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A net {aα} of positive, norm one elements of a C*-algebra A excises a state f of A ifThis notion has been used explicitly by the second author [4, 5, 6] for pure states, but the present paper will explore it more fully. The name is motivated by the following example. Let K be the unit disk in the complex plane, A = C(K) and f(a) = a(0). Define an(reiθ) = ϕn(r), whereNote that the sets {t ∊ K:an(t) ∊ 0} form rings about 0 with radii tending to 0. In this sense the sequence {an} “cuts out” the state f and, in the limit, isolates it from all other states.

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Let AT be the derivation on SBpf) defined by Ar(A')= TX-XT (T,Xe 93(^)). We prove that if T is an isometry or a normal operator, then the range of AT is orthogonal to the null space of Aj,. Also, we prove that if T is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of Ar is not all of S8(^f).