The Noncommutative Hahn-banach Theorems

The Hahn-Banach theorem in its simplest form asserts that a bounded linear functional defined on a subspace of a Banach space can be extended to a linear functional defined everywhere, without increasing its norm. There is an order-theoretic version of this extension theorem (Theorem 0.1 below) that is often more useful in context. The purpose of these lecture notes is to discuss the noncommutative generalizations of these two results and their relation to each other. We make use of several standard terms such as operator space, operator system, n-positive, n-contractive, completely positive, completely contractive, and refer the reader to [Pau02] for definitions. The original proof of the extension theorem for completely positive maps is found in [Arv69]. I will sketch a somewhat simplified proof that is based on the following theorem of M. G. Krein (see page 63 of [Nai70]).