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 ?1 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 trace?2,the tracial free product(M1,?1)*(M2,?2)is a type ?1 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 ?2,the tracial free product(M1,?1)*(M2,?2)is a type ?1 factor.