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| {{Expand German|Dichtheitssatz von Kaplansky|topic=sci|date=November 2012}}
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| In the theory of [[von Neumann algebra]]s, the '''Kaplansky density theorem''' states that if ''A'' is a *-subalgebra of the algebra ''B''(''H'') of bounded operators on a [[Hilbert space]] ''H'', then the strong closure of the unit ball of ''A'' in ''B''(''H'') is the unit ball of the strong closure of ''A'' in ''B''(''H''). This gives a strengthening of the [[von Neumann bicommutant theorem]], showing that an element ''a'' of the double commutant of ''A'', denoted by ''A′′'', can be strongly approximated by elements of ''A'' whose norm is no larger than that of ''a''.
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| The standard proof uses the fact that, when ''f'' is bounded, the [[continuous functional calculus]] ''a'' {{mapsto}} ''f''(''a'') satisfies, for a net {''a<sub>α</sub>''} of [[self adjoint operator]]s
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| :<math>\lim f(a_{\alpha}) = f (\lim a_{\alpha})</math>
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| in the [[strong operator topology]]. This shows that self-adjoint part of the unit ball in ''A′′'' can be approximated strongly by self-adjoint elements in the [[C*-algebra]] generated by ''A''. A matrix computation then removes the self-adjointness restriction and proves the theorem.
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| ==See also==
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| *[[Jacobson density theorem]]
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| ==References==
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| *V.F.R.Jones [http://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf von Neumann algebras]; incomplete notes from a course.
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| *M. Takesaki ''Theory of Operator Algebras I'' ISBN 3-540-42248-X
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| [[Category:Von Neumann algebras]]
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| [[Category:Theorems in functional analysis]]
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Latest revision as of 09:04, 8 October 2014
Friends call him Royal. The factor she adores most is flower arranging and she is attempting to make it a profession. Meter studying is where my primary income comes from but soon I'll be on my own. Kansas is our birth location and my parents live close by.
my blog post; auto warranty