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$ C^*$-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Authors: Hwa-Long Gau and Chi-Kwong Li
Journal: Proc. Amer. Math. Soc. 135 (2007), 2907-2914
MSC (2000): Primary 47A12, 47B15, 47B49, 15A60, 15A04, 15A18
Published electronically: May 8, 2007
MathSciNet review: 2317968
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Abstract | References | Similar Articles | Additional Information

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

$\displaystyle W(AB+BA) = W(\phi(A)\phi(B)+\phi(B)\phi(A)) \qquad \hbox{ for all } A, B \in \bV$

if and only if there is a unitary operator $ U \in B(H)$ such that $ \phi$ has the form

$\displaystyle X \mapsto \pm U^*XU \quad \hbox{ or } \quad X \mapsto \pm U^*X^tU,$

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.

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Additional Information

Hwa-Long Gau
Affiliation: Department of Mathematics, National Central University, Chung-Li 32001, Taiwan

Chi-Kwong Li
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23185

Keywords: Numerical range, Jordan product.
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
Article copyright: © Copyright 2007 American Mathematical Society

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