Tracial state space with non-compact extreme boundary

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.

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000043 ◽

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.

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Homomorphisms from AH-algebras

Journal of Topology and Analysis ◽

10.1142/s1793525317500091 ◽

2017 ◽

Vol 09 (01) ◽

pp. 67-125 ◽

Cited By ~ 4

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].

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Inductive Limits of Interval Algebras: The Tracial State Space

American Journal of Mathematics ◽

10.2307/2374993 ◽

1994 ◽

Vol 116 (3) ◽

pp. 605 ◽

Cited By ~ 17

Author(s):

K. Thomsen

Keyword(s):

State Space ◽

Inductive Limits ◽

Tracial State

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TRACES ARISING FROM REGULAR INCLUSIONS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788716000501 ◽

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.

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