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- W2904283506 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." @default.
- W2904283506 created "2018-12-22" @default.
- W2904283506 creator A5018559313 @default.
- W2904283506 date "2020-02-04" @default.
- W2904283506 modified "2023-09-24" @default.
- W2904283506 title "Matrix algebras over algebras of unbounded operators" @default.
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- W2904283506 doi "https://doi.org/10.1007/s43037-019-00052-y" @default.
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