Excising States of C*-Algebras

Abstract

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.

Pure States on Free Group C*-Algebras

“…Again by density of A in l 2 (F R ), for any a ∈ A and any > 0 there exists b ∈ A such that b 2 = 1 and ab 2 > a − . So α (b) 2 = b 2 = 1 by invariance of the trace, and hence α (a) ≥ α (a)α (b) 2 …”

“…By Theorem 2.4 it suffices to prove this for f . Choose an excising sequence {a n } for f 0 in C * r (F u ), as defined in [2]. Let p = lim a n in C * r (F R ) * * .…”

Consistency of a counterexample to Naimark's problem

“…18 Proof: Note that we are assuming (a ) and (b ) have the same index set. This is easy to arrange by replacing two different index sets with their product.…”

Orthogonal Pure States in Operator Theory

“…This result is sharper than existing results even in case of C * -algebras [1,2,3,4,5,7,8]. However, the main difficulty lies in the Jordan case when we have to develop new methods to overcame the lack of Hilbert-space representations which are ubiquitous in C * -case.…”