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 corresponding statement holds true for the commutant of $\tildeπ(C(X))$ under the assumption that a certain family of pure states of $\tildeπ(C^* (Σ))$ is total. Furthermore we establish that, if $C(X) \subsetneq C(X)'$, there exist both a $C^*$-subalgebra properly between $C(X)$ and $C(X)'$ which has the aforementioned intersection property, and such a $C^*$-subalgebra which does not have this property. We also discuss existence of a projection of norm one from $C^*(Σ)$ onto the commutant of $C(X)$.
We show that indecomposable weak Kac algebras are free over their Cartan subalgebras and prove a duality theorem for their actions. Using this result, for any biconnected weak Kac algebra we construct a minimal action on the hyperfinite II_1 factor. The corresponding crossed product inclusion of II_1 factors has depth 2 and an integer index. Its first relative commutant is, in general, non-trivial, so we derive some arithmetic properties of weak Kac algebras from considering reduced subfactors.
Let $α:G \curvearrowright M$ be a spatial action of countable abelian group on a "spatial" von Neumann algebra $M$ and $S$ be its unital subsemigroup with $G=S^{-1}S$. We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten $p$-class or the compact operators, of the w$^*$-semicrossed product of $M$ by $S$ when $M'$ contains no non-zero compact operators. We also prove a weaker result when $M$ is a von Neumann algebra on a finite dimensional Hilbert space and $(G,S)=(\mathbb{Z},\mathbb{Z}_{+})$, which extends a famous result due to Davidson (1977) for t…
Starting from a discrete $C^*$-dynamical system $(\mathfrak{A}, θ, ω_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(\mathfrak{A}\rtimes_α\mathbb{Z}, Φ_{θ, u},\om_o\circ E)$, $E:\mathfrak{A}\rtimes_α\mathbb{Z}\rightarrow\ga$ being the canonical conditional expectation of $\mathfrak{A}\rtimes_α\mathbb{Z}$ onto $\mathfrak{A}$, provided $\a\in\aut(\ga)$ commute with the $*$-automorphism $þ$ up tu a unitary $u\in\ga$. Here, $Φ_{θ, u}\in\aut(\mathfrak{A}\rtimes_α\mathbb{Z})$ can be considered as the fully noncommutative generalisation of …
For minimal unique ergodic diffeomorphisms $α_n$ of $S^{2n+1} (n>0)$ and $α_m$ of $S^{2m+1}(m>0)$, the $C^*$-crossed product algebra $C(S^{2n+1})\rtimes_{α_n} \mathbb{Z}$ is isomorphic to $C(S^{2m+1})\rtimes_{α_m} \mathbb{Z}$ even though $n\neq m$ . However, by cyclic cohomology, we show that smooth crossed product algebra $C^\infty(S^{2n+1})\rtimes_{α_n} \mathbb{Z}$ is not isomorphic to $C^\infty(S^{2m+1})\rtimes_{α_m} \mathbb{Z}$ if $n\neq m$.
Let $θ, θ'$ be irrational numbers and $A, B$ be matrices in $SL_2(\mathbb{Z})$ of infinite order. We compute the $K$-theory of the crossed product $\mathcal{A}_θ\rtimes_A \mathbb{Z}$ and show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are $*$-isomorphic if and only if $θ= \pmθ' \pmod{\mathbb{Z}}$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are Morita equivalent if and only if $θ…
The $K$-groups of the crossed product of the rotation C*-algebra $A_θ$ by free and amalgamated products of the cyclic groups $\mathbb Z_n$, for $n=2,3,4,6$, are calculated. The actions here arise from the canonical actions of these groups on the rotation algebra under the flip, cubic, Fourier, and hexic automorphisms, respectively. An interesting feature in this study is that although the inclusion $A_θ\to A_θ\rtimes \mathbb Z_n$ induces injective maps on their $K_0$-groups, the same is not the case for the inclusions $A_θ\rtimes \mathbb Z_d \to A_θ\rtimes \mathbb Z_n$ for $2\le d < n \le 6$ a…
Crossed products with noncommutative Bernoulli actions were introduced by Connes as the first examples of full factors of type III. This article provides a complete classification of the factors $(P,ϕ)^{\mathbb{F}_n} \rtimes \mathbb{F}_n$, where $\mathbb{F}_n$ is the free group and P is an amenable factor with an almost periodic state $ϕ$. We show that these factors are completely classified by the rank n of the free group $\mathbb{F}_n$ and Connes's Sd-invariant. We prove similar results for free product groups, as well as for classes of generalized Bernoulli actions.
