On operator-valued monotone independence (Q2926247)

Operator-valued independence, amalgamated independence and the like, all mean the same thing, namely, conditional (quantum) independence where the state in which subalgebras are (quantum) independent is replaced by a conditional expectation in which the subalgebras are conditionally independent. (This should not be confused with `conditional freeness' à la \textit{M. Bożejko} et al. [Pac. J. Math. 175, No. 2, 357--388 (1996; Zbl 0874.60010)]). The conditional version of monotone independence (introduced by \textit{N. Muraki} [Commun. Math. Phys. 183, No. 3, 557--570 (1997; Zbl 0874.60075)] and \textit{Y. G. Lu} [Probab. Math. Stat. 17, No. 1, 149--166 (1997; Zbl 0880.60065)] and developed very much by Muraki [loc.\,cit.]\ occurred, as noted in [\textit{M. Skeide}, Independence and product systems. In [\textit{M. Skeide}, in: Recent developments in stochastic analysis and related topics. Proceedings of the first Sino-German conference on stochastic analysis (a satellite conference of ICM 2002), Beijing, China, 29 August -- 3 September 2002. River Edge, NJ: World Scientific. 420--438 (2004; Zbl 1080.81034)], in all sorts of conditional noises and has been developed by \textit{M. Popa} [Pac. J. Math. 237, No. 2, 299--325 (2008; Zbl 1152.46054)].NEWLINENEWLINEThe authors present a multivariate moment-cumulant formula generalizing Muraki's work [loc.\,cit.]\ (scalar case) and Popa's work [loc.\,cit.]\ (single variable case).