Jordan Structures in Geometry and Analysis

Abstract

Jordan theory has developed rapidly in the last three decades, but very few books describe its diverse applications. Here, the author discusses some recent advances of Jordan theory in differential geometry, complex and functional analysis, with the aid of numerous examples and concise historical notes. These include: the connection between Jordan and Lie theory via the Tits–Kantor–Koecher construction of Lie algebras; a Jordan algebraic approach to infinite dimensional symmetric manifolds including Riemannian symmetric spaces; the one-to-one correspondence between bounded symmetric domains and JB*-triples; and applications of Jordan methods in complex function theory. The basic structures and some functional analytic properties of JB*-triples are also discussed. The book is a convenient reference for experts in complex geometry or functional analysis, as well as an introduction to these areas for beginning researchers. The recent applications of Jordan theory discussed in the book should also appeal to algebraists.

Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

“…For a general overview of JB-algebras, JB * -algebras and JB * -triples the reader is referred to the monographs [36,19,16,17].…”

Monoidal characterisation of groupoids and connectors

“…TROs were first studied in finite dimension by Hestenes [14], and have since been shown to essentially coincide with Hilbert C * -modules [32]. For further related ternary structures in algebra and geometry, see [7].…”

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

“…In this section we will assume familiarity with the notions of real, smooth Banach manifolds, smooth maps between (smooth) Banach manifolds, and Banach-Lie groups. As stated at the end of the introduction, the main references concerning the infinitedimensional differential geometry of Banach manifolds and Banach-Lie groups are [1,17,21,45,55], however, we think it is useful to recall here some notions regarding Banach-Lie subgroups of Banach-Lie groups. According to [55, p. 96 and p. 114], every closed subgroup K of a given Banach-Lie group G is a Banach-Lie group with respect to a unique Hausdorff topology in K such that the closed real subalgebra k = {a ∈ g : exp(ta) ∈ K ∀t ∈ R} (171) of the Lie algebra g of G is the Lie algebra of K. In general, the Hausdorff topology on K does not coincide with the relative topology inherited from the norm topology of G. A subgroup K of a Banach-Lie group G which is also a Banach submanifold of G is called a Banach-Lie subgroup of G. In particular, a Banach-Lie subgroup K of G is closed, it is a Banach-Lie group with respect to the relative topology inherited from the topology of G, and its Lie algebra k is given by equation (171) (see [55, p. 128]).…”