Cartan subalgebras in C*-algebras. Existence and uniqueness

Jean Renault; Xin Li

RECORD

Cartan subalgebras in C*-algebras. Existence and uniqueness

Xin Li, Jean Renault

Operator Algebras · Submitted 30 Mar 2017

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

The authors' main claim Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras.

how it is stated · asserted 0.91 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

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.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,308 input tokens.

What the abstract itself states, as probabilities: claims first 0.59 · states limitation 0.78 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.15 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analysed systematically using the theory of fibre bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear Roe algebras.

background contribution method result interpretation main claim

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 initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions.	contribution 0.98	0.86	0.08	asserted 0.91
s1	For homogeneous C*-algebras, these questions can be analysed systematically using the theory of fibre bundles.	method 0.90	0.20	0.04	asserted 0.69
s2	For group C-algebras, while we are able to find Cartan subalgebras in C-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras.	result 0.98	0.88	0.15	asserted 0.99
s3	Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras.	result 0.99	0.91	0.31	asserted 0.91
s4	However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear Roe algebras.	result 0.98	0.86	0.13	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analysed systematically using the theory of fibre bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear Roe algebras.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

8 arXiv papers cite this in 15 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 8 sentences without a confident label

Builds on it 1 paper

This in particular strengthens and unifies Corollaries 4.7 and 4.11 of [ref] (see Corollary [ref]).

Rigid sides of approximately finite dimensional simple operator algebras in non-separable category · Organization of the paper · 0.84

Compares with it 1 paper

The example covers partially the result of Li and Renault [ref] .

Homology of the Katsura-Exel-Pardo groupoid · Final remarks · 0.94

Cites it as background 5 papers

In [ref] , the relation between Cartan subalgebras in homogeneous -algebras over a space $X$ and principal $(_n ⊂_n)$-bundles over $X$ was pointed out.

Cartan subalgebras in dimension drop algebras · Preliminaries · 0.92
Recently, there has been growing interest in studying general existence and uniqueness questions for Cartan subalgebras in $C^*$-algebras [ref] .

Cartan subalgebras in uniform Roe algebras · Introduction · 0.94
Recently it has been a big interest in finding which classifiable $C^*$-algebras can be realized as étale groupoid $C^*$-algebras (see [ref] ).

Homology of the Katsura-Exel-Pardo groupoid · Introduction · 0.97

What it cites · 21 sentences without a confident label

Uses the method or tool of 1 paper

The first one uses the recent construction of Deeley, Putnam and Strung [ref] of the Jiang-Su algebra $𝒵$ as a groupoid C*-algebra.

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang-Su algebra · Introduction · 0.94

Cites as support 2 papers

It is shown in [ref] that these C*-algebras are tracially AF, hence AH algebras of slow dimension growth (by a result of H.

Some classifiable groupoid C*-algebras with prescribed K-theory · Non-uniqueness of Cartan subalgebras · 0.96
By [ref] , we conclude that $C^*_λ(Γ)$ has unique trace.

C*-simplicity and the unique trace property for discrete groups · Non-existence of Cartan subalgebras in certain reduced group C*-algebras · 0.97

Cites as background 3 papers

Existence and uniqueness of Cartan subalgebras in vN-algebras is an active field of research [ref] .

Strong solidity of free Araki-Woods factors · Introduction · 0.98
Classification of C*-algebras has seen tremendous advances recently [ref] .

Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras · Introduction · 1.00
Classification of C*-algebras has seen tremendous advances recently [ref] .

On the classification of simple amenable $C*$-algebras with finite decomposition rank, II · Introduction · 1.00

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.