$C^*$-isomorphisms, Jordan isomorphisms, and numerical range preserving maps
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Abstract:
Let $\mathbf V = B(H)$ or $S(H)$, where $B(H)$ is the algebra of a bounded linear operator acting on the Hilbert space $H$, and $S(H)$ is the set of self-adjoint operators in $B(H)$. Denote the numerical range of $A \in B(H)$ by $W(A) = \{ (Ax,x): x \in H, (x,x) = 1 \}$. It is shown that a surjective map $\phi : \mathbf V \rightarrow \mathbf V$ satisfies \[ W(AB+BA) = W(\phi (A)\phi (B)+\phi (B)\phi (A)) \qquad \text {for all $A$, $B \in \mathbf {V}$} \] if and only if there is a unitary operator $U \in B(H)$ such that $\phi$ has the form \[ X \mapsto \pm U^*XU \quad \mathrm {or} \quad X \mapsto \pm U^*X^tU, \] where $X^t$ is the transpose of $X$ with respect to a fixed orthonormal basis. In other words, the map $\phi$ or $-\phi$ is a $C^*$-isomorphism on $B(H)$ and a Jordan isomorphism on $S(H)$. Moreover, if $H$ has finite dimension, then the surjective assumption on $\phi$ can be removed.References
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Additional Information
- Hwa-Long Gau
- Affiliation: Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
- Email: hlgau@math.ncu.edu.tw
- Chi-Kwong Li
- Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23185
- MR Author ID: 214513
- Email: ckli@math.wm.edu
- Received by editor(s): May 12, 2006
- Received by editor(s) in revised form: June 1, 2006
- Published electronically: May 8, 2007
- Additional Notes: The research of the first author was supported by the National Science Council of the Republic of China
The research of the second author was supported by a USA NSF grant and an HK RCG grant. - Communicated by: Joseph A. Ball
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 2907-2914
- MSC (2000): Primary 47A12, 47B15, 47B49, 15A60, 15A04, 15A18
- DOI: https://doi.org/10.1090/S0002-9939-07-08807-7
- MathSciNet review: 2317968