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Z-stability of crossed products by strongly outer actions
Operator Algebras · Submitted 24 Dec 2009
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable.
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The abstract, sentence by sentence
We consider a certain class of unital simple stably finite C^*-algebras which absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable. As an application, we also prove that all strongly outer actions of Z^2 on Z are mutually cocycle conjugate.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider a certain class of unital simple stably finite C^*-algebras which absorb the Jiang-Su algebra Z tensorially. | background 0.58 | 0.41 | 0.05 | asserted 0.89 |
| s1 | Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable. | result 0.95 | 0.95 | 0.86 | asserted 1.00 |
| s2 | As an application, we also prove that all strongly outer actions of Z^2 on Z are mutually cocycle conjugate. | result 0.97 | 0.94 | 0.11 | asserted 1.00 |
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The paper
Original abstract
We consider a certain class of unital simple stably finite C^*-algebras which absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable. As an application, we also prove that all strongly outer actions of Z^2 on Z are mutually cocycle conjugate.
Human review
References
How the literature uses it
8 arXiv papers cite this in 42 sentences
Papers citing it · 26 sentences without a confident label
Builds on it 2 papers
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This is a continuation of our previous paper [ref] studying $𝒵$-stability of crossed product $C^*$-algebras and classification of group actions on $𝒵$.
Z-stability of crossed products by strongly outer actions II · Introduction · 0.91
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In the following lemma, (i) is a variant of [ref] and (ii) is a version of [ref] for the uniform topology on $(T(A))$.
Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary · Orthogonality on the compact extreme boundary · 0.82
Uses its method or tool 3 papers
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From $x'_aα_a(x'_b)=x'_bα_b(x'_a^-1)$, one obtains [equation] Besides, [equation] By $Δ_τ(x'_a)=Δ_τ(x_a)=0$, we have [equation] It follows from [ref] that there exists a unitary $v∈B$ such that [equation] Put $x”_g=v^*x'_gα_g(v)$ for all $g∈Γ$.
Z-stability of crossed products by strongly outer actions II · Cohomology vanishing · 0.93
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From the Krein-Milman theorem it follows that [equation] Because of Cuntz-Pedersen's theorem [ref] (see also the proof of [ref] ) we can obtain $u_j ∈A$, $j=1,2,...,M$ such that [equation] Since $(f_n)_n$ is a central sequence there exists $n_τ_0∈$ such that for $n≥n_τ_0$ and $σ∈T(A)$, [equation] By the definition of $f_0$ we have $f_i (σ) =δ_i,0$ for $σ∈B(τ_0, ε/16)$ and $i=0,1,...,N$, then for $σ∈B(τ_0,ε/16)$ and $n∈$ it follows that [equation] From this, for $n≥n_τ_0$ and $σ∈B(τ_0, ε/16)$ we have [equation]
Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary · Orthogonality on the compact extreme boundary · 0.89
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We now enlist the help of a Lemma taken from Matui-Sato [ref] .
Strongly outer product type actions · Strongly outer product type actions · 0.96
Compares with it 1 paper
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The original definition given in [ref] is slightly different from this version, but they are actually equivalent (see the discussion following [ref] ).
Z-stability of crossed products by strongly outer actions II · The property (SI) · 0.89
Cites it as support 2 papers
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By [ref] , any $A∈𝒯'$ has strict comparison of projections. * If $A∈𝒞$, then $A⊗B$ has tracial rank zero for any UHF algebra $B$ by [ref] , that is, $A⊗B$ belongs to $𝒯$. * Let $A∈𝒞'$ and let $B$ be a UHF algebra.
Classification of homomorphisms into simple Z-stable C^*-algebras · The target algebras · 0.97
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(iii) [ref] shows $$-stable.
Strongly outer product type actions · Crossed products by outer actions · 0.97
Cites it as background 3 papers
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In the next lemma, we let $𝒞_0$ denote the class of unital simple stably finite $C^*$-algebras introduced in [ref] .
Z^N-actions on UHF algebras of infinite type · Cohomology vanishing · 0.90
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First, let us recall the definition of property (SI) from [ref] .
Strict comparison and Z-absorption of nuclear C*-algebras · Proof of (iii) $⇒$ (iv) $⇒$ (i) of Theorem 1.1 · 0.98
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We remind the reader of the original definition of property (SI) from [ref] .
Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra · Excision in small central sequences and property (SI) · 0.99
What it cites · 63 sentences without a confident label
Builds on 2 papers
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In the same way, for a $^2$-action $α$ as in Corollary 5.6 or Corollary 5.7 of [ref] (including $α$ on a UHF algebra), the three conditions above are equivalent, too.
Z-actions on AH algebras and Z^2-actions on AF algebras · Preliminaries · 0.91
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Theorem [ref] also holds for cocycle actions $(α,u)$ in the following sense (see also [ref] ): If there exists a unital embedding of $𝒵$ into $(A_∞)^α$, then there exists an isomorphism $μ:A⊗𝒵→A$ such that $(α,u)$ is equivalent to $(μ∘(α⊗𝕀)∘μ^-1,μ(u⊗1))$.
Z^2-actions on Kirchberg algebras · $𝒵$-stability of crossed products · 0.92
Uses the method or tool of 5 papers
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For such $γ$, we can prove the assertion by the technique used in the proof of [ref] .
Classification of uniformly outer actions of Z^2 on UHF algebras · Uniqueness of $^2$-actions on the Jiang-Su algebra · 0.93
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Using the proposition above, one can prove this in a similar fashion to [ref] .
A generalization of the Jiang-Su construction · A Rohlin type theorem · 0.93
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By [ref] , there exist projections $e$ and $f$ in $A_∞$ such that [equation] and [equation] Define $u_k∈U(A)$ for $k=0,1,…$ by $u_0=1$ and $u_k+1=uβ(u_k)$.
Z-actions on AH algebras and Z^2-actions on AF algebras · Cocycle actions of $^2$ · 0.85
Compares with 2 papers
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Indeed, for any fixed $m∈$ one has [equation] and so the definition above is stronger than [ref] .
The Rohlin property for automorphisms of the Jiang-Su algebra · $𝒵$-stability of crossed products · 0.93
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The projection free tracial Rohlin property introduced in [ref] is quite similar to the weak Rohlin property defined above.
Crossed product C*-algebras by finite group actions with the projection free tracial Rokhlin property · Preliminaries · 0.99
Cites as support 2 papers
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Then [ref] tells us that $A⊗B$ has tracial rank zero.
Simple C*-algebras with locally finite decomposition rank · Preliminaries · 0.99
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In fact, by [ref] , $A$ has locally finite decomposition rank.
A Note On Subhomogeneous C*-Algebras · Preliminaries · 0.98
Cites as background 7 papers
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[ [ref] ] We say that a $C^*$-algebra $A$ has the property (SI) when for any central sequences $(e_n)_n$ and $(f_n)_n$ in $A$ satisfying $0≤e_n≤1$, $0≤f_n≤1$, [equation] there exists a central sequence $(s_n)_n$ in $A$ such that [equation]
The Rohlin property for automorphisms of the Jiang-Su algebra · $𝒵$-stability of crossed products · 0.80
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See Definition 1.2 and Lemma 1.16 of [ref] for the definition of the tracial Rohlin property for finite group actions.
The tracial Rokhlin property for actions of finite groups on C*-algebras · Closedness of $𝒞_0$ under taking crossed products · 0.87
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Winter [ref] , H.
Localizing the Elliott conjecture at strongly self-absorbing C*-algebras · Preliminaries · 0.81
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