Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C^*_r(G, Σ)$ for any twist $Σ$ over $G$. However, there are many groupoids which are not topologically principal, yet their (twisted) $C^*$-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle $c$ on $G$ and a subgroupoid $S \subseteq G$ under which $C^*_r(S, c)$ is Cartan in $C^*_r(G, c)$. When $G$ is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.
We construct Cartan subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan subalgebra.
When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify …
The study of different types of ideals in non self-adjoint operator algebras has been a topic of recent research. This study focuses on principal ideals in subalgebras of groupoid C*-algebras. An ideal is said to be principal if it is generated by a single element of the algebra. We look at subalgebras of r-discrete principal groupoid C*-algebras and prove that these algebras are principal ideal algebras. Regular canonical subalgebras of almost finite C*-algebras have digraph algebras as their building blocks. The spectrum of almost finite C*-algebras has the structure of an r-discrete princip…
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…
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 relations 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 natural definition of a Cartan subalgebra in a C*-algebra and an extension of K…
We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this param…
This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors.
We characterise Exel's noncommutative Cartan subalgebras in several ways using uniqueness of conditional expectations, relative commutants, or purely outer inverse semigroup actions. We describe in which sense the crossed product decomposition for a noncommutative Cartan subalgebra is unique. We relate the property of being a noncommutative Cartan subalgebra to aperiodic inclusions and effectivity of dual groupoids. In particular, we extend Renault's characterisation of commutative Cartan subalgebras.
We show that the continuous core of any type III free product factor has no Cartan subalgebra. This is a complement to previous works due to Houdayer--Ricard and Boutonnet--Houdayer--Raum.
It is a wide open problem to give an intrinsic criterion for a II_1 factor $M$ to admit a Cartan subalgebra $A$. When $A \subset M$ is a Cartan subalgebra, the $A$-bimodule $L^2(M)$ is "simple" in the sense that the left and right action of $A$ generate a maximal abelian subalgebra of $B(L^2(M))$. A II_1 factor $M$ that admits such a subalgebra $A$ is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II_1 factor $M$ to be s-thin and left open the question whether all s-thin II_1 factors admit a Cartan subalgebra. We answer this question negatively by construc…
We consider some conditions similar to Ozawa's condition (AO), and prove that if a non-injective factor satisfies such a condition and has the W*CBAP, then it has no Cartan subalgebras. As a corollary, we prove that $\rm II_1$ factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type $\rm III_1$ factors with such a condition are semisolid as a $\rm II_\infty$ factor.
These notes provide an account on four lectures given by Adrian Ioana at the Institut Henri Poincare in May 2011, regarding a number of "uniqueness of Cartan" type results obtained in a recent series of papers by Chifan-Peterson and Popa-Vaes. The text is purely expository.
We prove a generalized version of Renault's theorem for Cartan subalgebras. We show that the original assumptions of second countability and separability are not needed. This weakens the assumption of topological principality of the underlying groupoid to effectiveness.
The reduced $C^*$-algebra of the interior of the isotropy in any Hausdorff étale groupoid $G$ embeds as a $C^*$-subalgebra $M$ of the reduced $C^*$-algebra of $G$. We prove that the set of pure states of $M$ with unique extension is dense, and deduce that any representation of the reduced $C^*$-algebra of $G$ that is injective on $M$ is faithful. We prove that there is a conditional expectation from the reduced $C^*$-algebra of $G$ onto $M$ if and only if the interior of the isotropy in $G$ is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, $M$ is a C…
Supplementary results obtained after the completion of our previous paper are given together with discussing some examples. A quick review of the previous paper is also included.
In this paper, two sufficient and necessary conditions are given. The first one characterizes when the boundary path groupoid of a topological graph without singular vertices has closed interior of its isotropy group bundle, and the second one characterizes when the path groupoid of a row-finite k-graph without sources has closed interior of its isotropy group bundle. It follows that the associated topological graph algebra and the associated k-graph C*-algebra have Cartan subalgebras due to a result of Brown-Nagy-Reznikoff-Sims-Williams.
Let $C^*$-algebra that is acted upon by a compact abelian group. We show that if the fixed-point algebra of the action contains a Cartan subalgebra $D$ satisfying an appropriate regularity condition, then $A$ is the reduced $C^*$-algebra of a groupoid twist. We further show that the embedding $D \hookrightarrow A$ is uniquely determined by the twist. These results generalize Renault's results on Cartan subalgebras of $C^*$-algebras.
We construct ample groupoids from certain categories of paths, and prove that their $C^*$-algebras coincide with the continued fraction AF algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear $C^*$-algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.
Denote by $U_{\mathcal I}({\mathcal H})$ the group of all unitary operators in ${\bf 1}+{\mathcal I}$ where ${\mathcal H}$ is a separable infinite-dimensional complex Hilbert space and ${\mathcal I}$ is any two-sided ideal of ${\mathcal B}({\mathcal H})$. A Cartan subalgebra ${\mathcal C}$ of ${\mathcal I}$ is defined in this paper as a maximal abelian self-adjoint subalgebra of~${\mathcal I}$ and its conjugacy class is defined herein as the set of Cartan subalgebras $\{V{\mathcal C} V^*\mid V\in U_{\mathcal I}({\mathcal H})\}$. For nonzero proper ideals ${\mathcal I}$ we construct an uncounta…
We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us to deduce, among other things, that a crossed product $\mathcal O_2\rtimes_α\mathbb Z_p $ satisfies the UCT if there is some automorphism $γ$ of $\mathcal O_2$ with the property that $γ(\mathcal D_2)\subseteq \mathcal O_2\rtimes_α\mathbb Z_p$ is regular, where $\mathcal D_2$ denotes the canonical masa of $\mathcal O_2$. We prove that this condition is automa…
Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $σ$-finite factor $B$, the tensor product $N \otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa--Popa and Popa--Vaes's weakly compact action on the continuous core of $N \otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenabi…
We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full $C^*$-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.
