Lancaster

It has long been expected, and is now proved in many important cases, 
that quantum algebras are more rigid than their classical limits. That is, they 
have much smaller automorphism groups. This begs the question of whether this 
broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative 
projective geometry, from which we see that the correct object to study is a 
groupoid, rather than a group, and maps in this groupoid are the replacement 
for automorphisms. I will illustrate this with the example of quantum 
projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

 An old result of Dixmier, Day and others states that every continuous bounded representation of an amenable group on Hilbert space is similar to a unitary representation. In similar vein, one can ask if amenable subalgebras of $B(H)$ are always similar to self-adjoint subalgebras. This problem was open for many years, but it was recently shown by Farah and Ozawa that in general the answer is negative; their approach goes via showing that the Dixmier--Day result is false when $B(H)$ is replaced by the Calkin algebra.



In this talk, I will give some of the background, and then outline a simplified and more explicit version of their construction; this is taken from joint work with Farah and Ozawa (2014) . It turns out that the key mechanism behind these negative results is the large supply of projections in $\ell_\infty / c_0$, rather than the complicated structure of $B(H)$.

i) Does B(E) always contain a maximal left ideal which is not finitely generated?

ii) Is every finitely-generated maximal left ideal of B(E) necessarily of the form {T\in B(E): Tx = 0}? for some non-zero vector x in E?

Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals mentioned above, a positive answer to the second question would imply a positive answer to the first.

Our main results are:

Question i) has a positive answer for most (possibly all) infinite-dimensional Banach spaces;

Question ii) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains F(E); the answer to Question ii) is positive for many, but not all, Banach spaces. We also make some remarks on a more general conjecture that a unital Banach algebra is finite-dimensional whenever all its maximal left ideals are finitely generated; this is true for C*-algebras.

This is based on a recent paper with H.G. Dales, T. Kochanek, P. Koszmider and N.J. Laustsen (Studia Mathematica, 2013) and work in progress with N.J. Laustsen.

on Banach spaces are characterised by the Hille-Yosida theorem, in

practice it can be difficult to verify that this theorem's hypotheses

are satisfied. In this talk, it will be shown how to construct certain

quantum Markov semigroups (strongly continuous semigroups of

contractions on C* algebras) by realising them as expectation

semigroups of non-commutative Markov processes; the extra structure

possessed by such processes is sufficient to avoid the need to use

Hille and Yosida's result.