Extensions, Restrictions, and Representations of States on C*-algebras

In the first three sections the question of when a pure state g on a C*-subalgebra B of a C*-algebra A has a unique state extension is studied. It is shown that an extension/is unique if and only if inf||6(o — f(a)\)b\\ = 0 for each a in A, where the inf is taken over those b in B such that 0 < b < 1 and g(b) =* 1. The special cases where B is maximal abelian and/or A — B(H) are treated in more detail. In the remaining sections states of the form ri-> lim^T"*,,, xa), where {■*„}«£« •* a "* °* uTM1 vectors in H and % is an ultrafilter are studied. Introduction. In [10] Kadison and Singer studied the question: if 65 is a maximal abelian subalgebra of $ (%) (the set of bounded linear operators on a separable Hubert space) does each homomorphism of ÍB have a unique state extension to $ (3C)? They showed that if % is isomorphic to L°°(0, 1) then there are homomorphisms of 9> for which distinct state extensions exist. More recently in [13] Reid showed that if $ is a maximal abelian subalgebra of $ (%) which is isomorphic to /°°(N), where N denotes the positive integers, then there are (nontrivial) homomorphisms of satisfying certain requirements, and that if % is also weakly closed then this occurs if and only if & = % + [% +, &]~, where [<& +, &]~ is the norm closure of {BX XB: B G <$>, B > 0 and X G <£}. Received by the editors February 10, 1977. AMS (MOS) subject classifications (1970). Primary 46L05, 46L10, 46L25; Secondary 47A20, 47B47.