Journal of the
Korean Mathematical Society JKMS

In [6] and [7], we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann algebras 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 compatibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

Keywords: graph groupoids, crossed products, graph von Neumann algebras, $w^{*}$-probability spaces

MSC numbers: 05C62, 05C90, 17A50, 18B40, 46K10, 47A67, 47A99, 47B99