We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection $C\left(G\right)/C_{E^c}\left(G\right)\hookrightarrow L{²}_E\left(G\right)$ is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.

We prove that certain maximal ideals in Beurling algebras on the unit disc have approximate identities, and show the existence of functions with certain properties in these maximal ideals. We then use these results to prove that if T is a bounded operator on a Banach space X satisfying $\parallel T^n\parallel =O\left(n^{\beta }\right)$ as n → ∞ for some β ≥ 0, then ${\sum }_{n=1}^{\infty }\parallel {(1-T)}^{n}x\parallel /\parallel {(1-T)}^{n-1}x\parallel$ diverges for every x ∈ X such that ${(1-T)}^{\left[\beta \right]+1}x\ne 0$.

We construct the following: a perfect non Dirichlet set every proper closed subset of which is Kronecker, A weak Kronecker set which is not an $R$ set; an independent countable Dirichlet set which is not Kronecker; a collection of $q$-disjoint Kronecker sets whose union is independent but Helson $1/q$; A countable collection of disjoint Kronecker sets whose union is closed and independent but not Helson: a perfect independent Dirichlet set which is not Helson.

Let $H_n$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_n$. So $L^2\left(H_n\right)$ has a natural structure of G-module. We obtain a decomposition of $L^2\left(H_n\right)$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^2\left(H_n\right)$ where $L={\sum }_{j=1}^p(X_j^2+Y_j^2)-{\sum }_{j=p+1}^n(X_j^2+Y_j^2)$, and where $X_1,Y_1,...,X_n,Y_n,T$ denotes the standard basis of the Lie algebra of $H_n$. Finally, we obtain a spectral characterization of the...