Dilations of CP-maps commuting according to a graph

Alexander Vernik

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We study dilations of finite tuples of normal, completely positive and completely contractive maps (which we call CP-maps) acting on a von Neumann algebra, and commuting according to a graph G. We show that if G is acyclic, then a tuple commuting according to it has a simultaneous ∗-endomorphic dilation, which also commutes according to G. Conversely, if G has a cycle, we exhibit an example of a tuple of CP-maps commuting according to G, which does not have an ∗-endomorphic dilation to B(K), that also commutes according to G. To obtain these results we use dilation theory of representations of subproduct systems, as introduced and studied by Shalit and Solel. In the course of our investigations we also prove some results about those kinds of subproduct systems which arise from CP-maps commuting according to a graph.

Original languageEnglish
Pages (from-to)1291-1329
Number of pages39
JournalHouston Journal of Mathematics
Volume42
Issue number4
StatePublished - 1 Jan 2016

Keywords

  • Completely positive maps
  • Dilations
  • Product system
  • Subproduct system

ASJC Scopus subject areas

  • Mathematics (all)

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