Hanfeng Li

Soficity, amenability, and dynamical entropy

In a previous paper the authors developed an operator-algebraic approach to Lewis Bowens sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure-preserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable.

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C∗-algebra.

Dynamical entropy in Banach spaces

We introduce a version of Voiculescu-Brown approximation entropy for isometric automorphisms of Banach spaces and develop within this framework the connection between dynamics and the local theory of Banach spaces as discovered by Glasner and Weiss. Our fundamental result concerning this contractive approximation entropy, or CA entropy, characterizes the occurrence of positive values both geometrically and topologically. This leads to various applications; for example, we obtain a geometric description of the topological Pinsker factor and show that a C*-algebra is type I if and only if every multiplier inner *-automorphism has zero CA entropy. We also examine the behaviour of CA entropy under various product constructions and determine its value in many examples, including isometric automorphisms of ℓp for 1≤p≤∞ and noncommutative tensor product shifts.

Turbulence, representations, and trace-preserving actions

We establish criteria for turbulence in certain spaces of C -algebra repre- sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space (X; ) and on the hypernite II1 factor R. We also prove that the conjugacy action on the space of free actions of a countably innite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving ows on ( X; ) is generically turbulent.

Combinatorial Independence and Sofic Entropy

We undertake a local analysis of combinatorial independence as it connects to topological entropy within the framework of actions of sofic groups.

θ-Deformations as Compact Quantum Metric Spaces

Abstract.Let M be a compact spin manifold with a smooth action of the n-torus. Connes and Landi constructed θ-deformations Mθ of M, parameterized by n×n real skew-symmetric matrices θ. The Mθ’s together with the canonical Dirac operator (D,) on M are an isospectral deformation of M. The Dirac operator D defines a Lipschitz seminorm on C(Mθ), which defines a metric on the state space of C(Mθ). We show that when M is connected, this metric induces the weak-* topology. This means that Mθ is a compact quantum metric space in the sense of Rieffel.

On embeddings of full amalgamated free product C*–algebras

We examine the question of when the *-homomorphism λ: A * D B → A * D B of full amalgamated free product C*-algebras, arising from compatible inclusions of C*-algebras A C A, B C B and D C D , is an embedding. Results giving sufficient conditions for λ to be injective, as well as classes of examples where A fails to be injective, are obtained. As an application, we give necessary and sufficient conditions for the full amalgamated free product of finite-dimensional C*-algebras to be residually finite dimensional.

Family independence for topological and measurable dynamics

For a family F (a collection of subsets of Z_+), the notion of F-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and a m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent. Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented.

Bernoulli actions and infinite entropy

We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes the computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have a generating countable partition with finite entropy if the base has infinite entropy, which in the amenable case is well known and in the case that the acting group contains the free group on two generators was established by Bowen using a different argument.

A Hilbert C*-module admitting no frames

We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparovs stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules.