A polynomial f in non-commuting variables is trace-positive if the trace of f(A) is positive for all tuples A of symmetric matrices of the same size. The investigation of trace-positive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from stati...

Dedicated to Stephen Smale in recognition of his contributions to topology and dynamical systems Abstract We show that under some natural ergodicity assumptions extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L ∞-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank mod...

By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spinc manifolds. Especially, we get twisted Rokhlin congruences for 8k + 4 dimensional spinc manifolds.

We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért–Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the...

By studying modular invariance properties of some characteristic forms, which are related to elliptic genera, we obtain twisted cancellation formulas for characteristic forms. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spinc manifolds. In particular, we obtain twisted Rokhlin congruences for 8k + 4 dimensional spinc manifolds.

Let (X, Γ) be a free minimal dynamical system, where X is compact separable Hausdorff space and Γ discrete amenable group. It shown that, if has version of Rokhlin property (uniform property) C(X) ⋊ Cuntz comparison on open sets, then the radius crossed product C*-algebra at most half mean topological dimension Γ). These two conditions are to satisfied = ℤ or an extension Cantor system subexpon...

We define a new notion of weak containment for joinings, and we show that this notion implies an inequality between relative Rokhlin entropies. This leads to new upper bounds to Rokhlin entropy. We also use this notion to study how Pinsker algebras behave under direct products, and we study the Rokhlin entropy of restricted actions of finite-index subgroups.

Dedicated to Stephen Smale in recognition of his contributions to topology and dynamical systems Abstract We show that under some natural ergodicity assumptions extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L ∞-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank mod...

We classify actions of finite groups on some class of C*-algebras with the Rokhlin property in terms of the Cuntz semigroup. An obstruction is obtained for the Cuntz semigroup of a C*-algebra allowing such an action. We also classify certain inductive limit actions of finite groups on a class of C*-algebras containing AI-algebras. This classification is done via the equivariant Cuntz semigroup.

Let A be a unital separable C∗-algebra, and D a strongly selfabsorbing C∗-algebra (in the sense of Toms and Winter). We show that if A is D-absorbing, then the crossed product of A by a finite group is D-absorbing as well, assuming the action satisfying a Rokhlin property. With some extra restrictions on A and D, we obtain a similar result for crossed products by Z.