Derivations and Projections on Jordan Triples. An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis

TL;DR

This paper explores derivations, cohomology, and operator space theory in Jordan triples and related algebraic structures, aiming to deepen understanding of their properties, continuity, and applications in quantum functional analysis.

Contribution

It provides new insights into derivations, cohomology, and the structure of Jordan triples, especially in infinite dimensions and their connections to quantum functional analysis.

Findings

Derivations on Jordan triples are often automatically continuous.

Cohomology theory for Jordan algebras and triples is underdeveloped, especially in infinite dimensions.

Recent work links Jordan theory with operator space theory and quantum algebras.

Abstract

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. 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. One section in Part III is devoted to the subject of contractive projections, which play an important role in the structure theory of Jordan triples.