Let $α: Γ\curvearrowright A$ be an action of a discrete group $Γ$ on a unital C*-algebra $A$ by *-automorphisms and let $A \rtimes_{α,λ} Γ$ denote the corresponding reduced crossed product C*-algebra. Assuming that $Γ$ satisfies the approximation property, we establish a sufficient and (almost always) necessary condition on the action $α$ for the existence of a Galois correspondence between intermediate C*-algebras for the inclusion $A \subseteq A \rtimes_{α,λ} Γ$ and partial subactions of $α$. This condition, which we refer to as pointwise residual proper outerness, is a natural noncommutative generalization of freeness.
We define centrally large subalgebras of simple unital C*-algebras, strengthening the definition of large subalgebras in previous work. We prove that if A is any infinite dimensional simple separable unital C*-algebra which contains a centrally large subalgebra with stable rank one, then A has stable rank one. We also prove that large subalgebras of crossed product type are automatically centrally large. We use these results to prove that if X is a compact metric space which has a surjective continuous map to the Cantor set, and h is a minimal homeomorphism of X, then C* (Z, X, h) has stable r…
Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer $*$-automorphisms. In this paper we study the containment $M \subseteq M\rtimes_αG$ of $M$ inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the $M$-bimodules that are closed in the Bures topology and which coincide with the $w^*$-closed ones under a mild hypothesis on $G$. We use these results to obtain a general version of Mercer's theorem concerning the extension of certain isometric $w^*$-continuous …
We study uniform perturbations of crossed product C$^*$-algebras by amenable groups. Given a unital inclusion of C$^*$-algebras $C\subseteq D$ and sufficiently close separable intermediate C$^*$-subalgebras $A$, $B$ for this inclusion with a conditional expectation from $D$ onto $B$, if $A=C\rtimes G$ with $G$ discrete amenable, then $A$ and $B$ are isomorphic. Furthermore, if $C\subseteq D$ is irreducible, then $A=B$.
We find necessary and sufficient conditions for the subalgebra of analytic elements associated with a periodic C*-dynamical system to be a maximal norm-closed subalgebra. Our conditions are in terms of the Arveson spectrum of the action. We also describe equivalent properties of the system in terms of the strong Connes spectrum and the simplicity of the crossed product.
We study actions of locally compact groups on von Neumann factors and the associated crossed-product von Neumann algebras. In the setting of totally disconnected groups we provide sufficient conditions on an action $G\curvearrowright Q$ ensuring that the inclusion $Q \subset Q \rtimes G$ is irreducible and that every intermediate subfactor is of the form $Q \rtimes H$ for a closed subgroup $H<G$. This partially generalizes a result of Izumi-Longo-Popa [ILP98] and Choda [Ch78]. We moreover show that one can not hope to use their strategy for non-discrete groups.
This paper studies the relationship between minimal dynamical systems on the product of the Cantor set ($X$) and torus ($\T^2$) and their corresponding crossed product $C^*$-algebras. For the case when the cocycles are rotations, we studied the structure of the crossed product $C^*$-algebra $A$ by looking at a large subalgebra $A_x$. It is proved that, as long as the cocycles are rotations, the tracial rank of the crossed product $C^*$-algebra is always no more than one, which then indicates that it falls into the category of classifiable $C^*$-algebras. If a certain rigidity condition is sati…
We study uniform perturbations of intermediate C*-subalgebras of inclusions of simple C*-algebras. If a unital simple C*-algebra has a simple C*-subalgebra of finite index, then sufficiently close simple intermediate C*-subalgebras are unitarily equivalent. These C*-subalgebras need not to be nuclear. The unitary can be chosen in the relative commutant algebra. An imediate corollary is the following: If the relative commutant is trivial, then the set of intermediate C*-subagebras is a finite set.
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…
We describe the structure of the irreducible representations of crossed products of unital C*-algebras by actions of finite groups in terms of irreducible representations of the C*-algebras on which the groups act. We then apply this description to derive a characterization of irreducible representations of crossed-products by finite cyclic groups in terms of representations of the C*-algebra and its fixed point subalgebra. These results are applied to crossed-products by the permutation group on three elements and illustrated by various examples.
