Skew fields of non-commutative meromorphic functions affiliated to finite operator algebras

Abstract

We embed skew fields of meromorphic functions in several non-commuting variables into the topological $\ast -$algebra of closed linear operators affiliated to a finite von Neumann algebra, $\mathfrak{M}$, with faithful, normal tracial state, $\tau$. Namely, let ${\mathcal{O}}_d(r)$, $r\in (0,+\infty ]$, denote the rings of formal power series in $d\in \mathbb{N}$ non-commuting variables with "radius of convergence" at least $r$ and let $\Xi \in {\mathfrak{M}}^{1\times d}$ be a tuple of row-norm $<r$ with the property that there exist no non-zero trace-class operators $T_1,\cdots ,T_d\in \mathcal{B}(L^2(\mathfrak{M},\tau ))$ obeying ${\sum }_j[T_j,{\Xi }_j]=0$. Then the unital homomorphism ${\mathcal{O}}_d(r)\to \mathfrak{M}$ defined by the evaluation, ${\mathfrak{z}}_j\mapsto {\Xi }_j$, extends to an embedding of the universal skew field of fractions of ${\mathcal{O}}_d(r)$ into the $\ast -$algebra of closed operators affiliated to $\mathfrak{M}$. In their $L^2-$homology theory of von Neumann algebras, Connes and Shlyakhtenko defined a certain quantity, $0\le \Delta (\Xi )\le d$, as the Murray-von Neumann co-dimension of the Hilbert module generated by all $d-$tuples of finite rank operators obeying the above commutator condition with $\Xi \in {\mathfrak{M}}^{1\times d}$. As demonstrated by Mai, Speicher and Yin, this $\Delta$ measures, in a certain precise sense, the algebraic independence of the components of $\Xi$. Connes-Shlyakhtenko proved that $\Delta$ is an upper bound for Voiculescu's non-microstates free entropy dimension, ${\delta }^{\ast }(\Xi )$, and remark that it is of interest to determine whether these two quantities are equal. Using non-commutative function theory, we show that this is generally not the case, by constructing a triple of self-adjoint operators, $\Xi =({\Xi }_1,{\Xi }_2,{\Xi }_3)$, for which this inequality is strict, ${\delta }^{\ast }(\Xi )<\Delta (\Xi )$.