Molnár-dependence in a pre-Hilbert module over an \(H^*\)-algebra (Q1878633)

Let \(H\) be a right module over a proper \(H^*\)-algebra \(A\) with \(\tau(A)\) (the trace class of \(A\)) valued generalized inner product \((.,.)\). It is known that for any \(f,g\in H\) we have the following so-called strong Cauchy-Schwarz inequality \[ (\tau(f,g))^2 \leq \tau (f,f) \tau (g,g) \] where \(\tau\) denotes the trace-norm on \(\tau(A)\). In his paper [Acta Math. Hungar., 69, 301-325 (1995; Zbl 0888.46038)], \textit{B. Zalar} initiated the study of a relation \(\sim\) between the elements of \(H\) defined by \[ f\sim g \Longleftrightarrow (\tau(f,g))^2 =\tau (f,f) \tau (g,g). \] In analogy with the case of classical Hilbert spaces, the relation \(\sim\) can be viewed as a kind of dependence between the elements of \(H\). In his mentioned paper, Zalar posed the problem of describing all the elements \(g\in H\) for which \(g\sim f\) holds for a given \(f\in H\). Moreover, he asked whether the relation \(\sim\) is symmetric. The present author answers both questions completely.