A NOTE ON AMALGAMATED BOOLEAN, ORTHOGONAL AND CONDITIONALLY MONOTONE OR ANTI-MONOTONE PRODUCTS OF OPERATOR-VALUED C~*-ALGEBRAIC PROBABILITY SPACES

摘要

We directly prove that the amalgamated Boolean product (in M. Schiirmann, or R. Speicher and R. Worudi's sense, for the scalar-valued case), or the amalgamated orthogonal product (in R. Lenczewski's sense, for the scalar-valued case) of some bimodule maps (in particular, conditional expectations), and the amalgamated conditionally monotone or anti-monotone products (in T. Hasebe's sense for the scalar-valued case; see also, M. Popa's preprint , for an operator-valued case) of pairs of some bimodule maps preserve the (complete) positivity in C~*-algebraic setting. Our statements are formulated in terms of Schwarz maps. The proofs follow the method in, inspired by the scalar case technique due to M. Bozejko, M. Leinert, and R. Speicher from concerning the conditionally free product of states (see also).