Arveson's Extension Theorem for Conditionally Unital Completely Positive Maps

Conditionally unital completely positive maps are used to characterize generators of unital completely positive semigroups on $C^*$-algebras. In this work, a generalization of this notion is proposed that includes maps between different operator systems. In terms of this generalization, conditionally unital completely positive maps are infinitesimal increments of unital completely positive maps. The basic properties of conditionally unital completely positive maps are studied, the Choi–Jamiołkowski duality is established, and an Arveson-type extension theorem for completely bounded conditionally unital completely positive maps is proved in the case of maps with values in finite-dimensional $C^*$-algebras.