We introduce a non-commutative generalization of the notion of (approximately proper) equivalence relation and propose the construction of a "quotient space". We then consider certain one-parameter groups of automorphisms of the resulting C*-algebra and prove the existence of KMS states at every temperature. In a model originating from Thermodynamics we prove that these states are unique as well. We also show a relationship between maximizing measures (the analogue of the Aubry-Mather measures for expanding maps) and ground states. In the last section we explore an interesting example of phase transitions.
We study the Radon-Nikodym problem for approximately proper equivalence relations and more specifically the uniqueness of certain Gibbs states. One of our tools is a variant of the dimension group introduced in the study of AF algebras. As applications, we retrieve sufficient conditions for the uniqueness of traces on AF algebras and parts of the Perron-Frobenius-Ruelle theorem.
We introduce a generalization of the notion of approximately proper equivalence relations studied by Renault and with it we build an étale groupoid. Choosing a suitable set of continuous functions to play the role of a potential, we construct a cocycle in that groupoid and discuss the corresponding Radon-Nikodym problem.
Let $A$ be a unital separable amenable \CA and $C$ be a unital \CA with certain infinite property. We show that two full monomorphisms $h_1, h_2: A\to C$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]$ in $KL(A,C).$ Let $B$ be a non-unital but $σ$-unital \CA for which $M(B)/B$ has the certain infinite property. We prove that two full essential extensions are approximately unitarily equivalent if and only if they induce the same element in $KL(A, M(B)/B).$ The set of approximately unitarily equivalence classes of full essential extensions forms a group. If $A$ satisfies the …
Let $A$ be a unital separable \CA and $B=C\otimes {\cal K},$ where $C$ is a unital \CA. Let $τ: A\to M(B)/B$ be a weakly unital full essential extensions of $A$ by $B.$ We show that there is a bijection between a quotient group of $K_0(B)$ onto the set of strong unitary equivalence classes of weakly unital full essential extensions $σ$ such that $[σ]=[τ]$ in $KK^1(A, B).$ Consequently, when this group is zero, unitarily equivalent full essential extensions are strongly unitarily equivalent. When $B$ is a non-unital but $σ$-unital simple \CA with continuous scale, we also study the problem when…
In this paper, we prove a non-commutative version of the Weyl-von Neumann theorem for representations of unital, separable AH algebras into countably decomposable, semifinite, properly infinite, von Neumann factors, where an AH algebra means an approximately homogeneous ${\rm C}^{\ast}$-algebra. We also prove a result for approximate summands of representations of unital, separable AH algebras into finite von Neumann factors.
Given a Bratteli diagram D we consider the compact topological space formed by all infinite paths on D. Two such path are said to be tail-equivalent when they "have the same tail", i.e. when they eventually coincide. This equivalence relation is approximately proper and hence one may consider the C*-algebra associated to it according to a procedure recently introduced by the first named author and A. Lopes. The main result of this work is the proof that this algebra is isomorphic to the AF-algebra associated to the given Bratteli diagram.
Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is a sigma-finite II$_{\infty}$ factor von Neumann algebra, and $π,ρ:\mathcal{A}\rightarrow\mathcal{R}$ are unital $\ast$-homomorphisms such that, for every $a\in\mathcal{A}$, the range projections of $π\left( a\right) $ and $ρ\left( a\right) $ are Murray von Neuman equivalent in $\mathcal{R}% $. We prove that $π$ and $ρ$ are approximately unitarily equivalent modulo $\mathcal{K}_{\mathcal{R}}$, where $\mathcal{K}_{\mathcal{R}}$ is the norm closed ideal generated by the finite projections in $\mathcal{R}$. We also prove …
We prove that if two nonzero homomorphisms from the Cuntz algebra O_infinity to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows that O_infinity is classifiable in the sense of Rordam. In particular, Rordam's classification theorem for direct limits of matrix algebras over even Cuntz algebras extends to direct limits involving both matrix algebras over even Cuntz algebras and corners of O_infinity for which the K_0 group can be an arbitrary countable abelian gr…
We prove that if two homomorphisms from O_{\infty} to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows that O_{\infty} is classifiable in the sense of Rordam. In particular, Rordam's classification theorem for direct limits of matrix algebras over even Cuntz algebras extends to direct limits involving both matrix algebras over even Cuntz algebras and corners of O_{\infty}, for which the K_0-group can be an arbitrary countable abelian group with no even torsion.
We construct ergodic discrete probability measure preserving equivalence relations $\cR$ that has no proper ergodic normal subequivalence relations and no proper ergodic finite-index subequivalence relations. We show that every treeable equivalence relation satisfying a mild ergodicity condition and cost $>1$ surjects onto every countable group with ergodic kernel. Lastly, we provide a simple characterization of normality for subequivalence relations and an algebraic description of the quotient.
We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras. We also investigate the relationship between the Morita equivalence and isomorphism class of the C^{\ast} -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.
Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $ϕ, ψ: C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $ϕ$ and $ψ$ are approximately unitarily equivalent if and only if [ϕ]=[ψ] in KL(C,A) τ\circ ϕ=τ\circ ψfor all tracial states of A and ϕ^‡=ψ^‡, here ϕ^‡ and ψ^‡ are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A) induced by ϕand ψ, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.
