Excising States of <i>C</i>*-Algebras

Abstract

A net { a α } of positive, norm one elements of a C *-algebra A excises a state f of A if This 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 a n (re iθ ) = ϕ n (r) , where Note that the sets { t ∊ K:a n (t) ∊ 0} form rings about 0 with radii tending to 0. In this sense the sequence { a n } “cuts out” the state f and, in the limit, isolates it from all other states.