{"arxiv_id":"0912.4804","incoming":{"papers":8,"sentences":42,"groups":[{"kind":"extends","label":"Builds on it","papers":2,"sentences":3,"quotes":[{"arxiv_id":"1205.1590","title":"Z-stability of crossed products by strongly outer actions II","url":"/paper/1205.1590","section":"Introduction","sentence":"This is a continuation of our previous paper [ref] studying $𝒵$-stability of crossed product $C^*$-algebras and classification of group actions on $𝒵$.","intent_p":0.91,"contests_p":null},{"arxiv_id":"1209.3000","title":"Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary","url":"/paper/1209.3000","section":"Orthogonality on the compact extreme boundary","sentence":"In the following lemma, (i) is a variant of [ref] and (ii) is a version of [ref] for the uniform topology on $(T(A))$.","intent_p":0.82,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses its method or 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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]","intent_p":0.89,"contests_p":null},{"arxiv_id":"1403.5357","title":"Strongly outer product type actions","url":"/paper/1403.5357","section":"Strongly outer product type actions","sentence":"We now enlist the help of a Lemma taken from Matui-Sato [ref] .","intent_p":0.96,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":1,"sentences":1,"quotes":[{"arxiv_id":"1205.1590","title":"Z-stability of crossed products by strongly outer actions II","url":"/paper/1205.1590","section":"The property (SI)","sentence":"The original definition given in [ref] is slightly different from this version, but they are actually equivalent (see the discussion following [ref] ).","intent_p":0.89,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites it as 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$C^*$-algebras introduced in [ref] .","intent_p":0.9,"contests_p":null},{"arxiv_id":"1111.1637","title":"Strict comparison and Z-absorption of nuclear C*-algebras","url":"/paper/1111.1637","section":"Proof of (iii) $⇒$ (iv) $⇒$ (i) of Theorem 1.1","sentence":"First, let us recall the definition of property (SI) from [ref] .","intent_p":0.98,"contests_p":null},{"arxiv_id":"1209.5311","title":"Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra","url":"/paper/1209.5311","section":"Excision in small central sequences and property (SI)","sentence":"We remind the reader of the original definition of property (SI) from [ref] .","intent_p":0.99,"contests_p":null}]}],"unlabelled_sentences":26},"outgoing":{"papers":26,"sentences":88,"groups":[{"kind":"extends","label":"Builds on","papers":2,"sentences":5,"quotes":[{"arxiv_id":"0907.2474","title":"Z-actions on AH algebras and Z^2-actions on AF algebras","url":"/paper/0907.2474","section":"Preliminaries","sentence":"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.","intent_p":0.91,"contests_p":null},{"arxiv_id":"0902.0194","title":"Z^2-actions on Kirchberg algebras","url":"/paper/0902.0194","section":"$𝒵$-stability of crossed products","sentence":"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 $(μ∘(α⊗𝕀)∘μ^-1,μ(u⊗1))$.","intent_p":0.92,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses the method or tool of","papers":5,"sentences":7,"quotes":[{"arxiv_id":"0708.4073","title":"Classification of uniformly outer actions of Z^2 on UHF algebras","url":"/paper/0708.4073","section":"Uniqueness of $^2$-actions on the Jiang-Su algebra","sentence":"For such $γ$, we can prove the assertion by the technique used in the proof of [ref] .","intent_p":0.93,"contests_p":null},{"arxiv_id":"0903.5286","title":"A generalization of the Jiang-Su construction","url":"/paper/0903.5286","section":"A Rohlin type theorem","sentence":"Using the proposition above, one can prove this in a similar fashion to [ref] .","intent_p":0.93,"contests_p":null},{"arxiv_id":"0907.2474","title":"Z-actions on AH algebras and Z^2-actions on AF algebras","url":"/paper/0907.2474","section":"Cocycle actions of $^2$","sentence":"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 $u_k+1=uβ(u_k)$.","intent_p":0.85,"contests_p":null},{"arxiv_id":"0902.0194","title":"Z^2-actions on Kirchberg algebras","url":"/paper/0902.0194","section":"Cocycle actions of $^2$","sentence":"Then there exist $a,b∈U(A)$ such that [equation] and [equation] In particular, [equation] Using Lemma [ref] and Lemma [ref], one can prove this statement in the same way as [ref] .","intent_p":0.81,"contests_p":null},{"arxiv_id":"0708.4074","title":"Classification of outer actions of Z^N on O_2","url":"/paper/0708.4074","section":"Uniqueness of $^2$-actions on the Jiang-Su algebra","sentence":"We omit the detail, because it is exactly the same as [ref] .","intent_p":0.84,"contests_p":null}]},{"kind":"comparison","label":"Compares with","papers":2,"sentences":3,"quotes":[{"arxiv_id":"0908.0135","title":"The Rohlin property for automorphisms of the Jiang-Su algebra","url":"/paper/0908.0135","section":"$𝒵$-stability of crossed products","sentence":"Indeed, for any fixed $m∈$ one has [equation] and so the definition above is stronger than [ref] .","intent_p":0.93,"contests_p":null},{"arxiv_id":"0902.3324","title":"Crossed product C*-algebras by finite group actions with the projection free tracial Rokhlin property","url":"/paper/0902.3324","section":"Preliminaries","sentence":"The projection free tracial Rohlin property introduced in [ref] is quite similar to the weak Rohlin property defined above.","intent_p":0.99,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites as support","papers":2,"sentences":2,"quotes":[{"arxiv_id":"math/0602617","title":"Simple C*-algebras with locally finite decomposition rank","url":"/paper/math/0602617","section":"Preliminaries","sentence":"Then [ref] tells us that $A⊗B$ has tracial rank zero.","intent_p":0.99,"contests_p":null},{"arxiv_id":"math/0601069","title":"A Note On Subhomogeneous C*-Algebras","url":"/paper/math/0601069","section":"Preliminaries","sentence":"In fact, by [ref] , $A$ has locally finite decomposition rank.","intent_p":0.98,"contests_p":null}]},{"kind":"background","label":"Cites as background","papers":7,"sentences":8,"quotes":[{"arxiv_id":"0908.0135","title":"The Rohlin property for automorphisms of the Jiang-Su algebra","url":"/paper/0908.0135","section":"$𝒵$-stability of crossed products","sentence":"[ [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]","intent_p":0.8,"contests_p":null},{"arxiv_id":"math/0609782","title":"The tracial Rokhlin property for actions of finite groups on C*-algebras","url":"/paper/math/0609782","section":"Closedness of $𝒞_0$ under taking crossed products","sentence":"See Definition 1.2 and Lemma 1.16 of [ref] for the definition of the tracial Rohlin property for finite group actions.","intent_p":0.87,"contests_p":null},{"arxiv_id":"0708.0283","title":"Localizing the Elliott conjecture at strongly self-absorbing C*-algebras","url":"/paper/0708.0283","section":"Preliminaries","sentence":"Winter [ref] , H.","intent_p":0.81,"contests_p":null},{"arxiv_id":"0708.4074","title":"Classification of outer actions of Z^N on O_2","url":"/paper/0708.4074","section":"Introduction","sentence":"Izumi and the first-named author [ref] classified a large class 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