In this article we construct two minimal unique ergodic diffeomorphisms $α$ and $β$ on $S^3 \times S^{6} \times S^{8} $. We will show that $C(S^3 \times S^{6} \times S^{8}) \rtimes_α\mathbb{Z} $ and $C(S^3 \times S^{6} \times S^{8})\rtimes_β\mathbb{Z} $ are equivalent to each other, while $C^\infty (S^3 \times S^{6} \times S^{8})\rtimes_α\mathbb{Z} $ and $C^\infty(S^3 \times S^{6} \times S^{8} )\rtimes_β\mathbb{Z} $ are not.
For a locally compact quantum group $\mathbb{G}$, consider the convolution action of a quantum probability measure $μ$ on $L_\infty(\mathbb{G})$. As shown by Junge--Neufang--Ruan, this action has a natural extension to a Markov map on $\mathcal{B}(L_2(\mathbb{G}))$. We prove that the Poisson boundary of the latter can be realized concretely as the von Neumann crossed product of the Poisson boundary associated with $μ$ under the action of $\mathbb{G}$ induced by the coproduct. This yields an affirmative answer, for general locally compact quantum groups, to a problem raised by Izumi (2004) in t…
We show that for a closed embedding $\mathbb{H}\le \mathbb{G}$ of locally compact quantum groups (LCQGs) with $\mathbb{G}/\mathbb{H}$ admitting an invariant probability measure, a unitary $\mathbb{G}$-representation is type-I if its restriction to $\mathbb{H}$ is. On a related note, we also prove that if an action $\mathbb{G}\circlearrowright A$ of an LCQG on a unital $C^*$-algebra admits an invariant state then the full group algebra of $\mathbb{G}$ embeds into the resulting full crossed product (and into the multiplier algebra of that crossed product if the original algebra is not unital). W…
We show that if $(X.A)$ and $(Y,B)$ are two isomorphic Hilbert pro-$C^{\ast} $-bimodules, then the crossed product $A\times_{X}\mathbb{Z}$ of $A$ by $X$ and the crossed product $B\times_{Y}\mathbb{Z}$ of $B$ by $Y$ are isomorphic as pro-$C^{\ast}$-algebras. We also prove a property of "associativity" between " $\otimes_{\min}$" and "$\times_{X}$" $\ $as well as " $\otimes_{\max}$" and "$\times_{X}$". As an application of these results we show that the crossed product of a nuclear pro-$C^{\ast}$ -algebra $A$ by a full Hilbert pro-$C^{\ast}$-bimodule $X$ is a nuclear pro-$C^{\ast}$-algebra.
We lift an action of a torus $\mathbb{T}^n$ on the spectrum of a continuous trace algebra to an action of a certain crossed module of Lie groups that is an extension of $\mathbb{R}^n$. We compute equivariant Brauer and Picard groups for this crossed module and describe the obstruction to the existence of an action of $\mathbb{R}^n$ in our framework.
