Abstract
In (6) and (7), we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann al- gebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar- valued W ⁄ -probability on a given graph von Neumann algebra. We show that a diagonal graph W ⁄ -probability space (as a scalar-valued W ⁄ - probability space) and a graph W ⁄ -probability space (as an amalgamated W ⁄ -probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compat- ibility, we observe the scalar-valued freeness on a graph von Neumann algebra.
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