Bijections on the Unit Ball of $B(H)$ Preserving $^{\ast}$-Jordan Triple Product

Hejazian, Shirin; Safarizadeh, Mozhdeh

doi:10.22130/scma.2022.541050.1008

Document Type : Research Paper

Authors

Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159, Mashhad, Iran.

https://doi.org/10.22130/scma.2022.541050.1008

Abstract

Let $ \mathcal{B}_1$ denote the closed unit ball of $\mathcal B(H)$, the von Neumann algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 2$. Suppose that $\phi$ is a bijection on $ \mathcal{B}_1$ (with no linearity assumption) satisfying
\begin{equation*}
\phi(AB^{*}A)=\phi(A)\phi(B)^{*}\phi(A), \quad( A, B\in \mathcal{B}_1).
\end{equation*}
If $I$ and $\mathbb T$ denote the identity operator on $H$ and the unit circle in $\mathbb C$, respectively and if $\phi$ is continuous on $\{\lambda I: \lambda\in \mathbb T\}$, then we show that $\phi(I)$ is a unitary operator and $\phi(I)\phi$ extends to a linear or conjugate linear Jordan $^*$-automorphism on $\mathcal B(H)$. As a consequence, there is either a unitary or an antiunitary operator $U$ on $H$ such that $\phi(A)=\phi(I) UAU^*$, $(A\in {\mathcal B}_1)$ or $ \phi(A)=\phi(I) UA^*U^*$, $(A\in {\mathcal B}_1)$.

Keywords

20.1001.1.23225807.2022.19.3.9.8

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Sahand Communications in Mathematical Analysis

Bijections on the Unit Ball of $B(H)$ Preserving $^{\ast}$-Jordan Triple Product

References

References

Volume 19, Issue 3
September 2022
Pages 125-139

Bijections on the Unit Ball of $B(H)$ Preserving $^{\ast}$-Jordan Triple Product

References

References

Volume 19, Issue 3September 2022Pages 125-139

Volume 19, Issue 3
September 2022
Pages 125-139