Abstract:We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra $\mathcal L (X)$ (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on $\mathcal L (X)$ satisfying T• = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and $\mathcal L (X)$ is $w_{op… Show more
There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.
There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.
We generalize the theory of associative unitary normed algebras to the setting of noncommutative Jordan algebras. Special attention is devoted to the case of alternative algebras.
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.
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