Non associative \(C^*\)-algebras revisited (Q2760144)

This is a detailed survey of some recent developments about non-associative \(C^*\)-algebras, which also contains interesting new results concerning multipliers and isometries of non-associative \(C^*\)-algebras.NEWLINENEWLINENEWLINECan associativity be removed in \(C^*\)-algebras? This question was answered in the early 1980s by A. Rodríguez Palacios in two celebrated Theorems (1.3 and 1.4): Norm-unital complete normed complex algebras satisfying the Vidav-Palmer axiom (Gelfand-Naimark axiom, respectively) are nothing but unital non-commutative \(JB^*\)-algebras (unital \(C^*\)-algebras, respectively).NEWLINENEWLINENEWLINEIn recent years, the authors have revisited the theory of non-commutative \(JB^*\)-algebras and alternative \(C^*\)-algebras. Some of these results are reviewed in the survey: (1) The product \(p_A\) of every non-zero alternative \(C^*\)-algebra \(A\) is a vertex of the closed unit ball of the Banach space of all continuous bilinear mappings \(A\times A\to A\). If \(A\) is only assumed to be a non-commutative \(JB^*\)-algebra, then the above vertex property for \(p_A\) can fail. The question whether the vortex property for \(p_A\) characterizes alternative \(C^*\)-algebras among non-commutative \(JB^*\)-algebras remains an open problem. (2) The classification of prime non-commutative \(JB^*\)-algebras, extending the previous one of non-commutative \(JBW^*\)-factors; (3) the holomorphic characterization of non-commutative \(JB^*\)-algebras as those complete normed complex algebras having an approximate unit bounded and such that its open unit ball is a bounded symmetric domain.NEWLINENEWLINENEWLINESections 5 and 6 of the paper are devoted to proving new results. In Section 5 multipliers on non-commutative \(JB^*\)-algebras are introduced. It is proved that the set \(M(A)\) of multipliers of a non-commutative \(JB^*\)-algebra \(A\) becomes a new non-commutative \(JB^*\)-algebra. Indeed, \(M(A)\) is the largest non-commutative \(JB^*\)-algebra containing \(A\) as a closed essential ideal (Theorem 5.6), and \(M(A)\) is an alternative \(C^*\)-algebra if \(A\) is so. Section 6 deals with the non-associative discussion of the Kadison-Paterson-Sinclair theorem on surjective linear isometries between \(C^*\)-algebras.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00063].