In this short note we classify the Cartan subalgebras in all von Neumann algebras associated with graph product groups and their free ergodic measure preserving actions on probability spaces.
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II$_1$ factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to unitary conjugacy are not classifiable by countable structures, and d…
We introduce the Haagerup property for twisted groupoid $C^*$-dynamical systems in terms of naturally defined positive-definite operator-valued multipliers. By developing a version of `the Haagerup trick' we prove that this property is equivalent to the Haagerup property of the reduced crossed product $C^*$-algebra with respect to the canonical conditional expectation $E$. This extends a theorem of Dong and Ruan for discrete group actions, and implies that a given Cartan inclusion of separable $C^*$-algebras has the Haagerup property if and only if the associated Weyl groupoid has the Haagerup…
In this paper we exploit the fact that a Cuntz C$^*$-algebra is a groupoid C$^*$-algebra to facilitate the study of non-self-adjoint subalgebras of $O_n$. The Cuntz groupoid is not principal and the spectral theorem for bimodules does not apply in full generality. We characterize the bimodules (over a natural masa) which are determined by their spectra in the Cuntz groupoid; these are exactly the ones which are invariant under the guage automorphisms and exactly the ones which are generated by the Cuntz partial isometries which they contain. We investigate analytic subalgebras of $O_n, n$ fini…
We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,μ)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.
For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et. al, we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra.
We show that outer approximately represenbtable actions of a finite cyclic group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property if the corresponding crossed products satisfy the UCT and absorb a suitable UHF algebra tensorially. More concretely, we prove that for such an action there exists an inverse semigroup of homogeneous partial isometries that generates the ambient C*-algebra and whose idempotent semilattice generates a Cartan subalgebra. We prove a similar result for actions of finite cyclic groups with the Rokhlin property on UCT Kirchberg algebras absorbing a…
We prove a spectral theorem for bimodules in the context of graph C*-algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz-Krieger partial isometries which it contains iff it is invariant under the gauge automorphisms. We study 1-cocycles on the Cuntz-Krieger groupoid associated with a graph C*-algebra, obtaining results on when integer valued or bounded cocycles on the natural AF subgroupoid extend. To a finite graph with a total order, we associate a nest subalgebra of the graph C*-algebra and th…
A simple Steinberg algebra associated to an ample Hausdorff groupoid $G$ is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid $C^*$-algebra $C^*_r(G)$ is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate $*$-algebra $B(G)$ constructed using corners $1_U C^*_r(G) 1_U$ for all compact open subsets $U$ of the unit space of the g…
In this paper, we study a family of $C^*$-subalgebras defined by fixed points of generalized gauge actions of a Cuntz-Krieger algebra, by introducing a family of étale groupoids whose associated $C^*$-algebras are these $C^*$-subalgebras. We know that topological conjugacy classes of one-sided topological Markov shifts are characterized in terms of the isomorphism classes of these étale groupoids.
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined…
Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of T such that G:=E/T is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E;G,λ) is a quotient of the C*-algebra of E obtained by completing the space of T-equivariant functions on E. We show that C*(E;G,λ) is postliminal if and only if the orbit space of G is T_0 and that C*(E;G, λ) is liminal if and only if the orbit space is T_1. We also show that C*(E;G, λ) has bounded trace if and only if G is integrable and that C*(E;G, λ) is a Fell algebra if and only if G is Cartan. Let \…
In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $B\subseteq A$ of $\mathrm{C}^*$-algebras is isomorphic to the canonical inclusion of $\ell^\infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for `Roe Cartans' inside uniform Roe algebras up to automorphism when $X$ coarsely embeds into Hilbert space, and up to inner automorphism when $X$ has property A.
Suppose $G$ is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let $\go/G$ denote the orbit space of $G$ and $\cs(G)$ denote the groupoid $C^*$-algebra. Suppose that $G$ is a principal groupoid. We show that $C^*(G)$ is CCR if and only if $\go/G$ is a $T_1$ topological space, and that $C^*(G)$ is GCR if and only if $\go/G$ is a $T_0$ topological space. We also show that $C^*(G)$ is a Fell Algebra if and only if $G$ is a Cartan groupoid.
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 $Σ\rightarrow G$ is the twist associated with the embed…
We construct a group measure space II$_1$ factor that has two non-conjugate Cartan subalgebras. We show that the fundamental group of the II$_1$ factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is non-trivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.
We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $Γ$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a free ergodic probability measure preserving action of $Γ$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability …
Cartan inverse semigroups and twisted Steinberg algebras
University of Sydney（悉尼大学）；Victoria University of Wellington（惠灵顿维多利亚大学）；Queen’s University Belfast（贝尔法斯特女王大学）；MSU-Iligan Institute of Technology（棉兰老国立理工大学）
We establish a Galois correspondence for finite quantum groupoid actions on II_1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such a subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain C^*-algebras related to it.
Given a saturated Fell bundle A over an inverse semigroup S which is semi-abelian in the sense that the fibers over the idempotents of S are commutative, we construct a twisted etale groupoid L such that A can be recovered from L in a canonical way. As an application we recover most of Renault's recent result on the classification of Cartan subalgebras of C*-algebras through twisted etale groupoids.
We present examples of non-Hausdorff, etale, essentially principal groupoids for which three results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian within the reduced groupoid C*-algebra, (B) every nonzero ideal of the reduced groupoid C*-algebra has a nonzero intersection with the subalgebra of continuous functions on the unit space, and (C) the open support of a normalizer is a bissection.