We construct centrally large subalgebras in crossed products of $C (X, D)$ by automorphisms in which $D$ is simple, $X$ is compact metrizable, the automorphism induces a minimal homeomorphism of $X$, and a mild technical assumption holds. We use this construction to prove structural properties of the crossed product, such as (tracial) $Z$-stability, stable rank one, real rank zero, and pure infiniteness, in a number of examples. Our examples are not accessible via methods based on finite Rokhlin dimension, either because $D$ is not $Z$-stable or because $X$ is infinite dimensional.
Let $Γ^{+}$ be the positive cone of a totally ordered abelian discrete group $Γ$, and $α$ an action of $Γ^{+}$ by extendible endomorphisms of a $C^*$-algebra $A$. We prove that the partial-isometric crossed product $A\times_α^{\textrm{piso}}Γ^{+}$ is a full corner of a group crossed product $\mathcal{B}\times_βΓ$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(Γ,A)$ generated by a collection of faithful copies of $A$, and the action $β$ on $\mathcal{B}$ is induced by shift on $\ell^{\infty}(Γ,A)$. We then use this realization to show that $A\times_α^{\textrm{piso}}Γ^{+}$ has an essentia…
Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspo…
We prove six-term exact sequences of Pimsner-Voiculescu type for certain subalgebras of the Cuntz-Pimsner algebras. This sequence may, in particular, be applied to smooth subalgebras of the Quantum Heisenberg Manifolds in order to compute the generators of their cyclic cohomology. Further, our results include the known results for smooth crossed products. Our proof is based on a combination of arguments from the setting of (Cuntz-)Pimsner algebras and the Toeplitz proof of Bott-periodicity.
Recently, Adrian Ioana proved that all crossed products by free ergodic probability measure preserving actions of a nontrivial free product group Γ_1 * Γ_2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer N_M(A)'' of an amenable subalgebra A of an amalgamated free product von Neumann algebra M = M_1 *_B M_2. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in crossed products by Γ_1 * Γ_2.
We define the notion of strong spectral invariance for a dense Frechet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G\rtimes A is spectral invariant in the C*-crossed product G\rtimes B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one …
We present a systematic study of the structure of crossed products and fixed point algebras by compact group actions with the Rokhlin property on not necessarily unital C*-algebras. Our main technical result is the existence of an approximate homomorphism from the algebra to its subalgebra of fixed points, which is a left inverse for the canonical inclusion. Upon combining this with results regarding local approximations, we show that a number of classes characterized by inductive limit decompositions with weakly semiprojective building blocks, are closed under formation of crossed products by…
Suppose $Γ^{+}$ is the positive cone of a totally ordered abelian group $Γ$, and $(A,Γ^{+},α)$ is a system consisting of a $C^*$-algebra $A$, an action $α$ of $Γ^{+}$ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\times_α^{\piso}Γ^{+}$ is a full corner in the subalgebra of $Ł(\ell^{2}(Γ^{+},A))$, and that if $α$ is an action by automorphisms of $A$, then it is the isometric-crossed product $(B_{Γ^{+}}\otimes A)\times^{\iso}Γ^{+}$, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realiza…
In this paper we introduce a row version of Kadison and Kastler's metric on the set of C*-subalgebras of $\mathbb{B}(\mathcal{H})$. By showing C*-algebras have row length (in the sense of Pisier) of at most 2 we show that the row metric is equivalent to the original Kadison Kastler metric. Ino and Watatani have recently proved that in certain circumstances sufficiently close intermediate C*-algebras occur as small unitary perturbations. By adjusting their arguments to work with the row metric we are able to obtain universal constants independent of inclusions.
Consider an inclusion of diffuse von Neumann algebras A c M . We say that A c M has the absorbing amenability property if for any diffuse subalgebra B c A and any amenable intermediate algebra B c D c M we have that D is contained in A. We prove that the cup subalgebra associated to any subfactor planar algebra has the absorbing amenability property.
