We give a characterisation of the factoriality of the groupoid\break von Neumann algebra L(G) associated to a discrete measured groupoid (G,μ). We introduce the notion of groupoids with ``infinite conjugacy classes'' and show that this property together with ergodicity of the groupoid is equivalent with factoriality of L(G).

We define and study the notion of property (T) for Banach algebras, generalizing the one from C∗-algebras. For a second countable locally compact group G and a given family of Banach spaces E, we prove that our Banach algebraic property (TE) of the symmetrized pseudofunction algebras F∗E(G) characterizes the Banach property (TE) of Bader, Furman, Gelander and Monod for groups. In case G is a discrete group and E is the class of Lp-spaces for finite 1⩽p, we also achieve the analogue characterization using the symmetrized p-pseudofunction algebras F∗λp(G).

We present a complete study of rigidity for actions of quantum permutation groups on compact spaces. In particular, we describe explicitly all actions of the quantum permutation groups on classical compact spaces, and show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated with non-crossing partitions.

We continue the search, begun by T. Kato, for all pairs of real and bounded measurable functions {f,g} on the line that result in a positive commutator i[f(P),g(Q)]. Here P and Q are the usual Heisenberg operators: Q is multiplication by the coordinate x in L2(R,dx) and P=−id/dx on the appropriate domains. We prove a number of partial results including a connection with Loewner's celebrated theorem on matrix monotone functions.

We consider weakly closed transitive algebras of operators containing non-zero compact operators in real Banach spaces (Lomonosov algebras). It is shown that they are naturally divided in three classes: the algebras of real, complex and quaternion classes. The properties and characterizations of algebras in each class as well as some useful examples are presented. It is shown that in separable real Hilbert spaces there is a continuum of pairwise non-similar Lomonosov algebras of complex type and of quaternion type.

We obtain a slow exponential growth estimate for the spherical principal series representation rho_s of Lie group Sp(n, 1) at the edge (Re(s)=1) of Cowling's strip (|Re(s)|<1) on the Sobolev space H^alpha(G/P) when alpha is the critical value Q/2=2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy rho_s (s in [0, 1]) of the spherical principal series which is required for the first author's program for proving the Baum--Connes conjecture with coefficients for Sp(n,1).