Tracial approximation and ${\cal Z}$-stability

Lin, Huaxin

doi:10.48550/arxiv.2205.04013

Cited by 1 publication

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“…A part of the Toms-Winter conjecture states that the converse also holds for separable amenable simple C * -algebras, i.e., if A is a separable simple amenable C * -algebra with strict comparison, then A is Z-stable. In fact this is the only remaining unsolved part of the Toms-Winter conjecture (see [52], [38], [31], [48], [51], [53], [49], [13], and [37], for example). Return to the C * -algebra A mentioned above, let a, b ∈ A be positive elements.…”

Section: Introductionmentioning

confidence: 99%

Strict comparison and stable rank one

Lin¹

2023

Preprint

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Let A be a σ-unital finite simple C * -algebra which has strict comparison property. We show that if the canonical map Γ from the Cuntz semigroup to certain lower semi-continuous affine functions is surjective, then A has tracial approximate oscillation zero and stable rank one. Equivalently, if A has an almost unperforated and almost divisible Cuntz semigroup, then A has stable rank one and tracial approximate oscillation zero.We also have the following:Corollary 1.3. Let A be a non-elementary finite separable simple C * -algebra with almost unperforated and almost divisible Cu(A). Then A has stable rank one and tracial approximate oscillation zero. (see also Corollary 4.5). This paper is organized as follows. Section 2 is a preliminary. In Section 3, we consider the following question: Let A be a σ-unital simple C * -algebra with strict comparison. Suppose that a, b ∈ A + and d τ (a) < d τ (b) for all nonzero 2-quasitraces τ. Does it imply that there is x ∈ A such that x * x = a and xx * ∈ bAb ? We show that the answer is yes if d τ (a) is continuous on

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Section: Introductionmentioning

confidence: 99%

Strict comparison and stable rank one

Lin¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

Tracial approximation and ${\cal Z}$-stability

Cited by 1 publication

References 38 publications

Strict comparison and stable rank one

Strict comparison and stable rank one

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