The triangle equality in Hilbert $A$-modules

A. V. Kalinichenko^a, M. A. Pliev<sup>b

a</sup> North-Caucasian Institute of Mining and Metallurgy (State Technological University), 44 Nikolaeva str., Vladikavkaz, 362021 Russia
^b Southern Mathematical Institute of the Russian Academy of Sciences, 22 Markusa str., Vladikavkaz, 362027 Russia

Abstract: We show that for any two elements $x$, $y$ of Hilbert $A$-module $M$ over local $C^*$-algebra $A$ the generalized triangle equality $|x+y|=|x|+|y|$ holds if and only if $\langle x,y\rangle=|x||y|$.

Keywords: local $C^{\ast}$-algebra, Hilbert $A$-module, local Hilbert space, module compact operator, $\ast$-homomorphism, triangle equality.

UDC: 517.98: 519.46

Received: 29.08.2018
Revised: 29.08.2018
Accepted: 19.12.2018

DOI: 10.26907/0021-3446-2019-10-38-45

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2019, 63:10, 33–39

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