von Neumann Algebra

Given a Hilbert space H, a *-subalgebra A of B(H) is said to be a von Neumann algebra in H provided that A is equal to its bicommutant A^('') (Dixmier 1981). Here, B(H) denotes the algebra of bounded operators from H to itself.

A non-trivial corollary of the so-called bicommutant theorem says that a nondegenerate *-subalgebra of B(H) 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 A and of B(H) (Blackadar 2013), and due to its bijective equivalence is sometimes used as a definition for von Neumann algebras. In some literature, the assumption of A being unital (i.e., A containing the identity) is added to the hypotheses of this equivalence though, strictly speaking, the result holds in the somewhat more general case that A 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 A which admits a Banach space A_* 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 C^*-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." §430 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. 2: Advanced Theory. New York: Academic Press, 1986.Kadison, R. V. and Ringrose, J. R. Fundamentals of the Theory of Operator Algebras, Vol. 1: Elementary Theory. Providence, RI: Amer. Math. Soc., 1997.Takesaki, M. Theory of Operator Algebras I. Berlin: Springer-Verlag, 2001.

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von Neumann Algebra

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Moslehian, Mohammad Sal; Stover, Christopher; and Weisstein, Eric W. "von Neumann Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/vonNeumannAlgebra.html

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