Document Type : Research Paper

Authors

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

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*}
\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

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