Maps Preserving Zero ∗-Products on ℬ(ℋ)

Abstract

The conventional research topic in operator algebras involves exploring the structure of algebras and using homomorphic mappings to study the classification of algebras. In this study, a new invariant is developed based on the characteristics of the operator using the linear preserving method. The results show that the isomorphic mapping is used for preserving this invariant, which provides the classification information of operator algebra from a new perspective. Let H and K be Hilbert spaces with dimensions greater than two, and let B(H) and B(K) be the set of all bounded linear operators on H and K, respectively. For A,B∈B(H), the ∗, ∗-Lie, and ∗-Jordan products are defined by A∗B, A∗B−B∗A, and A∗B+B∗A, respectively. Let Φ:B(H)→B(K) be an additive unital surjective map. It is confirmed that if Φ preserves zero ∗, ∗-Lie, and ∗-Jordan products, then Φ is unitary or conjugate unitary isomorphisms.