2012/06/04 by Bernard Russo, Russo, Bernard
Mathematics · #17C55 (Secondary) #17C65 #46L57 (Primary) #46L70 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:17C55 #msc:17C65 #msc:46L57 #msc:46L70
paper · pdf · doi:10.48550/arxiv.1206.0694
This survey was updated in 2014, adding material that appeared since the original 63 page version was submitted in 2012. Now 72 pages, to appear in Proceedings of "V International Course of Mathematical Analysis in Andalusia," Almeria, Spain, September 12-16, 2011, to be published by World Scientific (http://www.worldscientific.com/worldscibooks/10.1142/9691)
openalex publication_date 2012/06/04 · arxiv created 2015/12/10 · arxiv updated 2015/12/11 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
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.