Matrix algebras over algebras of unbounded operators

Abstract

Let ${{\mathscr {M}}}$ be a $II_1$ factor acting on the Hilbert space ${{\mathscr {H}}}$, and ${\mathscr {M}} _{\text {aff}}$ be the Murray–von Neumann algebra of closed densely-defined operators affiliated with ${{\mathscr {M}}}$. Let $\tau$ denote the unique faithful normal tracial state on $\mathscr {M}$. By virtue of Nelson’s theory of non-commutative integration, ${\mathscr {M}} _{\text {aff}}$ may be identified with the completion of ${{\mathscr {M}}}$ in the measure topology. In this article, we show that $M_n({\mathscr {M}} _{\text {aff}}) \cong M_n({{\mathscr {M}}})_{\text {aff}}$ as unital ordered complex topological $*$-algebras with the isomorphism extending the identity mapping of $M_n({{\mathscr {M}}}) \rightarrow M_n({{\mathscr {M}}})$. Consequently, the algebraic machinery of rank identities and determinant identities are applicable in this setting. As a step further in the Heisenberg–von Neumann puzzle discussed by Kadison–Liu (SIGMA Symmetry Integrability Geom. Methods Appl., 10:Paper 009, 40, 2014), it follows that if there exist operators P, Q in ${\mathscr {M}} _{\text {aff}}$ satisfying the commutation relation $Q \; {{\hat{\cdot }}} \;P \; {\hat{-}} \;P \; {{\hat{\cdot }}} \;Q = {i\,}I$, then at least one of them does not belong to $L^p({{\mathscr {M}}}, \tau )$ for any $0 < p \le \infty$. Furthermore, the respective point spectrums of P and Q must be empty. Hence the puzzle may be recasted in the following equivalent manner - Are there invertible operators P, A in ${{\mathscr {M}}}_{\text {aff}}$ such that $P^{-1} \; {{\hat{\cdot }}} \;A \; {{\hat{\cdot }}} \;P = I \; {\hat{+}} \;A$? This suggests that any strategy towards its resolution must involve the study of conjugacy invariants of operators in ${{\mathscr {M}}}_{\text {aff}}$ in an essential way.