Linear orthogonality preservers of Hilbert \(C^*\)-modules over \(C^*\)-algebras with real rank zero (Q2845740)

Based on the papers of \textit{D. Ilišević} and \textit{A. Turnšek} [J. Math. Anal. Appl. 341, No. 1, 298--308 (2008; Zbl 1178.46055)] and \textit{C.-W. Leung, C.-K. Ng} and \textit{N.-C. Wong} [J. Aust. Math. Soc. 89, No. 2, 245--254 (2010; Zbl 1242.46068)], the authors give a conjecture as follows:NEWLINENEWLINELet \(A\) be a \(C^*\)-algebra and \(E\) and \(F\) be Hilbert \(A\)-modules with \(E\) being full. If \(\theta: E\to F\) is a \(\mathbb C\)-linear local map (i.e., \(\theta(x)a = 0\) whenever \(xa=0\) for all \(x\in E\), \(a\in A\)), which preserves orthogonality in the sense that for any \(x,y\in E\), \(\langle x,y\rangle= 0\) implies \(\langle \theta(x),\theta(y)\rangle =0\), then there is a central positive element \(u\) in the multiplier algebra \(M(A)\) of \(A\) such that \(\langle \theta (x), \theta(y) \rangle\;=\;u \langle x, y \rangle\) for all \(x,y\in E\).NEWLINENEWLINEThey then affirmatively answer this conjecture in three cases: (i) \(A\) is a \(C^*\)-algebra of real rank zero and \(\theta\) is an \(A\)-module map; (ii) \(A\) is a standard \(C^*\)-algebra; (iii) \(A\) is a \(W^*\)-algebra with no finite type II direct summand.