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Non-associative normed algebras. Volume 2. Representation theory and the Zel’manov approach. (English) Zbl 1390.17001
Encyclopedia of Mathematics and its Applications 167. Cambridge: Cambridge University Press (ISBN 978-1-107-04311-4/hbk; 978-1-108-67907-7/2-vol.set; 978-1-107-33781-7/ebook). xxvii, 729 p. (2018).
As said in the preface of Volume 1, the dividing line between the two volumes could be drawn between what can be done before and after involving the holomorphic theory of JB*-triples and the structure theory of non-commutative JB$$^*$$-algebras. Volume 1 answered the question on what can be said of a unital $$B^*$$-algebra when the associativity of the product is removed from the axioms, by proving the following two results:
Theorem 3.2.5. Any normed unital complex algebra $$A$$ endowed with a conjugate-linear vector space involution * such that $$1^*= 1$$ and $$\|a^*a\|=\|a\|^2$$ is alternative and * in an algebra involution on $$A$$.
Theorem 3.3.11. Any norm-unital complete normed algebra $$A$$ satisfying $$A = H(A) + iH(A)$$, where $$H(A)$$ denotes the set of all elements $$h \in A$$ such that $$f(h) \in \mathbb R$$ for every continuous linear functional $$f$$ of $$A$$ such that $$f(h) = 1 = \|f\|$$, is a noncommutative JB$$^*$$-algebra. Different unit-free versions of Theorem 3.2.5 are also proved in Volume 1 (see Theorem 3.5.68).
The main goal of the first chapter (Chapter 5) of Volume 2 is to prove what can be seen as a unit-free version of Theorem 3.3.11, namely that noncommutative JB$$^*$$-algebras are precisely those complete normed complex algebras having an approximate identity bounded by one, and whose open unit ball is a homogeneous domain. Some ingredients in the long proof of this result were already established in Volume 1. Among the new ingredients proved in this chapter we can find (1) Kaup’s holomorphic characterization of JB$$^*$$-triples as those complex Banach spaces whose open unit ball is a homogeneous domain (Theorem 5.6.68), and (2) the Barton-Horn-Timoney basic theory of JBW$$^*$$-triples establishing the separate $$w^*$$-continuity of the triple product of a JBW$$^*$$-triple (Theorem 5.7.20) and the uniqueness of the predual (Theorem 5.7.38). The proofs given by the authors of these (and others) results are not always the original ones, although sometimes (as in the case of result (1)) the latter underlie the former. On the other hand, the proof of result (2) is new and, contrarily to what happens in the original one, avoids any Banach space result on uniqueness of preduals. One of the deepest results in the theory of JB-algebras is the fact (proved in the celebrated book of Hanche-Olsen and Stormer as a consequence of the representation theory of JB-algebras) asserting that the closed subalgebra generated by two elements of a JB-algebra is a JC-algebra. This result allowed the authors to develop a basic theory on non-commutative JB$$^*$$-algebras (including Theorem 3.1.11) without any implicit or explicit additional reference to representation theory.
In Chapter 6, the authors conclude the basic theory of non-commutative JB$$^*$$-algebras, developing in depth the representation theory of these algebras, and, in particular, that of alternative $$C^*$$-algebras. Roughly speaking, this theory reduces the study of non-commutative JB$$^*$$-algebras (respectively, alternative $$C^*$$-algebras) to the knowledge of the so-called non-commutative JBW$$^*$$-factors (respectively, alternative $$W^*$$-factors). In particular, the study of alternative $$C^*$$-algebras reduces to that of (associative) $$C^*$$-algebras and the alternative $$C^*$$-algebra of complex octonions. Applying representation theory, and following a Zelmanovian approach, prime JB$$^*$$-algebras are described in Theorem 6.1.57, and prime noncommutative JB$$^*$$-algebras in Theorem 6.2.27.
Chapter 7 deals with the analytic treatment of Zelmanov’s prime theorems for Jordan systems, thus continuing the approach initiated in Section 6.1 for prime JB$$^*$$-algebras. The classical representation theory of JB$$^*$$-triples is revisited, and is applied (together with Zelmanovian techniques) to obtain the description of all prime JB$$^*$$-triples (see Theorem 7.1.41). Other applications of Zelmanov’s prime theorems to the study of normed Jordan algebras and triples are fully surveyed in Section 7.2.
The concluding chapter of the book (Chapter 8) develops some parcels of the theory of nonassociative normed algebras, not previously included in the book, and begins by revisiting one of the favorite topics of the authors in this field, namely the theory of $$H^*$$-algebras (see Section 8.1). Section 8.2 is devoted to show how, as in the case of $$H^*$$-algebras with zero annihilator, the study of certain normed (possibly non-star) algebras can be reduced to the knowledge of their minimal closed ideals (see Theorems 8.2.17 and 8.2.44). Other attractive topics, like the automatic continuity Theorem 8.3.9, the description of Banach Jordan algebras all elements of which have finite spectra (Theorem 8.3.21), and the non-associative study of the joint spectral radius of a bounded set in a normed algebra and of the so-called topologically nilpotent normed algebras (Section 8.4), conclude the book.
Finally, let me express a personal commentary: This book is the fruit of the work of a whole life, under the light and the magic of that city Washington Irving fell in love with.

##### MSC:
 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 17A15 Noncommutative Jordan algebras 17A80 Valued algebras 17C65 Jordan structures on Banach spaces and algebras 46H70 Nonassociative topological algebras 46K70 Nonassociative topological algebras with an involution 46L70 Nonassociative selfadjoint operator algebras
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