Isometries of Hilbert \(C^*\)-modules (Q2731960)

It was shown by \textit{E. C. Lance} [Bull. Lond. Math. Soc. 26, No.~4, 363-366 (1994; Zbl 0821.46073)] that the module structure together with the Banach space structure determines the Hilbert \(C^*\)-module structure. The author studies in the present paper, to what extent it is possible to recover the \(C^*\)-module structure from the Banach space structure alone. NEWLINENEWLINENEWLINELet \(X\), \(Y\) be full right Hilbert \(C^*\)-modules over \(C^*\)-algebras \(A\) and \(B\) respectively and let \(T:X\to Y\) be a linear surjective isometry. Denote by \({\mathcal K}(X)\) the \(C^*\)-algebra of `compact' operators on \(X\) and let \({\mathcal L}(X)=\left(\begin{smallmatrix}{\mathcal K}(X)&X\cr X^*&A\end{smallmatrix}\right)\) be the linking \(C^*\)-algebra of \(X\). The main result is that \(T\) can be extended to an isometry of \({\mathcal L}(X)\) onto \({\mathcal L}(Y)\). In this case \(T\) is a sum of two maps: a (bi)module map and a map that reverses the (bi)module action. If \(T\) is a 2-isometry (i.e. if the map \(I\otimes T:M_2\otimes X\to M_2\otimes Y\) is an isomorphism) then \(T\) preserves the \(C^*\)-module structure.