Kuroda's theorem for n$n$‐tuples in semifinite von Neumann algebras
Аннотация
Abstract The classical Kuroda–Bercovici–Voiculescu's theorem states that if is a commuting ‐tuple of self‐adjoint bounded operators on a Hilbert space , and if is a Banach ideal in not contained in the Lorentz‐ ideal , then for every , there exists a commuting ‐tuple of diagonal operators such that for all . In this paper, we obtain an extension of the Kuroda–Bercovici–Voiculescu's theorem to the setting of semifinite von Neumann algebras. Specifically, let be an ‐tuple of commuting self‐adjoint operators affiliated with a semifinite von Neumann algebra , let be a symmetric Banach function space on and let denote the non‐commutative symmetric space of measurable operators affiliated with . We prove that if (where is the Lorentz‐ function space), then for every , there exists a commuting ‐tuple of diagonal operators affiliated with such that for every .