Derivations and projections on Jordan triples: an introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis (Q2832868)

This is an expository article partially built on the material presented by the author in a minicourse at `V International Course of Mathematical Analysis in Andalusia' held in Almería in 2011, enlarged with recent results on operator spaces and JB\(^*\)-triple theory. The paper contains an almost encyclopedic panoramic view of the main results on derivations, Jordan derivations, local derivations, cohomology and contractive projections on associative Banach algebras, \(C^*\)-algebras and JB\(^*\)-triples from the perspective of a researcher who has been involved in many of the results surveyed in this paper. A second part of the paper is a challenging discussion on cohomology for associative and Lie Banach algebras with a road map to study what could be expected for the case of Banach triple systems. The last part of the paper is focussed on JB\(^*\)-triples, contractive projections, the Gelfand-Naimark theorem for JB\(^*\)-triples, and three recent contributions studying \(1\)-mixed injective atomic operator spaces, a holomorphic characterization of operator algebras by providing a necessary and sufficient condition (involving only the holomorphic structure of the Banach spaces underlying the operator space) to support a multiplication making it completely isometric and isomorphic to a unital operator algebra, and finally some recent works on enveloping TRO's and \(K\)-theory for JB\(^*\)-triples. The reader interested in a reference guide for these results should not miss this worthy contribution.NEWLINENEWLINEThe paper is divided into three main parts. Part I is devoted to the results on derivations on a wide range of algebras and triple systems in finite and infinite dimensions. The author begins with the studies on derivations on unital abelian \(C^*\)-algebras (that is, on algebras \(C(K)\) of continuous complex-valued functions on a compact Hausdorff space \(K\)), and continues dealing with associative, Jordan and Lie derivations on the algebra \(M_n (\mathbb{C})\) of \(n\times n\) square matrices with entries in the complex numbers, and triple derivations on \(M_{m,n} (\mathbb{C}),\) the space of rectangular \(m\) by \(n\) matrices with complex entries. After a substantial approach to finite-dimensional cases, the author considers derivations on general \(C^*\)-algebras, algebras of unbounded operators (Murray-von Neumann algebras and algebras of measurable operators), JC\(^*\)-algebras, JC\(^*\)-triples and JB\(^*\)-triples.NEWLINENEWLINEWhen dealing with derivations, there are two main aspects to study, one is the automatic continuity of these maps, while the second natural problem is to determine when they can be expressed as inner derivations. These two topics are treated in Section 3 in the different structures mentioned above. For example, beginning from a famous theorem of Sakai which assures the automatic continuity of every derivation on a \(C^*\)-algebra, the paper surveys other generalizations on automatic continuity due to Ringrose; Alaminos, Brešar and Villena; Cuntz; Hejazian and Niknam, together with a contribution by the author and this reviewer showing that every Jordan derivation from a \(C^*\)-algebra \(A\) into an arbitrary Jordan Banach module is continuous. The question whether every derivation on a determined \(C^*\)-algebra is inner or not is deeply illustrated with results by Sakai, Kadison, Johnson and the connections with the celebrated theorems on amenability and weak amenability established by Connes and Haagerup.NEWLINENEWLINEConcerning the wider setting of real and complex JB\(^*\)-triples, the paper introduces the reader to the results on automatic continuity of triple derivations by Barton-Friedman and Ho, Martínez, the author of the article under review and this reviewer. The existence or not of inner and outer derivations on different classes of JB\(^*\)-triples and \(\mathrm{JC}^*\)-algebras is also discussed.NEWLINENEWLINESection 3 concludes with a historical review of the theory of local derivations, from the results by Kadison and Johnson for von Neumann algebras and \(C^*\)-algebras, to the more recent studies on continuous local derivations on JBW\(^*\)-triples by Mackey, and on general JB\(^*\)-triples by Burgos, Fernández-Polo and the reviewer. The results, by Ayupov and Kudaybergenov and Kudaybergenov, Oikhberg, Peralta and Russo, proving the 2-local reflexivity of the sets of derivations and triple derivations on a von Neumann algebra are also surveyed.NEWLINENEWLINEPart II in the paper is called ``Cohomology''. The author motivates the study of cohomology for Jordan algebras in the well established cohomology theory of associative and Lie algebras. The