Numerical radius and product of elements in C*-algebras

Abstract

Let v(a) be the numerical radius of an element a in a unital C * -algebra A. In this paper, we show that v(aza * ) = v(z) for all z ∈ A if and only if a * a = 1. We also show that v(az) = v(z) for all z ∈ A if and only if a is a unitary element in the centre Z(A) of A. Furthermore, for two fixed elements a and b in A, we investigate when v(az) = v(zb) for all z ∈ A, and derive a number of consequences. ARTICLE HISTORY

A-numerical radius inequalities for semi-Hilbertian space operators

“…Motivated by theoretical study and applications, there have been many generalizations of the numerical radius (e.g., see [3,6,9,15,17,19,20,26,27,28,29,34]). One of these generalizations is the A-numerical radius of an operator T ∈ B(H) defined by…”

Operator inequalities via the triangle inequality

“…for a certain class of operators. We should remark that finding better bounds for the numerical radius has received a renowned interest in the last few years, as one can see in [2,3,4,6,7,8,10,11,13,14].…”

An extension of the $a$-numerical radius on $C^*$-algebras

“…There are many generalizations of the classical numerical range and numerical radius, and there has been a great deal of interest in their systematic properties and applications; see [3,6,9,10,15,20,25,26,27,32,33,34,35] and the references therein.…”