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Bicommutant Theorem


The bicommutant theorem is a theorem within the field of functional analysis regarding certain topological properties of function algebras. The theorem says that, given a Hilbert space H, a *-subalgebra A of L(H) which acts nondegenerately is dense in its bicommutant A^('') under the so-called sigma-strong operator topology. Here, L(H) denotes the algebra of bounded operators from H to itself.

The Bicommutant theorem is generally attributed to John von Neumann.

The theorem itself has a number of important corollaries, not the least among which is an equivalence by which one can classify a subalgebra A of L(H) as a von Neumann Algebra.


See also

Bicommutant, Commutant, sigma-Strong Operator Topology, von Neumann Algebra, W-*-Algebra

This entry contributed by Christopher Stover

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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.Royden, H. L. and Fitzpatrick, P. M. Real Analysis. Pearson, 2010.

Cite this as:

Stover, Christopher. "Bicommutant Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BicommutantTheorem.html

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