von Neumann Algebra (original) (raw)
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Given a Hilbert space 
, a 
-subalgebra 
of 
is said to be a von Neumann algebra in 
provided that 
is equal to its bicommutant 
(Dixmier 1981). Here, 
denotes the algebra of bounded operators from 
to itself.
A non-trivial corollary of the so-called bicommutant theorem says that a nondegenerate 
-subalgebra of 
is a von Neumann algebra if and only if it is strongly closed. This is further equivalent to a number of other analytic properties of 
and of 
(Blackadar 2013), and due to its bijective equivalence is sometimes used as a definition for von Neumann algebras. In some literature, the assumption of 
being unital (i.e., 
containing the identity) is added to the hypotheses of this equivalence though, strictly speaking, the result holds in the somewhat more general case that 
is merely nondegenerate.
One can easily show that every von Neumann algebra is a W-*-algebra and contrarily; as a result, some literature defines a von Neumann algebra as a C-*-algebra 
which admits a Banach space 
as a pre-dual. This convention, though not unheard of, is somewhat rare among literature on the topic.
See also
Bicommutant, Bicommutant Theorem, C-*-Algebra, Commutant,Nondegenerate Operator Action, W-*-Algebra
Portions of this entry contributed by Christopher Stover
Portions of this entry contributed by Mohammad Sal Moslehian
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References
Blackadar, B. "Operator Algebras: Theory of 
-Algebras and von Neumann Algebras." 2013. http://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf.Dixmier, J. Von Neumann Algebras. Amsterdam, Netherlands: North-Holland, 1981.Iyanaga, S. and Kawada, Y. (Eds.). "Von Neumann Algebras." ยง430 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1358-1363, 1980.Kadison, R. V. and Ringrose, J. R. Fundamentals of the Theory of Operator Algebras, Vol. 1: Elementary Theory. Providence, RI: Amer. Math. Soc., 1997.Kadison, R. V. and Ringrose, J. R.Fundamentals of the Theory of Operator Algebras, Vol. 2: Advanced Theory. New York: Academic Press, 1986.Takesaki, M. Theory of Operator Algebras I. Berlin: Springer-Verlag, 2001.
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Cite this as:
Moslehian, Mohammad Sal; Stover, Christopher; and Weisstein, Eric W. "von Neumann Algebra." From MathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/vonNeumannAlgebra.html