On the Radius of Comparison of a Commutative C*-algebra

2013 ◽  
Vol 56 (4) ◽  
pp. 737-744
Author(s):  
George A. Elliott ◽  
Zhuang Niu
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Cohomological Dimension ◽  
Compact Metric Space ◽  
C Algebra

Abstract.Let X be a compact metric space. A lower bound for the radius of comparison of the C*-algebra C(X) is given in terms of dimQ X, where dimQ X is the cohomological dimension with rational coefficients. If dimQX = dim X = d, then the radius of comparison of the C*-algebra C(X) is maxf0; (d/1)=2/1g if d is odd, and must be either d=2/1 or d=2/2 if d is even (the possibility d=2 - 1 does occur, but we do not know if the possibility d=2 - 2 can also occur).

scholarly journals CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

2009 ◽  
Vol 01 (03) ◽  
pp. 261-288 ◽  
Author(s):  
JOHN R. KLEIN ◽  
CLAUDE L. SCHOCHET ◽  
SAMUEL B. SMITH
Keyword(s):  
Gauge Group ◽  
Metric Space ◽  
Principal Bundle ◽  
Rational Homotopy ◽  
Compact Metric Space ◽  
Gauge Groups ◽  
C Algebra ◽  
C Algebras ◽  
Rational Cohomology ◽  
Rational Homotopy Type

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζdenote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

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 structure of the C*-algebra of a locally injective surjection

2011 ◽  
Vol 32 (4) ◽  
pp. 1226-1248 ◽  
Author(s):  
TOKE MEIER CARLSEN ◽  
KLAUS THOMSEN
Keyword(s):  
Metric Space ◽  
Matrix Algebra ◽  
Finite Covering ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Classification Result ◽  
Full Matrix ◽  
C Algebra ◽  
Simple Quotient ◽  
The Ideal

AbstractIn this paper we investigate the ideal structure of the C*-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, then it is also purely infinite.

scholarly journals Inverse limit theorem and expansion theorem for cohomological dimension

1983 ◽  
Vol 28 (1) ◽  
pp. 67-75
Author(s):  
Satya Deo ◽  
Subhash Muttepawar
Keyword(s):  
Limit Theorem ◽  
Metric Space ◽  
Inverse Limit ◽  
Cohomological Dimension ◽  
Dimension Function ◽  
Compact Metric Space ◽  
Expansion Theorem

For a given ground ring L, let dimL(X) denote the sheaf theoretic cohomological dimension of a space X. In this paper we prove an inverse limit theorem for this dimension function. Then we apply this theorem to show that for a large classs of rings the Freudenthal's expansion theorem, expressing a compact metric space X of dimL(X) as the inverse limit of an system of compact polyhedra Kα of dimL(Kα), is not valid.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

Convergence of successive approximations to a stochastic fixed point

1980 ◽  
Vol 17 (1) ◽  
pp. 297-299
Author(s):  
Arun P. Sanghvi
Keyword(s):  
Fixed Point ◽  
Recent Result ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Compact Metric Space

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γα : α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).

scholarly journals New results on topological dynamics of antitriangular maps

2001 ◽  
Vol 2 (1) ◽  
pp. 51 ◽  
Author(s):  
Francisco Balibrea ◽  
J.S. Cánovas ◽  
A. Linero
Keyword(s):  
Metric Space ◽  
Topological Dynamics ◽  
Compact Metric Space ◽  
Special Analysis ◽  
Continuous Maps

<p>We present some results concerning the topological dynamics of antitriangular maps, F:X<sup>2</sup>→ X<sup>2 </sup>with the formvF(x,y)=(g(y),f(x)), where (X,d) is a compact metric space and f,g : X→ X are continuous maps. We make an special analysis in the case of X = [0,1].</p>

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