Localizing the Elliott conjecture at strongly self-absorbing C*-algebras

Wilhelm Winter

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Localizing the Elliott conjecture at strongly self-absorbing C*-algebras

Wilhelm Winter

Operator Algebras · Submitted 2 Aug 2007

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z.

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What the abstract itself states, as probabilities: claims first 0.89 · states limitation 0.93 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.97 · mentions code or data availability 0.01.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept. As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z. Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C*-algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang-Su algebra itself.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C-algebra $D$; we also explain how the known classification theorems for nuclear C-algebras fit into this concept.	contribution 0.96	0.89	0.06	asserted 0.99
s1	As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z.	result 0.91	0.93	0.28	asserted 1.00
s2	Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely.	future work 0.91	0.85	0.22	hedged possibility 0.84
s3	When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense.	interpretation 0.99	0.59	0.04	asserted 0.98
s4	The strategy to attack the general case aims at combining these two approaches.	future work 0.86	0.52	0.05	asserted 0.79
s5	Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure.	result 0.76	0.92	0.18	asserted 1.00
s6	Also, in the monotracial case it does not rely on the existence or absence of projections in any way.	result 0.42	0.84	0.05	asserted 1.00
s7	In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C*-algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang-Su algebra itself.	result 0.39	0.91	0.23	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.99.

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arXiv listing abstract v3

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Original abstract

We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept. As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z. Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C*-algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang-Su algebra itself.

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19 arXiv papers cite this in 38 sentences

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Papers citing it · 22 sentences without a confident label

Builds on it 1 paper

It follows that the version of Proposition 4.5 of [ref] holds since proof requires no changes except that we change the word "isomorphism" to “" twice in the proof.

The Range of a Class of Classifiable Separable Simple Amenable C*-Algebras · The range · 0.93

Uses its method or tool 4 papers

Let ${u(t): t∈[0,1)}$ be a continuous path of unitaries in $B⊗Q$ with $u(0)=1_B⊗Q$ such that [equation] One then obtains a unitary suspended isomorphism which lifts $Γ$ along $Z_p,q$ (see [ref] ).

Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras · Classification of separable simple amenable s · 0.88
Let ${u(t): t∈[0,1)}$ be a continuous path of unitaries in $B⊗Q$ with $u(0)=1_B⊗Q$ such that [equation] One then obtains a unitary suspended $C([0,1])$-unital which lifts $Γ$ along $Z_p,q$ (see [ref] ).

The Range of a Class of Classifiable Separable Simple Amenable C*-Algebras · The range · 0.80
One of the technical tools used in the proof of [ref] and [ref] is a theorem which determines when two unital monomorphisms from a unital AH-algebra $C$ to a unital simple $A$ are asymptotically unitarily equivalent.

Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras · Introduction · 0.94

Cites it as support 1 paper

By [ref] , [ref] , [ref] , and [ref] , these C*-algebras are AH-algebras without dimension growth.

Mean dimension and AH-algebras with diagonal maps · Mean dimension zero and AH-algebras with diagonal maps · 0.97

Cites it as background 9 papers

With recent developments in the Elliott program ( [ref] , [ref] and [ref] ), the classifiable s now include the class of s which are rationally of tracial rank at most one.

Minimal dynamical systems on connected odd dimensional spaces · Introduction · 0.83
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.

Classification of homomorphisms into simple Z-stable C^*-algebras · Introduction · 0.87
Winter ( [ref] ) demonstrated a new approach to the Elliott program.

Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras · Introduction · 0.90

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