Nondegenerate Operator Action

A -algebra of operators on a Hilbert space is said to act nondegenerately if whenever for all , it necessarily implies that . Algebras which act nondegenerately are sometimes said to be nondegenerate.

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

is dense in .

Any -algebra containing the identity operator necessarily acts nondegenerately.