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 $*-$algebra of closed linear operators affiliated to a finite von Neumann algebra, $\mathfrak{M}$, with faithful, normal tracial state, $τ$. Namely, let $\mathscr{O} _d (r)$, $r \in (0, +\infty]$, denote the rings of formal power series in $d \in \mathbb{N}$ non-commuting variables with "radius of convergence" at least $r$ and let $Ξ\in \mathfrak{M} ^{1\times d}$ be a tuple of row-norm $<r$ with the property that there exist no non-zero trace-class operators $T_1, \cdots, T_d \in \mathscr{B} (L^2 (\mathfrak{M}, τ))$ obeying $\sum _j [T_j , Ξ_j] =0$. Then the unital homomorphism $\mathscr{O} _d (r) \rightarrow \mathfrak{M}$ defined by the evaluation, $\mathfrak{z} _j \mapsto Ξ_j$, extends to an embedding of the universal skew field of fractions of $\mathscr{O} _d (r)$ into the $*-$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, $0 \leq Δ(Ξ) \leq d$, 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 $Ξ\in \mathfrak{M} ^{1\times d}$. As demonstrated by Mai, Speicher and Yin, this $Δ$ measures, in a certain precise sense, the algebraic independence of the components of $Ξ$. Connes-Shlyakhtenko proved that $Δ$ is an upper bound for Voiculescu's non-microstates free entropy dimension, $δ^* (Ξ)$, 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, $Ξ= (Ξ_1,Ξ_2,Ξ_3)$, for which this inequality is strict, $δ^*(Ξ) < Δ(Ξ)$.