scholarly journals TRACES ARISING FROM REGULAR INCLUSIONS

2016 ◽  
Vol 103 (2) ◽  
pp. 190-230
Author(s):  
DANNY CRYTSER ◽  
GABRIEL NAGY
Keyword(s):  
State Space ◽  
Extension Property ◽  
Tracial State ◽  
Invariance Properties ◽  
Unit Space ◽  
Conditional Expectations ◽  
Graph Algebra ◽  
Natural Extensions ◽  
Tracial States ◽  
State Extension

We study the problem of extending a state on an abelian $C^{\ast }$-subalgebra to a tracial state on the ambient $C^{\ast }$-algebra. We propose an approach that is well suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient $C^{\ast }$-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid $C^{\ast }$-algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our framework, we are able to completely describe the tracial state space of a Cuntz–Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provide connections to related finiteness questions in graph $C^{\ast }$-algebras. Our investigation has close connections with the so-called unique state extension property and its variants.

scholarly journals Traces on cores of -algebras associated with self-similar maps

2013 ◽  
Vol 34 (6) ◽  
pp. 1964-1989 ◽  
Author(s):  
TSUYOSHI KAJIWARA ◽  
YASUO WATATANI
Keyword(s):  
Metric Spaces ◽  
Complete List ◽  
Branch Points ◽  
Tracial State ◽  
Continuous Type ◽  
Compact Metric Spaces ◽  
Self Similar ◽  
Tracial States ◽  
Discrete Type

AbstractWe completely classify the extreme tracial states on the cores of the ${C}^{\ast } $-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.

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.

Regularity of simple nuclear real C*-algebras under tracial conditions

2021 ◽  
pp. 1-9
Author(s):  
P. J. Stacey
Keyword(s):  
State Space ◽  
Finite Covering ◽  
Covering Dimension ◽  
Infinite Dimensional ◽  
Tracial State ◽  
C Algebras

Abstract The Toms–Winter conjecture is verified for those separable, unital, nuclear, infinite-dimensional real C*-algebras for which the complexification has a tracial state space with compact extreme boundary of finite covering dimension.

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].

scholarly journals Traces on topological-graph algebras

2017 ◽  
Vol 38 (5) ◽  
pp. 1923-1953
Author(s):  
CHRISTOPHER SCHAFHAUSER
Keyword(s):  
Probability Measures ◽  
Graph Algebras ◽  
Topological Graph ◽  
Gauge Invariant ◽  
Gauge Action ◽  
Tracial State ◽  
Suitable Sense ◽  
Affine Homeomorphism ◽  
Totally Disconnected ◽  
Tracial States

Given a topological graph $E$, we give a complete description of tracial states on the $\text{C}^{\ast }$-algebra $\text{C}^{\ast }(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\text{C}^{\ast }(E)$ and Radon probability measures on the vertex space $E^{0}$ which are, in a suitable sense, invariant under the action of the edge space $E^{1}$. It is shown that if $E$ has no cycles, then every tracial state on $\text{C}^{\ast }(E)$ is gauge invariant. When $E^{0}$ is totally disconnected, the gauge invariant tracial states on $\text{C}^{\ast }(E)$ are in bijection with the states on $\text{K}_{0}(\text{C}^{\ast }(E))$.

Sign in / Sign up

Export Citation Format

Share Document