Let $A$ be an amenable separable \CA and $B$ be a non-unital but $σ$-unital simple \CA with continuous scale. We show that two essential extensions $τ_1$ and $τ_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$ [τ_1]=[τ_2] {\rm in} KL(A, M(B)/B). $$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A, M(B)/B).$ Using $KL(A, M(B)/B),$ we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be appro…
Let E be an operator algebra on a Hilbert space with finite-dimensional generated C*-algebra. A classification is given of the locally finite algebras and the operator algebras obtained as limits of direct sums of matrix algebras over E with respect to star-extendible homomorphisms. The invariants in the algebraic case consist of an additive semigroup, with scale, which is a right module for the semiring $V_E = Hom_u(E \otimes \sK, E \otimes \sK)$ of unitary equivalence classes of star-extendible homomorphisms. This semigroup is referred to as the dimension module invariant. In the operator al…
We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff II_1 factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non implications were obtained in [Ef73,JS85,Va09,Ki12a,Ki12b]. We complete the picture with the remaining implications and counterexamples.
We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C*-algebras which behave like the proper actions on C*-algebras introduced by Rieffel in 1988. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel's theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different pro…
Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product $C^*$-algebras is discussed. For the Cantor minimal systems, a complete description is given for these relations via $K$-theory and $C^*$-algebras. For example, it is shown that two Cantor minimal systems are approximately $τ$-conjugate if and only if they are orbit equivalent and have the same periodic spectrum. It is also shown that two such systems are approximately $K$-conjugate if and only if the corresponding crossed product $C^*…
In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using $\KK$-theory and prove a result which asserts that there is a unitary of the form `identity + compact' which gives the unitary equivalence of two projections if the `essential codimension' of two projections vanishes for certain $C\sp*$-algebras employing the proper asymptotic unitary equivalence of $\KK$-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of $C(X)\otimes B$ where $X$ is $[0,1]$, $(-\infty, \infty)$, $[0,\infty)$, …
We prove that the solid ergodicity property is stable with respect to taking coinduction for a fairly large class of coinduced action. More precisely, assume that $Σ<Γ$ are countable groups such that $gΣg^{-1}\cap Σ$ is finite for any $g\inΓ\setminusΣ$. Then any measure preserving action $Σ\curvearrowright X_0$ gives rise to a solidly ergodic equivalence relation if and only if the equivalence relation of the associated coinduced action $Γ\curvearrowright X$ is solidly ergodic. We also obtain orbit equivalence rigidity for such actions by showing that the orbit equivalence relation of a rigid …
We present an elementary description of sofic equivalence relations, as well as some permanence properties for soficity. We answer a question by Conley, Kechris and Tucker-Drob of determining soficity in terms of the full group for aperiodic relations.
Suppose that a locally compact group $G$ acts freely and properly on the right of a locally compact space $T$. Rieffel proved that if $α$ is an action of $G$ on a $C^*$-algebra $A$ and there is an equivariant embedding of $C_0(T)$ in $M(A)$, then the action $α$ of $G$ on $A$ is proper, and the crossed product $A\rtimes_{α,r}G$ is Morita equivalent to a generalised fixed-point algebra $\Fix(A,α)$ in $M(A)^α$. We show that the assignment $(A,α)\mapsto\Fix(A,α)$ extends to a functor $\Fix$ on a category of $C^*$-dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel…
We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $Γ,Λ$ in a semisimple Lie group $G$ with finite center and no compact factors we prove that the action $Γ\curvearrowright G/Λ$ is rigid. If in addition $G$ has property (T) then we derive that the von Neumann algebra $L^{\infty}(G/Λ)\rtimesΓ$ has property (T). We also show that if the adjoint action of $G$ on the Lie algebra of $G$ - $\{0\}$ is amenable (e.g. if $G=SL_2(\Bbb R)$), then any ergodic subequivalence relation of the orbit equivalence relation of the action $Γ\curv…
We shall introduce the notions of the strong Morita equivalence for unital inclusions of unital $C^*$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^*$-algebras onto their unital $C^*$-subalgebras. Also, we shall study their basic properties.
We shall introduce the approximate representability and the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and discuss some basic properties of approximately representable coactions and coactions with the Rohlin property of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra. Also, we shall give an example of an approximately representable coaction of a finite dimensional $C^*$-Hopf algebra on a simple unital $C^*$-algebra which has also the Rohlin property and we shall give the 1-cohomology vanishing theorem for coactions of a …
Following Jansen and Waldmann, and Kajiwara and Watatani, we shall introduce notions of coactions of a finite dimensional $C^*$-Hopf algebra on a Hilbert $C^*$-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We shall investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras.
For an approximately central (AC) Powers-Rieffel projection $e$ in the irrational Flip orbifold C*-algebra $A_θ^Φ,$ where $Φ$ is the Flip automorphism of the rotation C*-algebra $A_θ,$ we compute the Connes-Chern character of the cutdown of any projection by $e$ in terms of K-theoretic invariants of these projections. This result is then applied to computing a complete K-theoretic invariant for the projection $e$ with respect to central equivalence (within the orbifold). Thus, in addition to the canonical trace, there is a $4\times6$ K-matrix invariant $K(e)$ arising from unbounded traces of t…
Let A be an approximately subhomogeneous (ASH) C*-algebra with slow dimension growth. We prove that if A is unital and simple, then the Cuntz semigroup of A agrees with that of its tensor product with the Jiang-Su algebra Z. In tandem with a result of W. Winter, this yields the equivalence of Z-stability and slow dimension growth for unital simple ASH algebras. This equivalence has several consequences, including the following classification theorem: unital ASH algebras which are simple, have slow dimension growth, and in which projections separate traces are determined up to isomorphism by th…
We study the saturation properties of several classes of $C^*$-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of $σ$-unital $C^*$-algebras; we extend their results by showing that some coronas of non-$σ$-unital $C^*$-algebras are countably degree-$1$ saturated. We then relate saturation of the abelian $C^*$-algebra $C(X)$, where $X$ is $0$-dimensional, to topological properties of $X$, particularly the saturation of $CL(X)$.
