# Journal of Operator Theory

Volume 68, Issue 1, Summer 2012 pp. 257-274.

Realization of conditionally monotone independence and monotone products of completely positive maps**Authors**: Mihai Popa

**Author institution:**Center for Advanced Studies in Mathematics at Ben Gurion University of the Negev, Department of Mathematics, P.O. B. 653, Be'er Sheva 84105, Israel and Institute of Mathematics, Romanian Academy, P.O. Box 1-764, Bucharest, -014700, Romania

**Summary:**The paper gives an operator algebras model for the conditional monotone independence, introduced by T. Hasebe. The construction is used to prove an embedding result for the N. Muraki's monotone product of $C^*$-algebras. Also, the formulas from the definition of conditional monotone independence are used to define the monotone product of maps which is shown to preserve complete positivity, similarly to the results from the case of free products.

**Keywords:**Completely positive maps, monotone and conditional monotone independence, monotone Fock spaces

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