results on cohomology of triple systems are very rare, or are in its first developing steps, or simply are nonexisting. The starting point for the study of cohomology for associative algebras appeared in a paper by Hochschild from 1945, and the subsequent contributions by Weibel. The cohomology for Lie algebras and the results due to Whitehead appeared quite shortly afterwards. The paper is very open at this stage, actually the cohomology theory for associative Banach algebras and Lie algebras is just an excuse of the author to encourage the reader to study continuous cohomology for some Banach triple systems where the known results are not conclusive, Section 6.2 actually offers a wide list of problems and road maps to explore the foundations of this continuous cohomology. The reader is presented with a list of challenging problems to entertain himself or herself for years.NEWLINENEWLINEPart III (Quantum functional analysis) contains an extremely well presented and motivated introduction to one of the main motivations to study JB\(^*\)-triples, that is, their stability under contractive projections. Another motivation is provided by the holomorphic classification of bounded symmetric domains in infinite-dimensional complex Banach spaces, as an alternative for the failure of the Riemann mapping theorem in dimension bigger than or equal to 2. Let us observe that the image of a contractive projection on \(B(H)\), the \(C^*\)-algebra of all bounded linear operators on a complex Hilbert space \(H\), can be isometrically isomorphic to the Hilbert space \(H\), in particular, the image of a contractive projection on a \(C^*\)-algebra need not be a \(C^*\)-algebra. Contractive projections on Banach spaces have been intensively studied in functional analysis, and the so-called contractive projection principle is a milestone result in the theory of JB\(^*\)-triples, and gives the second main motivation to study this class of Banach spaces. This result, independently pursued during the early eighties by Stacho, Kaup, and Friedman and the author, assures that the range of a contractive projection on a JB\(^*\)-triple is linearly isometric in a natural way to another JB\(^*\)-triple. The list of projectively stable sub-categories of Banach spaces given on page 191 is very illustrating.NEWLINENEWLINEThe author's personal point of view can be seen in the narration of the Gelfand-Naimark theorem for JB\(^*\)-triples, which gives a representation theorem for abstract JB\(^*\)-triples as JB\(^*\)-subtriples of direct sums of Cartan factors.NEWLINENEWLINEThere is a substantial interplay between the theory of JB\(^*\)-triples and the operator space theory developed by authors like Ruan, Stinespring, Arveson, Effros, Ozawa, Paulsen, Blecher, Le Merdy, and Helemskii. Part III has three main goals. The first one is devoted to survey a significant result by Neal and the author on the structure of contractively complemented Hilbertian operator spaces. We recall that an operator space \(Z\subseteq B(H)\) is mixed injective if, for every completely bounded linear map \(T\) from an operator space \(X\) to \(Z\) and every operator space \(Y\) containing \(X\), \(T\) admits a bounded extension \(\tilde T\) to \(Y\). In this case, there is a constant \(\lambda\geq 1\) such that \(\|\tilde T\|\leq \lambda\|T\|_{\text{cb}}\) and \(Z\) is actually called \(\lambda\)-mixed injective. The \(1\)-mixed injectives are, in other words, the ranges of contractive projections on \(B(H)\) and include all non-exceptional Cartan factors. An operator space \(X\) is completely semi-isometric to an operator space \(Y\) if there is a completely bounded linear homeomorphism \(T: X\to Y\) satisfying \(\|T\|_{\text{cb}}=\|T^{-1}\|=1\). In order to solve a problem posed by Oikhberg and Rosenthal in 2001, Neal and the author defined a family of Hilbertian operator spaces \(H^k_n\) (\(1\leq k\leq n\)) (generalizing the row and column Hilbert spaces) and prove that a \(1\)-mixed injective atomic subspace \(X\) of \(B(H)\) is isometrically completely contractive to an \(l^\infty\)-sum of non-exceptional Cartan factors and the \(H^k_n\).NEWLINENEWLINEThe second main topic considered in Part III is a another result by Neal and the author on a holomorphic characterization of operator algebras, where they gave a necessary and sufficient condition (involving only the holomorphic structure of the Banach spaces underlying the operator space) to support a multiplication making it completely isometric and isomorphic to a unital operator algebra. In the third main goal of Part III, the author reviews the recent works on enveloping TRO's and \(K\)-theory for JB\(^*\)-triples obtained by Bunce, Feely, Timoney, Bohle and Werner.NEWLINENEWLINEFor the entire collection see [Zbl 1353.46002].