$C^\ast$-extreme points of unital completely positive maps invariant under group action
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915747301687296 |
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| author | Kulkarni, Chaitanya J. |
| author_facet | Kulkarni, Chaitanya J. |
| contents | In this work, we study a sub-collection of unital completely positive maps from a unital $C^\ast$-algebra $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$ in the setting of $C^\ast$-convexity. Let $τ$ be an action of a group $G$ on the $C^\ast$-algebra $\mathcal{A}$ through $C^\ast$-automorphisms. We focus our attention to the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, which remain invariant under $τ$. We denote this collection by the notation $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. This collection forms a $C^\ast$-convex set. We characterize the set of $C^\ast$-extreme points of $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^\ast$-convexity for the set $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_15840 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $C^\ast$-extreme points of unital completely positive maps invariant under group action Kulkarni, Chaitanya J. Operator Algebras 46L05, 47L07, 46A55, 46B22, 46L55 In this work, we study a sub-collection of unital completely positive maps from a unital $C^\ast$-algebra $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$ in the setting of $C^\ast$-convexity. Let $τ$ be an action of a group $G$ on the $C^\ast$-algebra $\mathcal{A}$ through $C^\ast$-automorphisms. We focus our attention to the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, which remain invariant under $τ$. We denote this collection by the notation $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. This collection forms a $C^\ast$-convex set. We characterize the set of $C^\ast$-extreme points of $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^\ast$-convexity for the set $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. |
| title | $C^\ast$-extreme points of unital completely positive maps invariant under group action |
| topic | Operator Algebras 46L05, 47L07, 46A55, 46B22, 46L55 |
| url | https://arxiv.org/abs/2601.15840 |