Nondegenerate Operator Action

A nondegenerate operator action is an action of a *-algebra of bounded operators on a Hilbert space such that for all implies that . Here, the -operation is the adjoint operator. Algebras which act nondegenerately are sometimes said to be nondegenerate.

One can show that such an algebra is nondegenerate iff the subspace

is dense in (Blackadar 2017, p. 49).

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