol 69 (6) ◽  
pp. 1385-1421 ◽  
Author(s):  
Cornel Pasnicu ◽  
N. Christopher Phillips
Keyword(s):  
Hausdorff Space ◽  
Crossed Products ◽  
Topological Dimension ◽  
C Algebra ◽  
Dimension Zero ◽  
C Algebras ◽  
Ideal Property ◽  
Locally Compact Hausdorff Space ◽  
Totally Disconnected ◽  
The Ideal

AbstractFollowing up on previous work, we prove a number of results for C* -algebras with the weak ideal property or topological dimension zero, and some results for C* -algebras with related properties. Some of the more important results include the following:The weak ideal property implies topological dimension zero.For a separable C* -algebra A, topological dimension zero is equivalent to , to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C* -algebras B such that contains a nonzero projection.Extending the known result for , the classes of C* -algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups.If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗ B has the weak ideal property.If X is a totally disconnected locally compact Hausdorff space and A is a C0(X)-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable).Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C* -algebras, including all separable locally AH algebras.The weak ideal property does not imply the ideal property for separable Z-stable C* -algebras.We give other related results, as well as counterexamples to several other statements one might conjecture.

The Primitive Ideal Space of a C*-Algebra

1974 ◽  
Vol 26 (1) ◽  
pp. 42-49 ◽  
Author(s):  
John Dauns
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Complete Characterization ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Compact Sets ◽  
Locally Compact ◽  
C Algebra ◽  
Weak Star ◽  
Locally Compact Hausdorff Space

The commutative Gelfand-Naimark Theorem says that any commutative C*-algebra A is isomorphic to the ring C0(M, C) of all continuous complex-valued functions tending to zero outside of compact sets of a locally compact Hausdorff space M. A very important part of this theorem is an intrinsic and also a complete characterization of M as exactly the primitive ideal space of A in the hull-kernel (or weak-star) topology. In the non-commutative case, A ≌ Γ0(M, E)—the ring of sections tending to zero outside of compact subsets of a locally compact Hausdorff space M with values in the stalks or fibers E.

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).

scholarly journals CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

2020 ◽  
pp. 1-12
Author(s):  
BHARAT TALWAR ◽  
RANJANA JAIN
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hausdorff Space ◽  
Separable Hilbert Space ◽  
Locally Compact ◽  
C Algebras ◽  
Finite Sum ◽  
Locally Compact Hausdorff Space ◽  
Quotient Property

Abstract For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

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 Completely superharmonic measures for the infinitesimal generator A of a diffusion semi-group and positive eigen elements of A

1981 ◽  
Vol 83 ◽  
pp. 53-106 ◽  
Author(s):  
Masayuki Itô ◽  
Noriaki Suzuki
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Infinitesimal Generator ◽  
Hausdorff Space ◽  
Inductive Limit ◽  
Semi Group ◽  
Radon Measures ◽  
Countable Basis ◽  
Inductive Limit Topology ◽  
Locally Compact Hausdorff Space

Let X be a locally compact Hausdorff space with countable basis. We denote byM(X) the topological vector space of all real Radon measures in X with the vague topology,MK(X) the topological vector space of all real Radon measures in X whose supports are compact with the usual inductive limit topology.

Ranks of Algebras of Continuous C*-Algebra Valued Functions

2001 ◽  
Vol 53 (5) ◽  
pp. 979-1030 ◽  
Author(s):  
Masaru Nagisa ◽  
Hiroyuki Osaka ◽  
N. Christopher Phillips
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Tensor Products ◽  
List Type ◽  
Type Number ◽  
Locally Compact ◽  
The Real ◽  
C Algebra ◽  
Locally Compact Hausdorff Space

AbstractWe prove a number of results about the stable and particularly the real ranks of tensor products of C*-algebras under the assumption that one of the factors is commutative. In particular, we prove the following:(1)If X is any locally compact σ-compact Hausdorff space and A is any C*-algebra, then RR(C0(X) ⊗ A) ≤ dim(X) + RR(A).(2)If X is any locally compact Hausdorff space and A is any purely infinite simple C*-algebra, then RR(C0(X) ⊗ A) ≤ 1.(3)RR(C([0, 1]) ⊗ A) ≥ 1 for any nonzero C*-algebra A, and sr(C([0, 1]2) ⊗ A) ≥ 2 for any unital C*-algebra A.(4)If A is a unital C*-algebra such that RR(A) = 0, sr(A) = 1, and K1(A) = 0, then sr(C([0, 1]) ⊗ A) = 1.(5)There is a simple separable unital nuclear C*-algebra A such that RR(A) = 1 and sr(C([0, 1]) ⊗ A) = 1.

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.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

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