Automatic Norm Continuity of Weak Homeomorphisms

Automatic Continuity of Homomorphisms Between Banach algebras and Frechet algebras

Automatic continuity for homeomorphism groups and applications

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property : any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N ⊂ M is a submanifold, the group of homeomorphisms of M that preserve N also has this property. Various applications of automatic con...

On module extension Banach algebras

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. Then ${mathcal{S}}=A oplus X$, the $l^1$-direct sum of $A$ and $X$ becomes a module extension Banach algebra when equipped with the algebra product $(a,x).(a',x')=(aa',ax'+xa').$ In this paper, we investigate biflatness and biprojectivity for these Banach algebras. We also discuss on automatic continuity of derivations on ${mathcal{S...

Full Groups of Minimal Homeomorphisms

We study full groups ofminimal actions of countable groups by homeomorphisms on a Cantor space X, showing that these groups do not admit a compatible Polish group topology and, in the case of Z-actions, are coanalytic non-Borel inside Homeo(X). We then focus on the closure of the full group of a uniquely ergodic homeomorphism, elucidating underwhich conditions this group has a comeager (or, equ...

Automatic Continuity of Homomorphisms and Fixed Points on Metric Compacta

We prove that arbitrary homomorphisms from one of the groups Homeo(2), Homeo(2), Aut(Q, <), Homeo(R) or Homeo(S) into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result of V. G. Pestov, that any action of the discrete group Homeo+(R) by homeomorphisms on a compact...