Quantum graphs in infinite-dimensions: Hilbert--Schmidts and Hilbert modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914173802250240 |
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| author | Daws, Matthew |
| author_facet | Daws, Matthew |
| contents | We develop two approaches to Quantum (or Non-commutative) Graphs based on arbitrary von Neumann algebras $M\subseteq\mathcal B(H)$: one looking at operator bimodules of Hilbert--Schmidt (instead of bounded) operators, and the second looking at Quantum Adjacency Operators. Hilbert--Schmidt Quantum Graphs relate to Weaver's picture of Quantum Graphs in a complex way: by defining certain hull operations, we find a bijection between certain subsets of both objects. Given a nfs weight $φ$ on $M$ the operator-valued weight $φ^{-1}$ can be defined, as considered by Wasilewski for direct sums of matrix algebras. We show how to build a natural self-dual Hilbert $C^*$-module from this, which mediates a bijection between HS Quantum Relations and projections $e\in M\bar\otimes M^{\text{op}}$. When $e$ is integrable for the slice-map $\operatorname{id}\otimesφ^{\text{op}}$ there is a related normal CP map $A\colon M\to M$: this is a Quantum Adjacency Operator, which has a Kraus operator representation built from the HS Quantum Relation. When $e$ and its tensor swap map are both integrable, we find certain symmetries of $A$. We illustrate our theory by a careful consideration of certain examples, including detailed links with the finite-dimensional setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_23121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum graphs in infinite-dimensions: Hilbert--Schmidts and Hilbert modules Daws, Matthew Operator Algebras Functional Analysis Quantum Algebra We develop two approaches to Quantum (or Non-commutative) Graphs based on arbitrary von Neumann algebras $M\subseteq\mathcal B(H)$: one looking at operator bimodules of Hilbert--Schmidt (instead of bounded) operators, and the second looking at Quantum Adjacency Operators. Hilbert--Schmidt Quantum Graphs relate to Weaver's picture of Quantum Graphs in a complex way: by defining certain hull operations, we find a bijection between certain subsets of both objects. Given a nfs weight $φ$ on $M$ the operator-valued weight $φ^{-1}$ can be defined, as considered by Wasilewski for direct sums of matrix algebras. We show how to build a natural self-dual Hilbert $C^*$-module from this, which mediates a bijection between HS Quantum Relations and projections $e\in M\bar\otimes M^{\text{op}}$. When $e$ is integrable for the slice-map $\operatorname{id}\otimesφ^{\text{op}}$ there is a related normal CP map $A\colon M\to M$: this is a Quantum Adjacency Operator, which has a Kraus operator representation built from the HS Quantum Relation. When $e$ and its tensor swap map are both integrable, we find certain symmetries of $A$. We illustrate our theory by a careful consideration of certain examples, including detailed links with the finite-dimensional setting. |
| title | Quantum graphs in infinite-dimensions: Hilbert--Schmidts and Hilbert modules |
| topic | Operator Algebras Functional Analysis Quantum Algebra |
| url | https://arxiv.org/abs/2511.23121 |