Maps commuting with the λ-Aluthge transform for the semistar Jordan product

Abstract

Let H and K be two complex separable Hilbert spaces, such that and the algebra of bounded linear operators of H on itself. For every , the semistar Jordan product is denoted by and for every , the λ-Aluthge transform of A is denoted by . We show that a bijective map satisfies the following condition for some , if and only if there exists a unitary or anti-unitary operator , such that