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Nondegenerate Operator Action

A nondegenerate operator action is an action of a *-algebra A of operators on a Hilbert space H such that Txi=0 for all T in A implies that xi=0. Algebras A which act nondegenerately are sometimes said to be nondegenerate.

One can show that such an algebra A is nondegenerate if and only if the subspace

AH=span{Txi:T in A,xi in H}

is dense in H.

Any *-algebra A containing the identity operator I necessarily acts nondegenerately.

See also

Commutant, 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.

Referenced on Wolfram|Alpha

Nondegenerate Operator Action

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Nondegenerate Operator Action." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NondegenerateOperatorAction.html

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