Intermediate C ^ -algebras of Cartan embeddings

Let A be a C^*-algebra and let D be a Cartan subalgebra of A. We study the following question: if B is a C^*-algebra such that D \subseteq B \subseteq A, is D a Cartan subalgebra of B? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A onto B, an...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society. Series B 2021-01, Vol.8 (3), p.27-41
Main Authors: Brown, Jonathan H., Exel, Ruy, Fuller, Adam H., Pitts, David R., Reznikoff, Sarah A.
Format: Article
Language:English
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Research article
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Summary:Let A be a C^*-algebra and let D be a Cartan subalgebra of A. We study the following question: if B is a C^*-algebra such that D \subseteq B \subseteq A, is D a Cartan subalgebra of B? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A onto B, and the case when A is nuclear and D is a C^*-diagonal of A. In both cases there is a one-to-one correspondence between the intermediate C^*-algebras B, and a class of open subgroupoids of the groupoid G, where \Sigma \rightarrow G is the twist associated with the embedding D \subseteq A.
ISSN:2330-1511
2330-1511
DOI:10.1090/bproc/66