Let be -algebras and let be a linear map. It is said to be positive if for all , Kadison-Schwarz if for all , and completely positive if for each the map is positive. It is well-known that completely positive maps (in fact, even 2-positive maps) are automatically Kadison-Schwarz and that Kadison-Schwarz maps are positive, and in general none of the reversed implications hold.
In the reviewed paper the authors use elementary techniques to characterise bistochastic (i.e. unital and trace preserving) Kadison-Schwarz maps and use this characterisation together with the results of [Linear Algebra Appl. 347, No. 1–3, 159–187 (2002; Zbl 1032.47046)] to provide explicit examples of bistochastic Kadison-Schwarz maps acting on the 2 by 2 matrices which are not completely positive.