Lie Mappings and Related Operators
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In this chapter we will investigate how to obtain linear mappings which are compatible with the Lie structure of a c*-algebra from those preserving the associative structure, i.e., Lie derivations from derivations, Lie isomorphisms from isomorphisms. In the case of a boundedly centrally closed C*-algebra, this can simply be achieved through an additive perturbation by a centre- valued trace. Since the Lie structure is closely interrelated with the degree of non-commutativity of a C*-algebra, we need to be able to control the latter, and we shall use polynomial identities to this end (Section 6.1). A crucial result is the Amitsur-Levitzki theorem (6.1.1), which enables us to characterise those C*-algebras satisfying the second Kaplansky polynomial, in Theorem 6.1.7. It comes as no surprise that other operators such as commuting or commutativity preserving ones play a central role, and we shall also obtain representation theorems for these in Sections 6.2 and 6.5, respectively. In order to establish the description of Lie isomorphisms (Theorem 6.5.24) we need to decompose a Jordan isomorphism of a boundedly centrally closed C*- algebra into a sum of a multiplicative and an anti-multiplicative part. This is the topic of Section 6.3, notably Theorem 6.3.4 and Corollary 6.3.6, which extend the classical results of Kadison and Størmer for Jordan-*-isomorphisms. We also obtain Cusack’s theorem stating that every Jordan derivation on a semiprime algebra is a derivation (Theorem 6.3.11).
KeywordsBanach Algebra Prime Ring Polynomial Identity Semiprime Ring Jordan Derivation
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