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An algebraic approach to Wigner’s unitary-antiunitary theorem. (English) Zbl 0943.46033
Let $A= M_d(C)$ be the algebra of all $d\times d$ complex matrices, and let ${\germ H}$ be a left Hilbert $A$-module with generalized inner product $[.,.]:{\germ H}\times{\germ H}\to A$. Using some results from ring theory, the author proves the following generalization of Wigner’s unitary-antiunitary theorem: Suppose there exist $g,h\in{\germ H}$ such that $[g,h]= I$. Let $T:{\germ H}\to{\germ H}$ be a surjective map such that $|[Tf,Tf']|=|[f,f']|$ for all $f,f'\in{\germ H}$. If $d>1$, then there exists an $A$-unitary operator $U:{\germ H}\to{\germ H}$ and a phase function $\varphi:{\germ H}\to C$ (the values of $\varphi$ are of modulus 1) such that $Tf= \varphi(f)Uf$, $f\in{\germ H}$. If $d=1$ then $U$ is either unitary or antiunitary. Also, a Wigner type result for maps on prime $C^*$-algebras and a new proof of the real version of the Wigner result are given.

MSC:
46L08$C^*$-modules
46C05Hilbert and pre-Hilbert spaces: geometry and topology
46C50Generalizations of inner products
16N60Prime and semiprime associative rings
46H25Normed modules and Banach modules, topological modules
47L30Abstract operator algebras on Hilbert spaces
46L07Operator spaces and completely bounded maps