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- W1539546203 abstract "Let $M$ be a semi-finite factor and let $J(M)$ be the set of operators $T$ in $M$ such that $T=ETE$ for some finite projection $E$. In this paper we obtain a representation theorem for unitarily invariant norms on $J(M)$ in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on $J(M)$ coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical result cite{vN} on unitarily invariant norms on $M_n(cc)$. As another application, Ky Fan's dominance theorem cite{Fan} is obtained for semi-finite factors. Some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$ddot{text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are extended to general unitarily invariant norms related to semi-finite factors. We also prove that up to a scale the operator norm is the unique unitarily invariant norm associated to a type ${rm III}$ factor." @default.
- W1539546203 created "2016-06-24" @default.
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- W1539546203 date "2007-07-28" @default.
- W1539546203 modified "2023-09-27" @default.
- W1539546203 title "Unitarily invariant norms related to factors" @default.
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