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This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This second volume revisits JB*-triples, covers Zel'manov's celebrated work in Jordan theory, proves the unit-free variant of the Vidav–Palmer theorem, and develops the representation theory of alternative C*-algebras and non-commutative JB*-algebras. This completes the work begun in the first volume, which introduced these algebras and discussed the so-called non-associative Gelfand–Naimark and Vidav–Palmer theorems. This book interweaves pure algebra, geometry of normed spaces, and infinite-dimensional complex analysis. Novel proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, and an extensive bibliography.
Introduction In Subsection 4.1.1, we introduce the notion of a Jordan-invertible (in short J-invertible) element of a unital Jordan algebra, as well as that of the J-inverse of such an element. As we notice in §4.1.1, these notions generalize the classical ones of an invertible element and of the inverse of such an element in a unital alternative algebra A (cf. Definition 2.5.23), when A is seen as a Jordan algebra by passing to Asym. By considering the J-spectrum linked in the obvious way to the notion of a J-invertible element, we develop a spectral theory for normed unital Jordan algebras in parallel to that developed in the associative case (cf. Subsection 1.1.1). It is worth emphasizing the Jordan variant of Corollary 1.1.38 given by Theorem 4.1.10, as well as the Gelfand–Beurling formula stated in Theorem 4.1.17. We conclude the subsection by proving a wide non-associative generalization of Rickart's dense-range-homomorphism theorem in Theorem 4.1.19. Indeed, dense-range algebra homomorphisms from complete normed algebras to complete normed Jordan-admissible strongly semisimple algebras are automatically continuous (compare Theorem 3.6.7).
In Subsection 4.1.2, we introduce the notion of a topological J-divisor of zero in a normed Jordan algebra, noticing in Proposition 4.1.23 that this notion generalizes that of a one-sided topological divisor of zero in a normed alternative algebra A (cf. Definition 1.1.87), when A is seen as a normed Jordan algebra by passing to Asym.