# Matrix algebras over algebras of unbounded operators

## TL;DR

This paper explores the structure of matrix algebras over unbounded operator algebras affiliated with a II_1 factor, establishing isomorphisms and implications for the Heisenberg-von Neumann puzzle involving operator commutation relations.

## Contribution

It proves an isomorphism between matrix algebras over the Murray-von Neumann algebra and the algebra of matrices over the algebra, extending the identity map, and discusses implications for the Heisenberg-von Neumann puzzle.

## Key findings

- Established isomorphism between M_n(affiliated algebra) and matrix algebra over the algebra.
- Applied rank and determinant identities in the unbounded operator setting.
- Linked operator commutation relations to spectral and invertibility properties.

## Abstract

Let $\mathscr{M}$ be a $II_1$ factor acting on the Hilbert space $\mathscr{H}$, and $\mathscr{M}_{\textrm{aff}}$ 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, $\mathscr{M}_{\textrm{aff}}$ may be identified with the completion of $\mathscr{M}$ in the measure topology. In this article, we show that $M_n(\mathscr{M}_{\textrm{aff}}) \cong M_n(\mathscr{M})_{\textrm{aff}}$ as unital ordered complex topological $*$-algebras with the isomorphism extending the identity mapping of $M_n(\mathscr{M}) \to M_n(\mathscr{M})$. 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, 10 (2014), Paper 009), it follows that if there exist operators $P, Q$ in $\mathscr{M}_{\textrm{aff}}$ satisfying the commutation relation $Q \; \hat \cdot \; P \; \hat - \; P \; \hat \cdot \; Q = {i\mkern1mu} I$, 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 $\mathscr{M}_{\textrm{aff}}$ 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 $\mathscr{M}_{\textrm{aff}}$ in an essential way.