We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state $φ_0$ and any discrete group $Γ$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,φ_0)^{\mathbin{\bar{\otimes}}Γ}\rtimes Γ$ is solid relatively to $\mathcal{L}(Γ)$. In particular, if $\mathcal{L}(Γ)$ is solid then $M$ is solid and if $Γ$ is non-amenable and $A_0 \neq \mathbb{C}$ then $M$ is a full prime factor. This gives many new examples of solid or prime type $\mathrm{III}$ factors. Following Chifan and Ioana, we also obta…
For certain generating sets of the subfactor pair $M\subset M\rtimes G$ where $G$ is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form $M\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus\infty}$ for a large class of algebras $M$.
Let $φ:X\to X$ be a homeomorphism of a compact metric space $X$. For any continuous function $F:X\to \mathbb{R}$ there is a one-parameter group $α^{F}$ of automorphisms on the crossed product $C^*$-algebra $C(X)\rtimes_φ\mathbb{Z}$ defined such that $α^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and $U$ is the canonical unitary in the construction of the crossed product. In this paper we study the KMS states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS-states can be very rich and complicated. Our resul…
An automorphism $β$ of a $k$-graph $Λ$ induces a crossed product $C^* ( Λ) \rtimes_β\mathbb{Z}$ which is isomorphic to a $(k+1)$-graph algebra $C^* ( Λ\times_β\mathbb{Z})$. In this paper we show how this process interacts with $k$-graph $C^*$-algebras which have been twisted by an element of their second cohomology group. This analysis is done using a long exact sequence in cohomology associated to this data. We conclude with some examples
Larsen has recently extended Exel's construction of crossed products from single endomorphisms to abelian semigroups of endomorphisms, and here we study two families of her crossed products. First, we look at the natural action of the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group $Γ$, and the induced action on $C(Γ)$. We prove a uniqueness theorem for the crossed product, and we find a class of connected compact abelian groups $Γ$ for which the crossed product is purely infinite simple. Second, we consider some natural actions of the additive semigroup $\mathbb{N}^2$ …
In this paper we introduce the notion of multiplier of a Hilbert pro-$C^{\ast }$-bimodule and we investigate the structure of the multiplier bimodule of a Hilbert pro-$C^{\ast}$-bimodule. We also investigate the relationship between the crossed product $A\times _{X}\mathbb{Z}$ of a pro-$% C^{\ast }$-algebra $A$ by a Hilbert pro-$C^{\ast }$-bimodule $X$ over $A$, the crossed product $M(A)\times _{M(X)}\mathbb{Z}$ of the multiplier algebra $M(A)$ of $A$ by the multiplier bimodule $M(X)$ of $X$ and the multiplier algebra $M(A\times _{X}\mathbb{Z})$ of $A\times _{X}\mathbb{Z}$.
For $p \in [1, \infty),$ we define and study full and reduced crossed products of algebras of operators on $σ$-finite $L^p$ spaces by isometric actions of second countable locally compact groups. We give universal properties for both crossed products. When the group is abelian, we prove the existence of a dual action on the full and reduced $L^p$ operator crossed products. When the group is discrete, we construct a conditional expectation to the original algebra which is faithful in a suitable sense. For a free action of a discrete group on a compact metric space $X,$ we identify all traces on…
One of the major questions in the theory of Toeplitz operators on the Bergman space over the unit disk $\mathbb D$ in the complex plane $\mathbb C$ is a complete description of the commutant of a given Toeplitz operator, that is the set of all Toeplitz operators that commute with it. In \cite{l}, the first author obtained a complete description of the commutant of Toeplitz operator $T$ with any quasihomogeneous symbol $ϕ(r)e^{ipθ}, p>0$ in case it has a Toeplitz p-th root $S$ with symbol $ψ(r)e^{iθ}$, namely, commutant of $T$ is the closure of the linear space generated by powers $S^n$ which a…
Given a von Neumann algebra $M$ and a $W^{\ast}$-correspondence $E$ over $M$, we construct an algebra $H^{\infty}(E)$ that we call the Hardy algebra of $E$. When $M=\mathbb{C}=E$, then $H^{\infty}(E)$ is the classical Hardy space $H^{\infty}(\mathbb{T})$ of bounded analytic functions on the unit disc. We show that given any faithful normal representation $σ$ of $M$ on a Hilbert space $H$ there is a natural correspondence $E^σ$ over the commutant $σ(M)^{\prime}$, called the $σ$-dual of $E$, and that $H^{\infty}(E)$ can be realized in terms of ($B(H)$-valued) functions on the open unit ball $\ma…
Let $(A,α)$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $α$ of $A$. We describe the primitive ideal space of the partial-isometric crossed product $A\times_α^{\textrm{piso}}\mathbb{N}$ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.
