REFERENCES: COMPELTELY POSITIVE MAPS

Here are some references and telegraphic notes on completely positive maps of operator algebras. No attempt has been made to be thorough. • Vern Paulsen’s book [Pau02] is a general reference for basic material on completely positive and completely bounded maps of operator systems. • Stinespring’s theorem is the most basic result on the structure of com-pletely positive maps. It was first proved in [Sti55], in reply to a question of R. V. Kadison about finding a common generalization of the GNS construc-tion and the dilation theorem of Sz.-Nagy [SNF70], originally proved in 1953, concerning dilations of positive operator measures to spectral measures. It was essentially ignored for several years, was reiterated and somewhat sim-plified in [Arv69], and has now become part of the lore of the subject. • The extension theorem for completely positive maps was first proved in [Arv69], and subsequent proofs have not changed much (e.g., see [Pau02]). Wittstock’s subsequent generalization to completely bounded maps [Wit81] was simplified by Paulsen by essentially reducing it to the extension theorem for completely positive maps, and we will sketch the latter argument from [Pau02]. • The treatment of the Choi-Effros lifting theorem will follow the appendix of [Arv77], after a brief discussion of quasicentral approximate units, which were introduced in the same paper. • Time permitting, we will also discuss a recent noncommutative general-ization of the Perron-Frobenius theorem on positive n×n matrices to opera-tor algebras [Arv03]. A manuscript is available for downloading from eithe