  In this lecture, we present some results on Gaussian (or Rademacher) random
series of trace class operators, mainly due jointly with F. Lust-Piquard. We
will emphasize the probabilistic reformulation of these results, as well as the
open problems suggested by them. We start by a brief survey of what is known
about the problem of characterizing a.s. convergent (Gaussian or Rademacher)
series of random vectors in a Banach space. The main result presented here is
that for certain pairs of Banach spaces $E,F$ that include Hilbert spaces (and
type 2 spaces with the analytic UMD property), we have $$ R(E\hat\otimes F)
=R(E)\hat\otimes F + E\hat\otimes R(F) $$ where $R(E)$ denotes the space of
convergent Rademacher series with coefficients in $E$ and $E\hat\otimes F$
denotes the projective tensor product.

Functional Analysis;Operator Algebras;Probability