Numerical radius preserving operators on C * -algebras

Abstract

Let A be a unital C Ã -algebra. An element u of A is unitary and belongs to the centre of A if and only if jfuj 1 for every pure state f. Using this fact we show that a numerical radius preserving linear isomorphism on A is a C Ã -isomorphism multiplied by a fixed unitary element in the centre of A.

Maps preserving numerical radius distance on C*-algebras

“…It is easy to see that every conjugate C * -isomorphism also preserves numerical radius. Also note that multiplying by a central unitary does not change the numerical radius [2]. So the maps of the form described in the theorem preserve numerical radius distance.…”

“…Since Φ(I) is also an extreme point of B 1 (B, w), it is easily seen from Lemma 2 that Φ(I) is unitary. Thus, (Φ(I)) −1 is a central unitary element and, by [2], Ψ = (Φ(I)) −1 Φ is numerical radius preserving. Therefore, we may assume further that Φ(I) = I , and then prove that there is a central projection P ∈ A with Φ(P ) a central projection in B, a C * -isomorphism Φ 1 : P AP → Φ(P )BΦ(P ) and a conjugate C * -isomorphism Φ 2 : …”

Linear Maps Preserving Numerical Radius on Nest Algebras

“…The subject is related and has applications to many different branches of pure and applied science. In [2,3], Chan proved that if A is a unital C * -algebra then a surjective numerical radius isometry of A is a Jordan isomorphism multiplied by a fixed unitary element in the center of A. This is also true for weakly continuous, surjective numerical radius isometries of atomic nest algebras [4,5,14].…”

Numerical radius and product of elements in C*-algebras

“…The first one, borrowed form [6], characterizes central unitary elements of A in terms of pure states on A. Proof: Let c ∈ Z(A) be a unitary element, and let us show that v(cz) = v(z) for all z ∈ A.…”