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 for a large class of actions $Γ\curvearrowright \mathcal{A}$ of $C^*$-simple groups $Γ$ on unital $C^*$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-subalgebra $\mathcal{B}$, $C_λ^*(Γ)\subseteq \mathcal{B} \subseteq \mathcal{A}\rtimes_{r}Γ$, is of the form $\mathcal{A}_1\rtimes_{r}Γ$, where $\mathcal{A}_1$ is a unital $Γ$-$C^*$-subalgebra of $\mathcal{A}$.
For the $C^*$-crossed product $C^*(Σ)$ associated with an arbitrary topological dynamical system $Σ= (X, σ)$, we provide a detailed analysis of the commutant, in $C^* (Σ)$, of $C(X)$ and the commutant of the image of $C(X)$ under an arbitrary Hilbert space representation $\tildeπ$ of $C^* (Σ)$. In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system $Σ$, the commutant of $C(X)$ has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of $C^* (Σ)$. We also show that the corres…
We define smooth generalized crossed products and prove six-term exact sequences of Pimsner-Voiculescu type. This sequence may, in particular, be applied to smooth subalgebras of the Quantum Heisenberg Manifolds in order to compute the generators of their cyclic cohomology. Our proof is based on a combination of arguments from the setting of (Cuntz-)Pimsner algebras and the Toeplitz proof of Bott-periodicity.
Assume A is a Frechet algebra equipped with a smooth isometric action of a vector group V, and consider Rieffel's deformation A_J of A. We construct an explicit isomorphism between the smooth crossed products V\ltimes\A_J and V\ltimes\A. When combined with the Elliott-Natsume-Nest isomorphism, this immediately implies that the periodic cyclic cohomology is invariant under deformation. Specializing to the case of smooth subalgebras of C*-algebras, we also get a simple proof of equivalence of Rieffel's and Kasprzak's approaches to deformation.
We consider group actions of topological groups on C*-algebras of the types which occur in many physics models. These are singular actions in the sense that they need not be strongly continuous, or the group need not be locally compact. We develop a "crossed product host" in analogy to the usual crossed product for strongly continuous actions of locally compact groups, in the sense that its representation theory is in a natural bijection with the covariant representation theory of the action. We prove a uniqueness theorem for crossed product hosts, and analyze existence conditions. We also pre…
We investigate the structure of the automorphism of $\mathcal{O}_{2}$ which exchanges the two canonical isometries. Our main observation is that the fixed point C*-subalgebra for this action is isomorphic to $\mathcal{O}_{2}$ and we detail the relationship between the crossed-product and fixed point subalgebra.
We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S.
If A is a C*-algebra, G a locally compact group, K{\subset}G a compact subgroup and α:G{\to}Aut(A) a continuous homomorphism, let Ax_{α}G denote the crossed product. In this paper we prove that Ax_{α}G is nuclear (respectively type I or liminal) if and only if certain hereditary C*-subalgebras, S_{π}, I_{π}{\subset}Ax_{α}G π{\in}K, are nuclear (respectively type I or liminal). These algebras are the analogs of the algebras of spherical functions considered by R. Godement for groups with large compact subgroups. If K=G is a compact group or a compact quantum group, the algebras S_{π} are stably…
Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties of subalgebras with these mixing properties. We prove some basic results about mixing inclusions of von Neumann algebras and establish a connection between mixing properties and normalizers of von Neumann subalgebras. The special case of mixing subalgebras arising from inclusions of countable discrete groups finds applications to ergodic theory, in particular, a new generalization of a classical th…
We extend Matui's notion of almost finiteness to general etale groupoids and show that the reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result. (i) For any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank one. (ii) Any countable amenable group admits a minimal action on the Cantor set all whos…
