scholarly journals Homomorphisms from AH-algebras

2017 ◽  
Vol 09 (01) ◽  
pp. 67-125 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
State Space ◽  
Continuous Functions ◽  
Compact Metric Space ◽  
Tracial State ◽  
Affine Maps ◽  
Real Affine ◽  
Tracial Rank ◽  
Compatible Elements ◽  
Unital Simple ◽  
Approximate Version

Let [Formula: see text] be a general unital AH-algebra and let [Formula: see text] be a unital simple [Formula: see text]-algebra with tracial rank at most one. Suppose that [Formula: see text] are two unital monomorphisms. We show that [Formula: see text] and [Formula: see text] are approximately unitarily equivalent if and only if [Formula: see text] [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are continuous affine maps from tracial state space [Formula: see text] of [Formula: see text] to faithful tracial state space [Formula: see text] of [Formula: see text] induced by [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] and [Formula: see text] are induced homomorphisms s from [Formula: see text] into [Formula: see text], where [Formula: see text] is the space of all real affine continuous functions on [Formula: see text] and [Formula: see text] is the closure of the image of [Formula: see text] in the affine space [Formula: see text]. In particular, the above holds for [Formula: see text], the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements [Formula: see text], an affine map [Formula: see text] and a homomorphisms [Formula: see text], there exists a unital monomorphism [Formula: see text] such that [Formula: see text] and [Formula: see text].

Crossed products of C∗-algebras C(X,A) and their applications

2016 ◽  
Vol 27 (03) ◽  
pp. 1650029
Author(s):  
Jiajie Hua
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Crossed Products ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
C Algebras ◽  
Finite Dimensional ◽  
Tracial Rank ◽  
Stably Finite ◽  
Unital Simple

Let [Formula: see text] be an infinite compact metric space with finite covering dimension, let [Formula: see text] be a unital separable simple AH-algebra with no dimension growth, and denote by [Formula: see text] the [Formula: see text]-algebra of all continuous functions from [Formula: see text] to [Formula: see text] Suppose that [Formula: see text] is a minimal group action and the induced [Formula: see text]-action on [Formula: see text] is free. Under certain conditions, we show the crossed product [Formula: see text]-algebra [Formula: see text] has rational tracial rank zero and hence is classified by its Elliott invariant. Next, we show the following: Let [Formula: see text] be a Cantor set, let [Formula: see text] be a stably finite unital separable simple [Formula: see text]-algebra which is rationally TA[Formula: see text] where [Formula: see text] is a class of separable unital [Formula: see text]-algebras which is closed under tensoring with finite dimensional [Formula: see text]-algebras and closed under taking unital hereditary sub-[Formula: see text]-algebras, and let [Formula: see text]. Under certain conditions, we conclude that [Formula: see text] is rationally TA[Formula: see text] Finally, we classify the crossed products of certain unital simple [Formula: see text]-algebras by using the crossed products of [Formula: see text].

scholarly journals Rotation Algebras and the Exel Trace Formula

2015 ◽  
Vol 67 (2) ◽  
pp. 404-423 ◽  
Author(s):  
Jiajie Hua ◽  
Huaxin Lin
Keyword(s):  
Real Number ◽  
Trace Formula ◽  
Rank Zero ◽  
Tracial State ◽  
Tracial Rank ◽  
Image Position ◽  
Unital Simple ◽  
Tracial States

AbstractWe show that ifuandvare any two unitaries in a unitalC*–algebra such that ∥uv−vu∥ < 2 anduvu*v* commutes withuandv, then theC*–subalgebraAu,vgenerated by u and v is isomorphic to a quotient of some rotation algebraAθ, provided thatAu;vhas a unique tracial state. We also show that the Exel trace formula holds in any unitalC*–algebra. Let θ ∊ (−1/2, 1/2) be a real number. For any ∊ > 0; we prove that there exists ζ > 0 satisfying the following: if u and v are two unitaries in any unital simpleC*–algebra A with tracial rank zero such thatfor all tracial states τof A; then there exists a pair of unitariesandin A such that

scholarly journals UNITARIES IN A SIMPLE C*-ALGEBRA OF TRACIAL RANK ONE

2010 ◽  
Vol 21 (10) ◽  
pp. 1267-1281 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
State Space ◽  
Unitary Group ◽  
Affine Function ◽  
Connected Component ◽  
Finite Spectrum ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Tracial State ◽  
Rank One ◽  
Tracial Rank

