Although there is a natural embedding of a Banach space $V$ in its second
dual $V^{**},$ the latter is generally much larger than the former.
Nevertheless, as Grothendieck showed forty-five years ago, “locally”
$V^{**}$ resembles $V$ in the sense that one can approximate finite
dimensional subspaces of $V^{**}$ by finite dimensional subspaces of $V$.
This fact is generally referred to as the “local reflexivity” property of
Banach spaces, and has played a fundamental role in that theory.

There is an obvious analogue of the notion of local reflexivity in the
category of operator spaces and completely contractive mappings. Although a
wide category of $C^{*}$-algebras indeed has this property, it is known
that the analogue of this result is false in general. Turning to operator
spaces which are not $C^{*}$-algebras, it seemed only natural to assume
that an “elementary” example such as the operator space $\mathcal{S}_{1}$
of trace class operators must also be local reflexive. This has turned out
to be surprisingly difficult to prove. The affirmative result is due to
Junge, who used novel generalized ultraproduct methods to accomplish this.

In this lecture we will present a simplified proof based on a
characterization of local reflexivity due to Ruan and the speaker, together
with Pisier’s non-commmutative analogue of the Grothendieck-Pietsch
theorem, and a key ultraproduct result of Junge.

This pair of talks will be very expository; I will introduce
(almost) all the necessary background. My aim is to discuss the
construction of M(A), describe various results about its ideal
structure, and then develop — from scratch — just enough
K-theory to be able to sketch how such results can be obtained.
This part could also serve as propaganda and motivation for the
K-theoretically-challenged.