Non-commutative weak*-continuous operator extensions

Extension theorems such as the Hahn-Banach Extension Theorem are a central idea of functional analysis. In 1962, Gleason and Whitney proved an extension theorem for weak*-continuous, linear functionals on H^∞ (D) to positive, weak*-continuous functionals on L^∞ (T). Hoffman and Rossi in 1967 provided a related, albeit different, extension theorem: weak*-continuous characters on a unital subalgebra A of L^∞ (T) can be extended to positive functionals on L^∞ (T). They demonstrated that A+A^* being weak*-dense in H^∞ (D) was not necessary for a weak*-continuous character to have a positive, weak*-continuous extension. Arveson is credited with a non-commutative version of the Hahn-Banach Extension Theorem: a completely positive, linear map from a unital, self-adjoint subspace X of a C^-algebra A into B(H), where H is an arbitrary Hilbert space, can be extended to a completely positive map from A into B(H). In this work, we look at the methods Hoffman and Rossi used to develop their extension theorem, and investigate if they can be used in a non-commutative setting. We provide a non-commutative generalization of their Extended Krein-Smulian Theorem, which was a central part of their proof. We also demonstrate that some special subalgebras of the non-commutative space, M_2 (L^∞ (m)) will have weak-continuous functionals on them with weak*-continuous and positive extensions on all of M_2 (L^∞ (m)). Furthermore, we generalize the work of Hoffman and Rossi in a non-commutative fashion to some weak*-continuous characters on singly generated dual algebras.