Pseudo-multiplicative unitaries associated with inclusions of finite-dimensional \(C^*\)-algebras (Q5957194)

The author constructs a pseudo-multiplicative unitary from an inclusion of finite-dimensional \(C^*\)-algebras \(A_0\subset A_1\). The construction is based on the use of Hilbert modules, and requires the inclusion of \(C^*\)-algebras to satisfy two special conditions. The first condition is of technical nature, while the second is conjectured to be equivalent to the condition that the extension \(A_0\subset A_1\) is of depth 2. If this is the case, then the reviewer believes that the results of the present paper are very closely related to and might have some bearings on the algebraic theory of depth 2 ring extensions developed by L.\ Kadison and collaborators [cf. \textit{L. Kadison}, New Examples of Frobenius Extensions. AMS, Providence-RI (1999; Zbl 0929.16036), \textit{L. Kadison} and \textit{D. Nikshych}, Adv. Math. 163, 258-286 (2001; Zbl 1029.46098), \textit{L. Kadison} and \textit{K. Szlachányi}, Dual bialgebroids for depth two ring extensions (2001), ArXiv: math.RA/0108067]. The construction is illustrated by an example based on \(2\times 2\)-matrices.