Jordan isomorphisms and maps preserving spectra of certain operator products

Jinchuan Hou¹ ; Chi-Kwong Li² ; Ngai-Ching Wong³

¹ Department of Mathematics Taiyuan University of Technology Taiyuan 030024, P.R. of China
² Department of Mathematics The College of William & Mary Williamsburg, VA 13185, U.S.A.
³ Department of Applied Mathematics National Sun Yat-sen University and National Center for Theoretical Sciences Kaohsiung 80424, Taiwan and Department of Mathematics The Chinese University of Hong Kong Hong Kong

Studia Mathematica, Tome 184 (2008) no. 1, pp. 31-47

Résumé

Let $\mathcal{A}_1, \mathcal{A}_2$ be (not necessarily unital or closed) standard operator algebras on locally convex spaces $X_1, X_2$, respectively. For $k \ge 2$, consider different products $T_1* \cdots *T_k$ on elements in ${\cal A}_i$, which covers the usual product $T_1* \cdots *T_k = T_1\cdots T_k$ and the Jordan triple product $T_1*T_2 = T_2T_1T_2$. Let ${\mit\Phi} :\mathcal{A}_1\rightarrow\mathcal{A}_2$ be a (not necessarily linear) map satisfying $\sigma({\mit\Phi}(A_1)*\cdots *{\mit\Phi}(A_k)) =\sigma(A_1*\cdots *A_k)$ whenever any one of $A_i$'s has rank at most one. It is shown that if the range of ${\mit\Phi}$ contains all rank one and rank two operators then ${\mit\Phi}$ must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.

Détail

DOI : 10.4064/sm184-1-2

Keywords: mathcal mathcal necessarily unital closed standard operator algebras locally convex spaces respectively consider different products * cdots *t elements cal which covers usual product * cdots *t cdots jordan triple product *t mit phi mathcal rightarrow mathcal necessarily linear map satisfying sigma mit phi * cdots * mit phi sigma * cdots *a whenever has rank shown range mit phi contains rank rank operators mit phi jordan isomorphism multiplied root unity similar results self adjoint operators acting hilbert spaces obtained