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Two basic examples of von Neumann algebras are as follows. The ring ''L''∞(R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, which acts by pointwise multiplication on the Hilbert space ''L''2(R) of square integrable functions. The algebra ''B''(''H'') of all bounded operators on a Hilbert space ''H'' is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least 2. von Neumann algebras, under the original name of rings of operators, were first studied by von Neumann in 1929; he and Francis Murray developed the basic theory in a series of papers starting in 1936. DEFINITIONS There are three common ways to define von Neumann algebras.
Von Neumann Double Commutant Theorem says that the first two definitions are equivalent.
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#The Predual Of The Von Neumann Algebra ''B''(''H'') Of Bounded Operators On A Hilbert Space ''H'' Is The Banach Space Of All
| "http://wwwinformationdelightinfo/information/entry/trace_class" class="copylinks">Trace Class operators with the trace norm ''A''= Tr(''A'') The Banach space of trace class operators is itself the dual of the C-algebra of compact operators (which is not a von Neumann algebra) |
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