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Classification of homomorphisms into simple Z-stable C^*-algebras
Operator Algebras · Submitted 25 Feb 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
We classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C^*-algebra, or any unital simple separable nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C^*-algebra such that A\otimes B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C^*-algebra whose projections separate traces and whose extremal traces are finitely many.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence. | contribution 0.88 | 0.95 | 0.92 | asserted 1.00 |
| s1 | The domain algebra C is allowed to be any unital separable commutative C^*-algebra, or any unital simple separable nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of tracial rank zero for a UHF algebra B. | contribution 0.59 | 0.42 | 0.05 | asserted 0.85 |
| s2 | The target algebra A is allowed to be any unital simple separable Z-stable C^*-algebra such that A\otimes B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C^*-algebra whose projections separate traces and whose extremal traces are finitely many. | contribution 0.67 | 0.53 | 0.08 | asserted 0.90 |
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The paper
Original abstract
We classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C^*-algebra, or any unital simple separable nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C^*-algebra such that A\otimes B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C^*-algebra whose projections separate traces and whose extremal traces are finitely many.
Human review
How the literature uses it
3 arXiv papers cite this in 6 sentences
Papers citing it · 4 sentences without a confident label
Uses its method or tool 1 paper
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By applying [ref] (see also [ref] ), we get a finite subset $G⊂C$, $δ>0$, a finite subset $P$ of projections in $C$ and a finite subset $U$ of unitaries in $C$.
Z-stability of crossed products by strongly outer actions II · Homotopy of unitaries · 1.00
What it cites · 28 sentences without a confident label
Builds on 1 paper
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By using the theorems above, we obtain the following generalization of [ref] .
Classification of homomorphisms and dynamical systems · $C^*$-algebras of real rank zero · 0.99
Uses the method or tool of 1 paper
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If $u∈A$ is a unitary satisfying [equation] then there exists a path of unitaries $w:[0,1]→A$ such that [equation] and [equation] We can prove this in a similar fashion to [ref] by using the theorem above.
Approximate Homotopy of Homomorphisms from $C(X)$ into a Simple $C^*$-algebra · Homotopy of unitaries · 0.95
Compares with 1 paper
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But the proof given in [ref] is quite lengthy, and so we provide a simpler and self-contained proof for the reader's convenience.
Classification of homomorphisms and dynamical systems · Introduction · 0.91
Cites 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.
Z-stability of crossed products by strongly outer actions · The target algebras · 0.97
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Then $A⊗B$ has real rank zero by [ref] and has stable rank one by [ref] (or [ref] ).
The stable and the real rank of Z-absorbing C*-algebras · The target algebras · 1.00
Cites as background 5 papers
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The classification theorems in [ref] assert that any nuclear $C^*$-algebras $A,B∈𝒞$ satisfying the UCT are isomorphic if and only if their $K$-groups are isomorphic as graded ordered groups.
Localizing the Elliott conjecture at strongly self-absorbing C*-algebras · Introduction · 0.87
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The classification theorems in [ref] assert that any nuclear $C^*$-algebras $A,B∈𝒞$ satisfying the UCT are isomorphic if and only if their $K$-groups are isomorphic as graded ordered groups.
Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras · Introduction · 0.87
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Lin [ref] classified unital homomorphisms from AH algebras into simple separable $C^*$-algebras of tracial rank no more than one.
Approximate Unitary Equivalence in Simple C^*-algebras of Tracial Rank One · Introduction · 0.81
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