Block quantum dynamical semigroups of completely positive definite kernels
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| Format: | Preprint |
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2024
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| author | Dey, Santanu Saini, Dimple Trivedi, Harsh |
| author_facet | Dey, Santanu Saini, Dimple Trivedi, Harsh |
| contents | Kolmogorov decomposition for a given completely positive definite kernel is a generalization of Paschke's GNS construction for the completely positive map. Using Kolmogorov decomposition, to every quantum dynamical semigroup (QDS) for completely positive definite kernels over a set $S$ on given $C^*$-algebra $\mathcal{A},$ we shall assign an inclusion system $F = (F_s)_{s\ge 0}$ of Hilbert bimodules over $\mathcal{A}$ with a generating unit $ξ^σ=(ξ^σ_s)_{s\ge 0}.$ Consider a von Neumann algebra $\mathcal{B}$, and let $\mathfrak{T}=(\mathfrak{T}_s)_{s\ge 0}$ be a QDS over a set $S$ on the algebra $M_2(\mathcal{B})$ with $\mathfrak{T}_s=\begin{pmatrix}\mathfrak{K}_{s,1} & \mathfrak{L}_s\\\mathfrak{L}_s^*& \mathfrak{K}_{s,2} \end{pmatrix}$ which acts block-wise. Further, suppose that $(F^i_s )_{s\ge 0}$ is the inclusion system affiliated to the diagonal QDS $(\mathfrak{K}_{s,i})_{s\ge 0}$ along with the generating unit $(ξ^σ_{s,i} )_{s\ge 0},$ $σ\in S,i\in \{1,2\}$, then we prove that there exists a unique contractive (weak) morphism $V = (V_s)_{s\ge 0}:F^2_s \to F^1_s$ such that $\mathfrak{L}_s^{σ,σ'}(b)=\langle ξ_{s,1}^σ,V_s bξ_{s,2}^{σ'}\rangle$ for every $σ',σ\in S$ and $b\in \mathcal{B}.$ We also study the semigroup version of a factorization theorem for $\mathfrak{K}$-families. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Block quantum dynamical semigroups of completely positive definite kernels Dey, Santanu Saini, Dimple Trivedi, Harsh Operator Algebras 46L08, 46L57, 81S22 Kolmogorov decomposition for a given completely positive definite kernel is a generalization of Paschke's GNS construction for the completely positive map. Using Kolmogorov decomposition, to every quantum dynamical semigroup (QDS) for completely positive definite kernels over a set $S$ on given $C^*$-algebra $\mathcal{A},$ we shall assign an inclusion system $F = (F_s)_{s\ge 0}$ of Hilbert bimodules over $\mathcal{A}$ with a generating unit $ξ^σ=(ξ^σ_s)_{s\ge 0}.$ Consider a von Neumann algebra $\mathcal{B}$, and let $\mathfrak{T}=(\mathfrak{T}_s)_{s\ge 0}$ be a QDS over a set $S$ on the algebra $M_2(\mathcal{B})$ with $\mathfrak{T}_s=\begin{pmatrix}\mathfrak{K}_{s,1} & \mathfrak{L}_s\\\mathfrak{L}_s^*& \mathfrak{K}_{s,2} \end{pmatrix}$ which acts block-wise. Further, suppose that $(F^i_s )_{s\ge 0}$ is the inclusion system affiliated to the diagonal QDS $(\mathfrak{K}_{s,i})_{s\ge 0}$ along with the generating unit $(ξ^σ_{s,i} )_{s\ge 0},$ $σ\in S,i\in \{1,2\}$, then we prove that there exists a unique contractive (weak) morphism $V = (V_s)_{s\ge 0}:F^2_s \to F^1_s$ such that $\mathfrak{L}_s^{σ,σ'}(b)=\langle ξ_{s,1}^σ,V_s bξ_{s,2}^{σ'}\rangle$ for every $σ',σ\in S$ and $b\in \mathcal{B}.$ We also study the semigroup version of a factorization theorem for $\mathfrak{K}$-families. |
| title | Block quantum dynamical semigroups of completely positive definite kernels |
| topic | Operator Algebras 46L08, 46L57, 81S22 |
| url | https://arxiv.org/abs/2401.16846 |