What are the possible operator algebra products which a given operator space
can be equipped with? This is the question which I had been bearing since
GPOTS~2002. In late 2002, V.~I.~Paulsen defined quasi-multipliers for operator
spaces, and suggested to me to study them as a part of study of multipliers for
operator spaces which was initiated by D.~P~.Blecher, and developed by himself
and V.~I.~Paulsen and some others including me. Then, accidentally I discovered
that quasi-multipliers can be used to characterize operator algebra products.
That is, the possible operator algebra products which a given operator space
can be equipped with are precisely the bilinear mappings that are implemented
by the contractive quasi-multipliers of the operator space. These facts are
presented in the joint paper [Kaneda-Paulsen 2003, to appear in Journal of
Functional Analysis]. Moreover, I found that there is a beautiful geometrical
characterization of operator algebra products using the Haagerup tensor
product. That is, operator algebra products are described only in terms of
matrix norms and completely contractive mappings. The last result is elegant
enough to obtain a generalization of Blecher-Ruan-Sinclair Theorem as a simple
corollary, and also can be considered as the ``quasi'' version of
Blecher-Effros-Zarikian's tau trick theorem in which they characterized
one-sided multipliers only in terms of matrix norms and completely contractive
mappings. In my characterization, using the Haagerup tensor product is
essential, and this fact reminds us that the Haagerup tensor product is a very
meaningful concept in study of operator spaces. All these results were
presented in GPOTS~2003. After my talk, G.~K.~Pedersen asked a good question:
What are the extreme points of the unit ball of a quasi-multiplier space? This
gave me a further direction to study quasi-multipliers, and I have been
studying this topic with ideas inspired by a famous characterization of the
extreme points of $C^*$-algebras by R.~V.~Kadison. In this GPOTS~2004, I will
present the best answer I have so far to Pedersen's question.