This article is cited in 6 papers

Brief communications

Ideal $F$-norms on $C^*$-algebras

A. M. Bikchentaev

Kazan (Volga Region) Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia

Abstract: We show that every noncompactness measure on a $W^*$-algebra is an ideal $F$-pseudonorm. We establish the criterion of right Fredholm property of an element with respect to $W^*$-algebra. We prove that the element $-I$ realizes maximum distance from the positive element to the subset of all isometries of unital $C^*$-algebra, here $I$ is the unit of $C^*$-algebra. We also consider differences of two finite products of elements from the unit ball of $C^*$-algebra and obtain an estimate of their ideal $F$-pseudonorms. The paper is concluded with the convergence criterion in complete ideal $F$-norm for two series of elements from $W^*$-algebra.

Keywords: $C^*$-algebra, $W^*$-algebra, trace, Hilbert space, linear operator, Fredholm operator, isometry, unitary operator, compact operator, ideal, ideal $F$-norm, measure of noncompactness.

UDC: 517.983+517.986

Received: 13.10.2014

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2015, 59:5, 58–63

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