Fundamental group of finite von Neumann algebras with finite dimensional normal trace space

Abstract: We introduce the fundamental group $F(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space. The form of $F(\mathcal{M})$ is completely determined. Moreover, there exists a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space such that $F(\mathcal{M})=G$ for any conceivable groups $G$ as fundamental group.