Byung-Jay Kahng, Canisius College
Invariant weights on a locally compact quantum
Abstract: Motivated by the purely algebraic notion of ``weak
multiplier Hopf algebras'', we develop the definition of a class of
locally compact quantum groupoids in the C*-algebra framework.
Existence of a certain canonical idempotent element plays an
important role. As in the quantum group case, we require left
and right Haar weights but the antipode is not explicitly
defined. This class would contain all locally compact quantum
groups, and form a self-dual category.
this talk, we will focus on how to formulate the left and right
invariance conditions, similar to but different from the quantum
group case. We will gather some alternative forms of the
invariant conditions. Then we will explore the central roles
these invariant weights play in the quantum groupoid theory, in the
construction of the regular representations (in terms of certain
partial isometries) and the antipode map.
This is based on an on-going joint work with Alfons Van Daele
Brandon Seward, Courant Institute
Positive entropy actions of countable groups factor onto
Abstract: I will show that if a free ergodic action of a
countable group has positive Rokhlin entropy (or, less
generally, positive sofic entropy) then it factors onto all
Bernoulli shifts of lesser or equal entropy. This extends to all
countable groups the well known Sinai factor theorem from
classical entropy theory. A consequence of this theorem is that
every positive-entropy free ergodic action of a non-amenable
group satisfies the measurable von Neumann conjecture.