We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_θ$, for any real angle $θ$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of $\mathcal{A}_θ$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.
We present a natural imbedding of the crossed product $\mathcal A \rtimes_ξ\mathbb Z^d$ into the $C^\ast$-algebra of adjointable operators over the standard Hilbert $\mathcal A$-module $\mathcal H_{\mathcal A}$. By replacing the representations on Hilbert spaces with this canonical imbedding, we define Fredholm modules and corresponding Chern-Connes characters that are intrinsic to the $C^\ast$-dynamical system $(\mathcal A,ξ,\mathbb Z^d)$. The compression of the Dirac operator against projectors from $\mathcal A \rtimes_ξ\mathbb Z^d$ produces generalized Fredholm operators over $\mathcal H_{\…
We introduce crossed products of a $C^*$-algebra $A$ by a completely positive map $\varrho:A\to A$ relative to an ideal in $A$. They generalize various crossed products by endomorphisms when $\varrho$ is multiplicative. When $A$ is commutative they include $C^*$-algebras associated to Markov operators by Ionescu, Muhly, Vega, and to topological relations by Brenken, but in general they are not modeled by topological quivers popularized by Muhly and Tomforde. We show that Exel's crossed product $A\rtimes_{α,L} \mathbb{N}$, generalized to the case where $A$ is not necessarily unital, is the cros…
We show that for any locally compact Hausdorff space $Y$ with finite covering dimension and for any continuous flow $\mathbb{R} \curvearrowright Y$, the resulting crossed product $C^*$-algebra $C_0(Y) \rtimes \mathbb{R}$ has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of $C^*$-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of $\mathbb{Z}$. Some novel tec…
The adiabatic groupoid $\mathcal{G}_{ad}$ of a smooth groupoid $\mathcal{G}$ is a deformation relating $\mathcal{G}$ with its algebroid. In a previous work, we constructed a natural action of $\mathbb{R}$ on the C*-algebra of zero order pseudodifferential operators on $\mathcal{G}$ and identified the crossed product with a natural ideal $J(\mathcal{G})$ of $C^*(\mathcal{G}_{ad})$. In the present paper we show that $C^*(\mathcal{G}_{ad})$ itself is a pseudodifferential extension of this crossed product in a sense introduced by Saad Baaj. Let us point out that we prove our results in a slightly …
We construct explicit examples of half-sided modular inclusions ${\mathcal N}\subset{\mathcal M}$ of von Neumann algebras with trivial relative commutants. After stating a general criterion for triviality of the relative commutant in terms of an algebra localized at infinity, we consider a second quantization inclusion ${\mathcal N}\subset{\mathcal M}$ with large relative commutant and construct a one-parameter family ${\mathcal N}_κ\subset{\mathcal M}_κ$, $κ\geq0$, of half-sided inclusions such that ${\mathcal N}_0={\mathcal N}$, ${\mathcal M}_0={\mathcal M}$ and ${\mathcal N}_κ'\cap{\mathcal…
Green's theorem gives a Morita equivalence $C_0(G/H,A)\rtimes G\sim A\rtimes H$ for a closed subgroup $H$ of a locally compact group $G$ acting on a $C^*$-algebra $A$. We prove an analogue of Green's theorem in the case $G=\mathbb{Z}$, where the automorphism generating the action is replaced by a Hilbert $C^*$-bimodule.