We show that for a locally compact group G there is a one-to-one correspondence between G-invariant weak*-closed subspaces E of the Fourier-Stieltjes algebra B(G) containing B_r(G) and quotients C*_E(G) of C*(G) which are intermediate between C*(G) and the reduced group algebra C*_r(G). We show that the canonical comultiplication on C*(G) descends to a coaction or a comultiplication on C*_E(G) if and only if E is an ideal or subalgebra, respectively. When αis an action of G on a C*-algebra B, we define "E-crossed products" B\rtimes_{α,E} G lying between the full crossed product and the reduced…
We consider a fixed free and proper action of a locally compact group $G$ on a space $T$, and actions $α:G\to \Aut A$ on $C^*$-algebras for which there is an equivariant embedding of $(C_0(T),\rt)$ in $(M(A),α)$. A recent theorem of Rieffel implies that $α$ is proper and saturated with respect to the subalgebra $C_0(T)AC_0(T)$ of $A$, so that his general theory of proper actions gives a Morita equivalence between $A\rtimes_{α,r}G$ and a generalised fixed-point algebra $A^α$. Here we investigate the functor $(A,α)\mapsto A^α$ and the naturality of Rieffel's Morita equivalence, focusing in parti…
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…
Let $G$ be a finite group and $H$ a normal subgroup. $D(H;G)$ is the crossed product of $C(H)$ and ${\Bbb C}G$ which is only a subalgebra of $D(G)$, the quantum double of $G$. One can construct a $C^*$-subalgebra ${\mathcal{F}}_{_H}$ of the field algebra $\mathcal{F}$ of $G$-spin models, such that ${\mathcal{F}}_{_H}$ is a $D(H;G)$-module algebra. The concrete construction of $D(H;G)$-invariant subalgebra ${\mathcal{A}}_{_{(H,G)}}$ of ${\mathcal{F}}_{_H}$ is given. By constructing the quasi-basis of conditional expectation $z_{_H}$ of ${\mathcal{F}}_{_H}$ onto ${\mathcal{A}}_{_{(H,G)}}$, the $…
Uniformly recurrent subalgebras in finite von Neumann algebras
Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）；Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学与索邦大学，法国国家科学研究中心，IMJ-PRG）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
We construct a class of II_1 factors M that admit unclassifiably many Cartan subalgebras in the sense that the equivalence relation of being conjugate by an automorphism of M is complete analytic, in particular non Borel. We also construct a II_1 factor that admits uncountably many non isomorphic group measure space decompositions, all involving the same group G. So G is a group that admits uncountably many non stably orbit equivalent actions whose crossed product II_1 factors are all isomorphic.
In 1990, Rieffel defined a notion of proper action of a group $H$ on a $C^*$-algebra $A$. He then defined a generalized fixed point algebra $A^α$ for this action and showed that $A^α$ is Morita equivalent to an ideal of the reduced crossed product. We generalize Rieffel's notion to define proper groupoid dynamical systems and show that the generalized fixed point algebra for proper groupoid actions is Morita equivalent to a subalgebra of the reduced crossed product. We give some nontrivial examples of proper groupoid dynamical systems and show that if $(\A, G, α)$ is a groupoid dynamical syste…
For a discrete group $G$, we develop a `$G$-balanced tensor product' of two coactions $(A,δ)$ and $(B,ε)$, which takes place on a certain subalgebra of the maximal tensor product $A\otimes_{\max} B$. Our motivation for this is that we are able to prove that given two actions of $G$, the dual coaction on the crossed product of the maximal-tensor-product action is isomorphic to the $G$-balanced tensor product of the dual coactions. In turn, our motivation for this is to give an analogue, for coaction functors, of a crossed-product functor originated by Baum, Guentner, and Willett, and further de…
We define and study large and stably large subalgebras of simple unital C*-algebras. The basic example is the orbit breaking subalgebra of a crossed product by Z, as follows. Let X be an infinite compact metric space, let h be a minimal homeomorphism of X, and let Y be a closed subset of X. Let u be the standard unitary in C* (Z, X, h). The Y-orbit breaking subalgebra is the subalgebra of C* (Z, X, h) generated by C (X) and all elements f u for f in C (X) such that f vanishes on Y. If intersects each orbit of h at most once, then the Y-orbit breaking subalgebra is large in C* (Z, X, h). Large …
