Slides of a talk given to the Functional Analysis seminar in Besancon.
Some of these problems have since been solved.
“e in support” implies convergence
Consider a on a finite quantum group such that where
,
with
. This has a positive density of trace one (with respect to the Haar state
), say
,
where is the Haar element.
An element in a direct sum is positive if and only if both elements are positive. The Haar element is positive and so . Assume that
(if
, then
for all
and we have trivial convergence)
Therefore let
be the density of .
Now we can explicitly write
.
This has stochastic operator
.
Let be an eigenvalue of
of eigenvector
. This yields
and thus
.
Therefore, as is also an eigenvector for
, and
is a stochastic operator (if
is an eigenvector of eigenvalue
, then
, contradiction), we have
.
This means that the eigenvalues of lie in the ball
and thus the only eigenvalue of magnitude one is
, which has (left)-eigenvector the stationary distribution of
, say
.
If is symmetric/reversible in the sense that
, then
is self-adjoint and has a basis of (left)-eigenvectors
and we have, if we write
,
,
which converges to (so that
).
If is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers
converge and thus so do the
Total Variation Decrasing
Uses Simeng Wang’s . Result holds for compact Kac if the state has a density.
Periodic
is concentrated on a coset of a proper normal subgroup of 
is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by
Supported on Subgroup implies Reducible
I have a proof that reducible is equivalent to supported on a pre-subgroup.
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May 1, 2020 at 2:05 pm
Research: What Next? | J.P. McCarthy: Math Page
[…] to the problem of the Ergodic Theorem, and in May I visited Uwe in Besancon, where I gave a talk outlining some problems that I wanted to solve. The main focus was on proving this Ergodic Theorem for Finite Quantum Groups, and thankfully that […]