We bound the nuclear dimension of crossed products associated to some partial actions of finite groups or $\mathbb{Z}$ on finite dimensional locally compact Hausdorff second countable spaces. Our results apply to globalizable partial actions, finite group partial actions, minimal partial automorphisms, and partial automorphisms acting on zero-dimensional spaces, or a class of one dimensional spaces, containing $1$-dimensional CW complexes. This extends work on global systems by Hirshberg and Wu.
Suppose $A$ is a $C^*$-algebra and $H$ is a $C^*$-correspondence over $A$. If $H$ is regular in the sense that the left action of $A$ is faithful and is given by compact operators, then we compute the $K$-theory of $\mathcal{O}_A(H) \rtimes \mathbb{T}$ where the action is the usual gauge action. The case where $A$ is an AF-algebra is carefully analyzed. In particular, if $A$ is AF, we show $\mathcal{O}_A(H) \rtimes \mathbb{T}$ is AF. Combining this with Takai duality and an AF-embedding theorem of N. Brown, we show the conditions AF-embeddability, quasidiagonality, and stable finiteness are eq…
We study the matricial field (MF) property for certain reduced crossed product C*-algebras and their traces. Using classification techniques and induced K-theoretic dynamics, we show that reduced crossed products of ASH-algebras of real rank zero by free groups are MF if and only if they are stably finite. We also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties. Combining these results with recent progress in Elliott's Classification Program, it follows that if $A$ is a separable, simple, unital, nuclear, monotracial C*-al…
Let $A$ be a unital simple $A\T$-algebra of real rank zero. Given an isomorphism $γ_1: K_1(A)\to K_1(A),$ we show that there is an automorphism $\af: A\to A$ such that $\af_{*1}=γ_1$ which has the tracial Rokhlin property. Consequently, the crossed product $A\rtimes_{\af}\Z$ is a simple unital AH-algebra with real rank zero. We also show that automorphism with Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.
We provide a reference for basic categorial properties of the categories of (possibly non-unital) $\mathbb{C}$-linear $*$-categories or $C^{*}$-categories, and (not necessarily unit-preserving) functors. Generalizing the classical case of algebras with $G$-action, we extend the construction of crossed products to categories with $G$-action. We will show that the crossed product functor preserves exact sequences and excisive squares and sends weak equivalences to equivalences.
We introduce a subclass of recursive subhomogeneous algebras, in which each of the pullback maps is diagonal in a suitable sense. We define the notion of a diagonal map between two such algebras and show that every simple inductive limit of these algebras with diagonal bonding maps has stable rank one. As an application, we prove that for any infinite compact metric space $T$ and minimal homeomorphism $h\colon T\to T$, the associated dynamical crossed product $\mathrm{C^*}(\mathbb{Z},T,h)$ has stable rank one. This affirms a conjecture of Archey, Niu, and Phillips. We also show that the Toms-W…
Let $X$ be a Cantor set, and let $A$ be a unital separable simple amenable $C$*-algebra with tracial rank zero which satisfies the Universal Coefficient Theorem, we use $C(X,A)$ to denote the set of all continuous functions from $X$ to $A$, let $α$ be an automorphism on $C(X,A)$. Suppose that $C(X,A)$ is $α$-simple and $[α]=[\mbox{id}_{1\otimes A}]$ in $KL(1\otimes A,1\otimes A)$, we show that $C(X,A)\rtimes_α\mathbb{Z}$ has tracial rank zero.
We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including: $D$-absorbing C*-algebras, where $D$ is a strongly self-absorbing C*-algebra, stable C*-algebras, C*-algebras with finite …
We construct a sequence of $n-1$ cyclic exact sequences that can be used to compute the $K$-theory of the $C^\star$-algebra crossed product $A \ltimes {\mathbb Z}_n$.
