2017/09/30 by Bhat, B. V. Rajarama, Mukherjee, Mithun
#46L08 #46L30 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1710.00180
D. Bures defined a metric β on states of a C^*-algebra and this concept has been generalized to unital completely positive maps ϕ: \mathcal A → \mathcal B, where \mathcal B is either an injective C^*-algebra or a von Neumann algebra. We introduce a new distance γ for the same classes of unital completely positive maps. We use in our definition the distance between representations on the same Hilbert C^*-module in contrast to the Bures metric which uses one representation and distinct vectors. This metric can be expressed in terms of a class of completely positive maps on free products of C^*-algebras and in this setting γ looks like Wasserstein metric on probability measures. Surprisingly, when the range algebra \mathcal B is injective, γ and β are related by the following explicit formula: β2= 2-√(4- γ2) . A deep result of Choi and Li on constrained dilation is the main tool in proving this formula.