Ergodic decomposition in the space of unital completely positive maps
| Autor: | Bhattacharya, Angshuman, Kulkarni, Chaitanya J. |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1142/S0219025724500127 |
| Popis: | The classical decomposition theory for states on a C*-algebra that are invariant under a group action has been studied by using the theory of orthogonal measures on the state space \cite{BR1}. In \cite{BK3}, we introduced the notion of \textit{generalized orthogonal measures} on the space of unital completely positive (UCP) maps from a C*-algebra $A$ into $B(H)$. In this article, we consider a group $G$ that acts on a C*-algebra $A$, and the collection of $G$-invariant UCP maps from $A$ into $B(H)$. This article examines a $G$-invariant decomposition of UCP maps by using the theory of generalized orthogonal measures on the space of UCP maps, developed in \cite{BK3}. Further, the set of all $G$-invariant UCP maps is a compact and convex subset of a topological vector space. Hence, by characterizing the extreme points of this set, we complete the picture of barycentric decomposition in the space of $G$-invariant UCP maps. We establish this theory in Stinespring and Paschke dilations of completely positive maps. We end this note by mentioning some examples of UCP maps admitting a decomposition into $G$-invariant UCP maps. Comment: V5, 29 pages, some more results have been added, accepted for publication in Infinite Dimensional Analysis, Quantum Probability and Related Topics |
| Databáze: | arXiv |
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