Strong Bi-skew Commutativity Preserving Maps on von Neumann Algebras

Let M be a von Neumann algebra with no central summands of type I 1 . Assume that Φ : M → M is a surjective map and Φ ( I ) is an unitary operator. It is shown that Φ is strong bi-skew commutativity preserving (that is, Φ satisfies Φ ( A ) Φ ( B ) ∗ - Φ ( B ) Φ ( A ) ∗ = A B ∗ - B A ∗ for all A , B...

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Bibliographic Details
Published in:Bulletin of the Iranian Mathematical Society 2023-04, Vol.49 (2), Article 15
Main Authors: Qi, Xiaofei, Chen, Shaobo
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Original Paper
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Summary:Let M be a von Neumann algebra with no central summands of type I 1 . Assume that Φ : M → M is a surjective map and Φ ( I ) is an unitary operator. It is shown that Φ is strong bi-skew commutativity preserving (that is, Φ satisfies Φ ( A ) Φ ( B ) ∗ - Φ ( B ) Φ ( A ) ∗ = A B ∗ - B A ∗ for all A , B ∈ M ) if and only if there exists a self-adjoint central operator Z ∈ M with Z 2 = I such that Φ ( A ) = Z A Φ ( I ) for all A ∈ M . The strong bi-skew commutativity preserving maps on prime algebras with involution are also characterized.
ISSN:1017-060X
1735-8515
DOI:10.1007/s41980-023-00759-7