Journal of Operator Theory

Volume 79, Issue 1, Winter 2018 pp. 201-211.

Dimensions of complex Hilbert spaces are determined by the commutativity relation

Authors: Bojan Kuzma
Author institution:University of Primorska, Koper, SI-6000 Slovenia, \textit{and} IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia

Summary: Let $\mathcal H$ and $\mathcal K$ be complex Hilbert spaces. Assuming the set-theore\-tical axiom on generalized continuum hypothesis it is shown that if the commutativity relation in $\mathscr B(\HH)$ , the algebra of bounded linear operators on $\mathcal H$ , is the same as in $\mathscr B(\mathcal K )$ , then $\dim\mathcal H =\dim\mathcal K$ .

DOI: http://dx.doi.org/10.7900/jot.2017feb13.2169
Keywords: Hilbert space, Banach algebra, commutativity, commuting graph

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