Matrix algebras over algebras of unbounded operators

Abstract

Let $\mathcal{M}$ be a $I{I}_1$ factor acting on the Hilbert space $\mathcal{H}$, and ${\mathcal{M}}_{\text{aff}}$ be the Murray-von Neumann algebra of closed densely-defined operators affiliated with $\mathcal{M}$. Let $\tau$ denote the unique faithful normal tracial state on $\mathcal{M}$. By virtue of Nelson's theory of non-commutative integration, ${\mathcal{M}}_{\text{aff}}$ may be identified with the completion of $\mathcal{M}$ in the measure topology. In this article, we show that $M_n({\mathcal{M}}_{\text{aff}})\cong M_n(\mathcal{M}{)}_{\text{aff}}$ as unital ordered complex topological $\ast$-algebras with the isomorphism extending the identity mapping of $M_n(\mathcal{M})\to M_n(\mathcal{M})$. 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, 10 (2014), Paper 009), it follows that if there exist operators $P,Q$ in ${\mathcal{M}}_{\text{aff}}$ satisfying the commutation relation $Q\hat{\cdot }P\widehat{-}P\hat{\cdot }Q=iI$, then at least one of them does not belong to $L^p(\mathcal{M},\tau )$ 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 ${\mathcal{M}}_{\text{aff}}$ such that $P^{-1}\hat{\cdot }A\hat{\cdot }P=I\widehat{+}A$? This suggests that any strategy towards its resolution must involve the study of conjugacy invariants of operators in ${\mathcal{M}}_{\text{aff}}$ in an essential way.

22 pages, abstract changed and minor corrections, to appear in Banach J. Math. Anal

References in corpus (1)

• The Heisenberg Relation - Mathematical Formulations

Cited by in corpus (1)

• On Murray-von Neumann algebras -- I: Topological, Order-Theoretic and Analytical Aspects