$K_ 0$ of certain subdiagonal subalgebras of von Neumann algebras

Abstract:

We show that ${K_0}$ of any finite maximal subdiagonal subalgebra of a separably acting finite von Neumann algebra is isomorphic to ${K_0}$ of the diagonal of the subalgebra. It results that ${K_0}$ of any finite, $\sigma$-weakly closed, maximal triangular subalgebra of a separably acting finite von Neumann algebra is isomorphic to ${K_0}$ of the diagonal of the subalgebra, provided that the diagonal of the subalgebra is a Cartan subalgebra of the von Neumann algebra. In addition, given any separably acting type ${\text {II}_1}$ factor $\mathcal {M}$, we explicitly compute ${K_0}$ of those triangular subalgebras $\mathcal {T}$ of $\mathcal {M}$ that have the property that there exists a UHF subalgebra $\mathcal {A}$ of $\mathcal {M}$ and a standard triangular UHF algebra $\mathcal {S}$ in $\mathcal {A}$ such that $\mathcal {A}$ is $\sigma$-weakly dense in $\mathcal {M}$ and $\mathcal {T}$ is the $\sigma$-weak closure of $\mathcal {S}$.