Mathematics, Department of


Date of this Version



Copyright by THETA, 2017


J. OPERATOR THEORY 78:2(2017), 357–416 doi: 10.7900/jot.2016sep15.2128


We give general structure theory for pairs (C,D) of unital C*- algebras where D is a regular and abelian C*-subalgebra of C.

When D is maximal abelian in C, we prove existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D such that EjD = idD; E is a useful replacement for a conditional expectation when no expectation exists. When E is faithful, (C,D) has numerous desirable properties: e.g. the linear span of the normalizers has a unique minimal C*- norm; D norms C; and isometric isomorphisms of norm-closed subalgebras lying between D and C extend uniquely to their generated C8-algebras.