Nondegenerate Operator Action
A *-algebra A of operators on a Hilbert space H is said to act nondegenerately if whenever Txi=0 for all T in A, it necessarily 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-*-AlgebraThis entry contributed by Christopher Stover
Explore with Wolfram|Alpha
WolframAlpha
More things to try:
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 ActionCite this as:
Stover, Christopher. "Nondegenerate Operator Action." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NondegenerateOperatorAction.html