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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2311.16170 |
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| _version_ | 1866910627754147840 |
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| author | Ghosh, Indrajit Nayak, Soumyashant |
| author_facet | Ghosh, Indrajit Nayak, Soumyashant |
| contents | In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let $\mathscr{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$, and let $\mathscr{M}_{\text{aff}}$ denote the set of unbounded operators of the form $T = AB^{\dagger}$ for $A, B \in \mathscr{M}$ with $\ker(B)\subseteq\ker(A)$ , where $(\cdot)^{\dagger}$ denotes the Kaufman inverse. We show that $\mathscr{M}_{\text{aff}}$ is closed under product, sum, Kaufman-inverse and adjoint, and has the structure of a right near-semiring; Moreover, the above quotient representation of an operator in $\mathscr{M}_{\text{aff}}$ is essentially unique. The Murray-von Neumann affiliated operators for $\mathscr{M}$ turn out to be precisely the closed operators in $\mathscr{M}_{\text{aff}}$. Let $Φ$ be a unital normal homomorphism between represented von Neumann algebras $(\mathscr{M}; \mathcal{H})$ and $(\mathscr{N}; \mathcal{K})$. With the help of the quotient representation, we obtain a canonical extension of $Φ$ to a mapping $Φ_{\text{aff}} : \mathscr{M}_{\text{aff}} \to \mathscr{N}_{\text{aff}}$ which respects sum, product, Kaufman-inverse, and adjoint. Thus $\mathscr{M}_{\text{aff}}$ is intrinsically associated with $\mathscr{M}$ and transforms functorially as we change representations of $\mathscr{M}$. Furthermore, $Φ_{\text{aff}}$ preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein-von Neumann extensions of densely-defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via `abstract nonsense'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_16170 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic aspects and functoriality of the set of affiliated operators Ghosh, Indrajit Nayak, Soumyashant Operator Algebras Mathematical Physics Functional Analysis Representation Theory 46L10 (Primary) 47C15, 47L60, 46M15, 47B02 (Secondary) In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let $\mathscr{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$, and let $\mathscr{M}_{\text{aff}}$ denote the set of unbounded operators of the form $T = AB^{\dagger}$ for $A, B \in \mathscr{M}$ with $\ker(B)\subseteq\ker(A)$ , where $(\cdot)^{\dagger}$ denotes the Kaufman inverse. We show that $\mathscr{M}_{\text{aff}}$ is closed under product, sum, Kaufman-inverse and adjoint, and has the structure of a right near-semiring; Moreover, the above quotient representation of an operator in $\mathscr{M}_{\text{aff}}$ is essentially unique. The Murray-von Neumann affiliated operators for $\mathscr{M}$ turn out to be precisely the closed operators in $\mathscr{M}_{\text{aff}}$. Let $Φ$ be a unital normal homomorphism between represented von Neumann algebras $(\mathscr{M}; \mathcal{H})$ and $(\mathscr{N}; \mathcal{K})$. With the help of the quotient representation, we obtain a canonical extension of $Φ$ to a mapping $Φ_{\text{aff}} : \mathscr{M}_{\text{aff}} \to \mathscr{N}_{\text{aff}}$ which respects sum, product, Kaufman-inverse, and adjoint. Thus $\mathscr{M}_{\text{aff}}$ is intrinsically associated with $\mathscr{M}$ and transforms functorially as we change representations of $\mathscr{M}$. Furthermore, $Φ_{\text{aff}}$ preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein-von Neumann extensions of densely-defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via `abstract nonsense'. |
| title | Algebraic aspects and functoriality of the set of affiliated operators |
| topic | Operator Algebras Mathematical Physics Functional Analysis Representation Theory 46L10 (Primary) 47C15, 47L60, 46M15, 47B02 (Secondary) |
| url | https://arxiv.org/abs/2311.16170 |