$C^*$-extreme contractive completely positive maps
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arXiv
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| Format: | Preprint |
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2024
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| author | R, Anand O. Sumesh, K. |
| author_facet | R, Anand O. Sumesh, K. |
| contents | In this paper we generalize a specific quantized convexity structure of the generalized state space of a $C^*$-algebra and examine the associated extreme points. We introduce the notion of $P$-$C^*$-convex subsets, where $P$ is any positive operator on a Hilbert space $\mathcal{H}$. These subsets are defined with in the set of all completely positive (CP) maps from a unital $C^*$-algebra $\mathcal{A}$ into the algebra $B(\mathcal{H})$ of bounded linear maps on $\mathcal{H}$. In particular, we focus on certain $P$-$C^*$-convex sets, denoted by $\mathrm{CP}^{(P)}(\mathcal{A},B(\mathcal{H}))$, and analyze their extreme points with respect to this new convexity structure. This generalizes the existing notions of $C^*$-convex subsets and $C^*$-extreme points of unital completely positive maps. We significantly extend many of the known results regarding the $C^*$-extreme points of unital completely positive maps into the context of $P$-$C^*$-convex sets we are considering. This includes abstract characterization and structure of $P$-$C^*$-extreme points. Further, using these studies, we completely characterize the $C^*$-extreme points of the $C^*$-convex set of all contractive completely positive maps from $\mathcal{A}$ into $B(\mathcal{H})$, where $\mathcal{H}$ is finite-dimensional. Additionally, we discuss the connection between $P$-$C^*$-extreme points and linear extreme points of these convex sets, as well as Krein-Milman type theorems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05008 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $C^*$-extreme contractive completely positive maps R, Anand O. Sumesh, K. Operator Algebras Functional Analysis In this paper we generalize a specific quantized convexity structure of the generalized state space of a $C^*$-algebra and examine the associated extreme points. We introduce the notion of $P$-$C^*$-convex subsets, where $P$ is any positive operator on a Hilbert space $\mathcal{H}$. These subsets are defined with in the set of all completely positive (CP) maps from a unital $C^*$-algebra $\mathcal{A}$ into the algebra $B(\mathcal{H})$ of bounded linear maps on $\mathcal{H}$. In particular, we focus on certain $P$-$C^*$-convex sets, denoted by $\mathrm{CP}^{(P)}(\mathcal{A},B(\mathcal{H}))$, and analyze their extreme points with respect to this new convexity structure. This generalizes the existing notions of $C^*$-convex subsets and $C^*$-extreme points of unital completely positive maps. We significantly extend many of the known results regarding the $C^*$-extreme points of unital completely positive maps into the context of $P$-$C^*$-convex sets we are considering. This includes abstract characterization and structure of $P$-$C^*$-extreme points. Further, using these studies, we completely characterize the $C^*$-extreme points of the $C^*$-convex set of all contractive completely positive maps from $\mathcal{A}$ into $B(\mathcal{H})$, where $\mathcal{H}$ is finite-dimensional. Additionally, we discuss the connection between $P$-$C^*$-extreme points and linear extreme points of these convex sets, as well as Krein-Milman type theorems. |
| title | $C^*$-extreme contractive completely positive maps |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2412.05008 |