An Application Of A Sequence Of Orthogonal Unitaries In The Tracial Free Product Of Finite Von Neumann Algebras

Let M1 be a countably decomposable finite von Neumann algebra with a faithful normal trace τ1 We prove that,if there is a sequence {uk:k ∈ M} of orthogonal unitaries in M1,then for any finite von Neumann algebra M2(≠C)with a faithful normal traceτ2,the tracial free product(M1,τ1)*(M2,τ2)is a type Ⅱ1 factor.As a corollary,we obtain that,if there is a von Neumann subalgebra N of M1 such that N has no minimal projection,then for any finite von Neumann algebra M2(≠C)with a faithful normal trace τ2,the tracial free product(M1,τ1)*(M2,τ2)is a type Ⅱ1 factor.