We survey various aspects of the problem of automatic continuity of homomorphisms between Polish groups.

Let X be a locally convex Hausdorff space (briefly, lcHs) and L(X) denote the space of all continuous linear operators of X into itself. The space L(X) is denoted by Ls(X) when it is equipped with the topology of pointwise convergence in X (i.e. the strong operator topology). By a spectral measure in X is meant a σ-additive map P : Σ → Ls(X), defined on a σ-algebra Σ of subsets of some set Ω, w...

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...