Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras

Zhang, Ting; Qi, Xiaofei

doi:10.3390/axioms14020093

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Open AccessArticle

Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras

by

Ting Zhang

^1,† and

Xiaofei Qi

^1,2,*,†

¹

School of Mathematics and Statistics, Shanxi University, Taiyuan 030006, China

²

Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan 030006, China

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Axioms 2025, 14(2), 93; https://doi.org/10.3390/axioms14020093

Submission received: 10 December 2024 / Revised: 22 January 2025 / Accepted: 24 January 2025 / Published: 26 January 2025

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

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Abstract

Let

A

be a self-adjoint standard operator algebra on a real or complex Hilbert space of dimension

\geq 2

, and let

k \in {1, 2, 3}

. The k-skew commutator for

A, B \in A

is defined by

_{*} {[A, B]}_{1} = A B - B A^{*}

and

_{*} {[A, B]}_{k} =_{*} {[A,_{*} {[A, B]}_{k - 1}]}_{1}

. Assume that

Φ : A \to A

is a map whose range contains all rank-one projections. In this paper, we prove that

Φ

is strong k-skew-commutativity preserving, that is,

_{*} {[Φ (A), Φ (B)]}_{k} =_{*} {[A, B]}_{k}

for all

A, B \in A

if and only if one of the following statements holds: (i)

Φ

is either the identity map or the negative identity map whenever

k \in {1, 3}

; (ii)

Φ

is the identity map whenever

k = 2

.

Keywords: bounded linear operators; self-standard operator algebras; k-skew commutators; preservers; Hilbert spaces

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MDPI and ACS Style

Zhang, T.; Qi, X. Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras. Axioms 2025, 14, 93. https://doi.org/10.3390/axioms14020093

AMA Style

Zhang T, Qi X. Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras. Axioms. 2025; 14(2):93. https://doi.org/10.3390/axioms14020093

Chicago/Turabian Style

Zhang, Ting, and Xiaofei Qi. 2025. "Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras" Axioms 14, no. 2: 93. https://doi.org/10.3390/axioms14020093

APA Style

Zhang, T., & Qi, X. (2025). Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras. Axioms, 14(2), 93. https://doi.org/10.3390/axioms14020093

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