Characterizations of commutativity forC*-algebras

Abstract

Let si be a C*-algebra acting on the Hilbert space H and let SP be the self-adjoint elements of si. The following characterization of commutativity is due to I. Kaplansky (see Dixmier [3, p. 58] characterize commutativity for si in terms of the usual order structure on £f. We show that Kaplansky's theorem reduces the proofs of these order characterizations to simple computations.

An order characterization of commutativity for 𝐶*-algebras

“…For the commutativity of C * -algebras, a lot of results have been obtained from various points of view, for example, the nilpotent and ideal characterizations given by I. Kaplansky (see [2] and [7]), the numerical characterizations given by M. J. Crabb, J. Duncan and C. M. McGregor (see [1]) and J. Duncan and P. J. Taylor (see [4]), the order characterizations given by T. Ogasawara (see [11]), S. Sherman (see [12]), M. J. Crabb, J. Duncan and C. M. McGregor (see [1]) and M. Fukamiya, M. Misonou and Z. Takeda (see [5]), * -representation characterizations given by C. F. Skau (see [13]) and S. Wright (see [14]) and spectral characterizations given by R. Nakamoto (see [10]) and Y. Kato (see [8]). …”

Banach space characterizations of unitaries: A survey

“…(Theorem 2.1), as well as some results, both in the theory of Banach algebras [8,13,14,18,43,54,57] and in the one of Banach spaces [4,34,[49][50][51], originated in that characterization. It is worth mentioning that the Bandyopadhyay-Jarosz-Rao paper [4] is motivated by the recent Akemann-Weaver rediscovery [2] of the Bohnenblust-Karlin characterization, and that, in its turn, some results in [4] become rediscoveries of previous ones in [14,51].…”

On commutativity of C*-algebras

“…These also enable us to generalize Theorem 2 and 3 in [1]. Finally, a particular case of our Theorem 5 shows that si is commutative if and only if every T esi satisfies the first order growth condition (Gi).…”