Maps Preserving Zero ∗-Products on ℬ(ℋ)

Meili Wang, Jing Zhang (), Yipeng Li and Lina Shangguan
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Meili Wang: Department of Applied Mathematics, Xi’an University of Science and Technology, Xi’an 710054, China
Jing Zhang: Modern Industrial Innovation Practice Center, Dongguan Polytechnic College, Dongguan 523808, China
Yipeng Li: Department of Applied Mathematics, Xi’an University of Science and Technology, Xi’an 710054, China
Lina Shangguan: Department of Applied Mathematics, Xi’an University of Science and Technology, Xi’an 710054, China

Mathematics, 2023, vol. 11, issue 20, 1-19

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.

Keywords: Hilbert space; ?-product; unitary isomorphic (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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