Let A be a \(C^*\)-algebra, f a state on A. This paper studies the notion of excision, viz., a net \(\{a_{\alpha}\}\) of positive, norm one elements in A excises f if \(\lim_{\alpha}\| a_{\alpha}aa_{\alpha}-f(a)a^ 2_{\alpha}\| =0\) for every a in A.
The authors prove that a state f can be excised if and only if it is in the weak\({}^*\)-closure of the pure states on A. A set of states, L, which are \(weak^*\)-limit points of a sequence \(\{f_ n\}\) of pure states satisfying \(f_ n(a_ n)=1\) (where \(\{a_ n\}\) is an orthogonal, positive sequence of norm one elements in a \(C^*\) algebra A with unit) is investigated. It is noted that if A is Abelian, then L is contained in the pure states. This fails if A is not abelian. However, if A is the algebra of all bounded operators on a Hilbert space, and \(\{a_ n\}\) consists of finite rank projections, then L is contained in the pure states. This last result is generalized to the context of the multiplier algebra of a non-unital, \(\sigma\)-unital \(C^*\) algebra A with sequence \(\{a_ n\}\subset A\) “tending to infinity” rapidly enough. In a rather general setting, by invoking the Continuum Hypothesis, the authors show that L has non-void intersection with the pure states. Whether L is contained in the pure states is left unanswered in this general setting.
A technical tool usefully employed by the authors is the notion of an \(\ell^{\infty}\)-embedding of a family \(\{b_{\alpha}\}\) of mutually orthogonal, positive, norm one elements of A, viz., sums of the form \(\sum_{\alpha}b_{\alpha}^{1/2}a_{\alpha}b_{\alpha}^{1/2}\) make sense in A for any bounded family \(\{a_{\alpha}\}\). The paper concludes with results on the pure states of the multiplier algebra and certain maximal abelian subalgebras therein.