Cartan Subalgebras in $C^*$-Algebras

By J. Renault. Irish Mathematical Society Bulletin, 2008

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According to J. Feldman and C. Moore's well- known theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable standard measured equivalence re- lations and Cartan pairs, i.e., a von Neumann algebra (on a separable Hilbert space) together with a Cartan subalgebra. A. Kumjian gave a C -algebraic analogue of this theorem in the early eighties. After a short survey of maximal abelian self-adjoint subalgebras in operator algebras, I present a nat- ural denition of a Cartan subalgebra in a C -algebra and an extension of Kumjian's theorem which covers graph algebras and some foliation algebras.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f35ddc287cc24d1d75d5d76518459e1650be9006

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Citations in the corpus

Cartan subalgebras in C*-algebras. Existence and uniqueness (2017, background): “By [52], every Cartan pair (A,B) consisting of a Cartan subalgebra B in a C*-algebra A is of the form (C r (G,Σ), C0(G )), where the groupoid G is as above, G is its unit space and Σ is a twist over G.”
Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras (2007, background): “We refer the reader to [18] for background material on étale groupoids.”
Continuous orbit equivalence rigidity (2015, background): “It would also be very interesting to study the notion of Cartan subalgebras of C∗algebras [24] (see also Remark 2.4).”
Reconstruction of groupoids and C*-rigidity of dynamical systems (2017, methodology): “If D ⊆ A is Cartan as in [39], then H(A,D) is the Weyl groupoid of [39].”
\'Etale groupoids and their $C^*$-algebras (2017, methodology): “Moreover, results of Renault [39] show that we can characterise twisted C-algebras associated to étale effective groupoids amongst arbitrary C-algebras in purely C-algebraic terms: they are the ones that admit a Cartan subalgebra.”
Homology and topological full groups of étale groupoids on totally disconnected spaces (2009, methodology): “It suffices to show (1).”
Graph algebras and orbit equivalence (2014, methodology): “We show that G(C∗(E),D(E)) and GE are isomorphic as topological groupoids for an arbitrary graph E, which can be deduced from Renault’s results in [22] only when every cycle in E has an exit.”
Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras (2016, methodology): “This construction goes back to the work of Feldman and Moore [15] on von Neumann factors and was continued by Kumjian [17] and Renault [26] for C∗-algebras.”
Etale groupoids arising from products of shifts of finite type (2015, background): “We refer the reader to [33, 34] for background material on étale groupoids.”
Partial transformation groupoids attached to graphs and semigroups (2016, background): “Recall (see [34]) that a topological groupoid G is called topologically principal if the set of points in G with trivial isotropy is dense in G.”