Derivations and Projections on Jordan Triples. An introduction to
nonassociative algebra, continuous cohomology, and quantum functional
analysis
This paper is an elaborated version of the material presented
by the author
in a three hour minicourse at "V International Course of Mathematical
Analysis
in Andalusia," Almeria, Spain, September 12-16, 2011. Part I is devoted to an
exposition of the properties of derivations on various algebras and triple
systems in finite and infinite dimensions, the primary questions addressed
being whether the derivation is automatically continuous and to what
extent it
is an inner derivation. One section in Part I is devoted to the subject of
contractive projections, which play an important role in the structure theory
of Jordan triples and in Part III. Part II discusses cohomology theory of
algebras and triple systems, in both finite and infinite dimensions. Although
the cohomology of associative and Lie algebras is substantially developed, in
both finite and infinite dimensions, the same could not be said for Jordan
algebras. Moreover, the cohomology of triple systems has a rather sparse
literature which is essentially non-existent in infinite dimensions. Thus,
one
of the goals of this paper is to encourage the study of continuous cohomology
of some Banach triple systems. Part III discusses three topics, two very
recent, which involve the interplay between Jordan theory and operator space
theory (quantum functional analysis). The first one, a joint work of the
author, discusses the structure theory of contractively complemented
Hilbertian
operator spaces, and is instrumental to the third topic, which is concerned
with some recent work on enveloping TROs and K-theory for JB*-triples The
second topic presents some very recent joint work by the author concerning
quantum operator algebras.
(A revised version of 72 pages will appear in Proceedings of V |CIDAMA,
Almeria, Spain, September 12-16, 2011. World Scientific (May 2016))
Bernard Russo < brusso@math.uci.edu >