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This paper is an elaborated version of the material presented by the authorin a three hour minicourse at "V International Course of Mathematical Analysisin Andalusia," Almeria, Spain, September 12-16, 2011. Part I is devoted to anexposition of the properties of derivations on various algebras and triplesystems in finite and infinite dimensions, the primary questions addressedbeing whether the derivation is automatically continuous and to what extent itis an inner derivation. Part II discusses cohomology theory of algebras andtriple systems, in both finite and infinite dimensions. Although the cohomologyof associative and Lie algebras is substantially developed, in both finite andinfinite dimensions, the same could not be said for Jordan algebras. Moreover,the cohomology of triple systems has a rather sparse literature which isessentially non-existent in infinite dimensions. Thus, one of the goals of thispaper is to encourage the study of continuous cohomology of some Banach triplesystems. Part III discusses three topics, two very recent, which involve theinterplay between Jordan theory and operator space theory (quantum functionalanalysis). The first one, a joint work of the author, discusses the structuretheory of contractively complemented Hilbertian operator spaces, and isinstrumental to the third topic, which is concerned with some recent work onenveloping TROs and K-theory for JB*-triples. The second topic presents somevery recent joint work by the author concerning quantum operator algebras. Onesection in Part III is devoted to the subject of contractive projections, whichplay an important role in the structure theory of Jordan triples.

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