Ergodic decomposition in the space of unital completely positive maps
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| Format: | Preprint |
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2023
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| author | Bhattacharya, Angshuman Kulkarni, Chaitanya J. |
| author_facet | Bhattacharya, Angshuman Kulkarni, Chaitanya J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_13771 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ergodic decomposition in the space of unital completely positive maps Bhattacharya, Angshuman Kulkarni, Chaitanya J. Operator Algebras Primary (2020) 46A55, 46B22, 47A35, Secondary (2020) 46L45, 47A20, 46L08 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. |
| title | Ergodic decomposition in the space of unital completely positive maps |
| topic | Operator Algebras Primary (2020) 46A55, 46B22, 47A35, Secondary (2020) 46L45, 47A20, 46L08 |
| url | https://arxiv.org/abs/2308.13771 |