Banach space characterizations of unitaries: A survey

Abstract

We survey Banach space characterizations of unitary elements of C * -algebras, JB * -triples, and JB-algebras. In the case of the existence of a pre-dual, appropriate specializations of these characterizations are also reviewed.

Dual operator systems

“…The appropriate variant of the above lemma holds also in dual Banach spaces. This follows from results of [1] (see also the survey [20]). At the time of writing the first version of our paper, we were not aware of the reference [1] and we are grateful to Professor Angel Rodrigues Palacios for bringing this fact to our attention.…”

Similarities and differences between real and complex Banach spaces: an overview and recent developments

“…As shown in [218,Theorem 3.1] and [56,Theorem 4.2.24], the conclusion in Theorem 5.23 remains true when the C * -algebra A is replaced by a JB * -triple. However, in the real setting the conclusions are rather different.…”

“…Many results have been derived from the celebrated paper of Kadison [136][42], on surjective linear isometries of C * -algebras; one of them is an implicit Banach space characterization of unitary elements in unital C * -algebras. It is well explained by Rodríguez-Palacios [218] that the mentioned characterization can be deduced from results of Kadison as well as Bohnenblust and Karlin [43], and an explicit statement was included by Akemann and Weaver [3]. (1) u is unitary;…”

Metric characterisation of unitaries in JB$^*$-algebras

“…An appropriate versions of the just commented result in the setting of JB *algebras and JB * -triples was established by A. Rodríguez Palacios in [22] (see section 2 for the missing notions). We recall that a complex (respectively, real) Jordan algebra M is a (not-necessarily associative) algebra over the complex (respectively, real) field whose product is abelian and satisfies (a…”

“…(3) u is a vertex of the closed unit ball of M , (see [22,Theorem 3.1] and [5,Theorem 4.2.24], where the result is proved in the more general setting of JB * -triples).…”