正交酉元列在有限von Neumann代数的迹自由积中的应用

令M_1为一个有限的von Neumann代数,τ_1为其上的一个忠实正规迹态.我们将证明,如果M_1中存在一列两两正交的酉元列{u_k:k∈N},则对任意具有忠实正规迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子.作为推论可以得出,如果M_1有一个von Neumann子代数N不包含最小投影,则对任意具有忠实迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子.

An Application of a Sequence of Orthogonal Unitaries in the Tracial Free Product of Finite von Neumann Algebras

Let M₁ be a finite von Neumann algebra with a faithful normal trace τ₁ and let M₁^o={a ∈ M,τ₁(a)=0}. We prove that, if there is a sequence {u_k:k ∈ M } of orthogonal unitaries in M₁^o, 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. 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.