Cuntz-Pimnser algebras, completely positive maps and Morita equivalence

Let be a completely positive map on and let be the associated GNS--correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of , , is strongly Morita equivalent to the Cuntz algebra , where is the index of .

     

On orbit equivalence of borel automorphisms

LetE andF be two Borel sets of the countable productZ of the two point space {0,1}. Assume thatE andF are invariant sets for the odometer transformationR and thatE andF are of measure zero with respect to the unique finiteR-invariant measure onZ. We show thatE andF areR-orbit equivalent in a strict sense.     

可测空间的算子扩张及其上的等价关系

讨论可测空间（U，σU))的两种不同的扩张，使其满足任意并（交）的封闭性，并证明二者是等价的，找到了利用可测空间（U，σ（U））的算子的扩张空间（U，σ^*（U))中的上（下）方逼近算子定义的一个等价关系。     

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph $C^{?}$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.     

Complete analytic equivalence relations

We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasi-orders.

     

The von Neumann entropy and unitary equivalence of quantum states

K. He, J. Hou and M. Li have recently given a sufficient and necessary condition for unitary equivalence of quantum states. This condition is based on the von Neumann entropy. In this note, we first give a short proof of their result, and then we improve it.     

Bernoulli actions of sofic groups have completely positive entropy

We prove that every Bernoulli action of a sofic group has completely positive entropy with respect to every sofic approximation net. We also prove that every Bernoulli action of a finitely generated free group has the property that each of its nontrivial factors with a finite generating partition has positive f-invariant.     

On the absence of invariant measures with locally maximal entropy for a class of shifts of finite type

We prove that for a class of shifts of finite type, , any invariant measure which is not a measure of maximal entropy can be perturbed a small amount in the weak* topology to an invariant measure of higher entropy. Namely, there are no invariant measures which are strictly local maxima for the entropy function.

     

Piecewise linear maps, Liapunov exponents and entropy

Let be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA,x∈LA, then the Liapunov exponent λ(x) of fA,x is equal to a measure theoretic entropy hmA,x of fA,x, where mA,x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxxλ(x)=maxxhmA,x=log(λ₁), where λ₁ is the maximal eigenvalue of A.     

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories.     

Derived equivalence of algebras

The derived equivalence and stable equivalence of algebrasR _A ^m and R _B ^m are studied. It is proved, using the tilting complex, thatR _A ^m andR _B ^m are derived-equivalent whenever algebrasA andB are derived-equivalent.     

A converse to Dye's theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the ordering.

     

On equivalence of derived categories

Let A be an Abelian category and B be a thick subcategory of A. Let D ^b(B) denote the derived category of cohomologically bounded chain complexes of objects in A and D _B ^b (A) denote the derived category of cohomologically bounded chain complexes of objects in A with cohomology in B. We give two if and only if conditions for equivalence of D(B) and D _B ^b (A), and we give an example where D ^b (B) and D _B ^b (A) are not equivalent.     

Unitary equivalence to a complex symmetric matrix: An algorithm

We present a necessary and sufficient condition for a 3×3 matrix to be unitarily equivalent to a symmetric matrix with complex entries, and an algorithm whereby an arbitrary 3×3 matrix can be tested. This test generalizes to a necessary and sufficient condition that applies to almost every n×n matrix. The test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix.     

Independence equivalence classes of cycles

The independence equivalence class of a graph G is the set of graphs that have the same independence polynomial as G. Beaton, Brown and Cameron (2019) found the independence equivalence classes of even cycles, and raised the problem of finding the independence equivalence class of odd cycles. The problem is completely solved in this paper.     

Measure equivalence rigidity and bi-exactness of groups

We get three types of results on measurable group theory; direct product groups of Ozawa's class S groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class S groups. As consequences, Monod-Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A?G, B?Γ of non-amenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass-Serre rigidity results on amalgamated free products of non-amenable exact direct product groups.     

Independence equivalence classes of paths

The independence equivalence class of a graph G is the set of graphs that have the same independence polynomial as G. A graph whose independence equivalence class contains only itself, up to isomorphism, is independence unique. Beaton, Brown and Cameron [2] showed that paths with an odd number of vertices are independence unique and raised the problem of finding the independence equivalence class of paths with an even number of vertices. The problem is completely solved in this paper.     

Topological entropy of standard type monotone twist maps

We study invariant measures of families of monotone twist maps with periodic Morse potential . We prove that there exist a constant such that the topological entropy satisfies . In particular, for . We show also that there exist arbitrary large such that has nonuniformly hyperbolic invariant measures with positive metric entropy. For large , the measures are hyperbolic and, for a class of potentials which includes , the Lyapunov exponent of the map with invariant measure grows monotonically with .

     

Automorphic equivalence in the varieties of representations of Lie algebras

Abstract

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.     

Superrigidity and countable Borel equivalence relations

We formulate a Borel version of a corollary of Furman's superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.