Let A be a unital separable simple infinite dimensional C*-algebra with tracial rank not more than one and with the tracial state space T(A) and let U(A) be the unitary group of A. Suppose that u ∈ U0(A), the connected component of U(A) containing the identity. We show that, for any ϵ > 0, there exists a self-adjoint element h ∈ As.a such that [Formula: see text] We also study the problem when u can be approximated by unitaries in A with finite spectrum. Denote by CU(A) the closure of the subgroup of unitary group of U(A) generated by its commutators. It is known that CU(A) ⊂ U0(A). Denote by [Formula: see text] the affine function on T(A) defined by [Formula: see text]. We show that u can be approximated by unitaries in A with finite spectrum if and only if u ∈ CU(A) and [Formula: see text] for all n ≥ 1. Examples are given for which there are unitaries in CU(A) which cannot be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence of amenability.

CROSSED PRODUCTS OF CERTAIN C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450010 ◽  
Author(s):  
JIAJIE HUA ◽  
YAN WU
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Cantor Set ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let X be a Cantor set, and let A be a unital separable simple amenable [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|1⊗A] = [ id |1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and [Formula: see text] for all u ∈ U(A) (where α‡ and id‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊α ℤ is a unital simple [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋n → X × 𝕋n be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

A CLASSIFICATION OF NON-SIMPLE C*-ALGEBRAS OF TRACIAL RANK ONE: INDUCTIVE LIMITS OF FINITE DIRECT SUMS OF SIMPLE TAI C*-ALGEBRAS

2011 ◽  
Vol 03 (03) ◽  
pp. 385-404 ◽  
Author(s):  
CHUNLAN JIANG
Keyword(s):  
Compatibility Conditions ◽  
Inductive Limits ◽  
Direct Sums ◽  
State Spaces ◽  
Tracial State ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Unital Simple

In this paper, we will classify the class of C*-algebras which are inductive limits of finite direct sums of unital simple separable nuclear C*-algebras with tracial rank no more than one (or equivalently TAI algebras) with torsion K1-group which satisfy the UCT. The invariant consists of ordered total K-theory and the tracial state spaces of cutdown algebras (with certain compatibility conditions).

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 Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

Stability of rotation relations in -algebras

2020 ◽  
pp. 1-33
Author(s):  
Jiajie Hua ◽  
Qingyun Wang
Keyword(s):  
Real Number ◽  
Continuous Branch ◽  
Tracial Rank ◽  
Unital Simple

Abstract Let $\Theta =(\theta _{j,k})_{3\times 3}$ be a nondegenerate real skew-symmetric $3\times 3$ matrix, where $\theta _{j,k}\in [0,1).$ For any $\varepsilon>0$ , we prove that there exists $\delta>0$ satisfying the following: if $v_1,v_2,v_3$ are three unitaries in any unital simple separable $C^*$ -algebra A with tracial rank at most one, such that $\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}\end{align*}$ for all $\tau \in T(A)$ and $j,k=1,2,3,$ where $\log _{\theta }$ is a continuous branch of logarithm (see Definition 4.13) for some real number $\theta \in [0, 1)$ , then there exists a triple of unitaries $\tilde {v}_1,\tilde {v}_2,\tilde {v}_3\in A$ such that $\begin{align*}\tilde{v}_k\tilde{v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.\end{align*}$ The same conclusion holds if $\Theta $ is rational or nondegenerate and A is a nuclear purely infinite simple $C^*$ -algebra (where the trace condition is vacuous). If $\Theta $ is degenerate and A has tracial rank at most one or is nuclear purely infinite simple, we provide some additional injectivity conditions to get the above conclusion.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

scholarly journals Cellular automata over generalized Cayley graphs

2017 ◽  
Vol 28 (3) ◽  
pp. 340-383 ◽  
Author(s):  
PABLO ARRIGHI ◽  
SIMON MARTIEL ◽  
VINCENT NESME
Keyword(s):  
Cellular Automata ◽  
Metric Space ◽  
Cayley Graphs ◽  
Continuous Functions ◽  
Point Of View ◽  
Automata Theory ◽  
Time Varying ◽  
Compact Metric Space ◽  
Bounded Degree ◽  
Translation Invariant

It is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions over a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to an origin; and the fact that they have a well-defined notion of translation. We propose a notion of graphs, which preserves or generalizes these features. Whereas Cayley graphs are very regular, generalized Cayley graphs are arbitrary, although of a bounded degree. We extend cellular automata theory to these arbitrary, bounded degree, time-varying graphs. The obtained notion of cellular automata is stable under composition and under inversion.

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