Recovering automorphisms of quantum spaces
Abstract
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).
17:00
Constructing amenable operator algebras
Abstract
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 $B(H)$.
17:00
Maximal left ideals of operators acting on a Banach space
Abstract
We address the following two questions regarding the maximal left ideals of the Banach algebras B(E) of bounded operators acting on an infinite-dimensional Banach space E:
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.
The construction of quantum dynamical semigroups by way of non-commutative Markov processes
Abstract
Although generators of strongly continuous semigroups of contractions
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.
A Laman theorem for non-Euclidean bar-joint frameworks.
Abstract
Laman's theorem characterises the minimally infinitesimally rigid frameworks in Euclidean space $\mathbb{R}^2$ in terms of $(2,3)$-tight graphs. An interesting problem is to determine whether analogous results hold for bar-joint frameworks in other normed linear spaces. In this talk we will show how this can be done for frameworks in the non-Euclidean spaces $(\mathbb{R}^2,\|\cdot\|_q)$ with $1\leq q\leq \infty$, $q\not=2$. This is joint work with Stephen Power.