Murray-von Neumann dimension for strictly semifinite weights
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| Format: | Preprint |
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2024
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| _version_ | 1866910889011052544 |
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| author | Guinto, Aldo Garcia Lorentz, Matthew Nelson, Brent |
| author_facet | Guinto, Aldo Garcia Lorentz, Matthew Nelson, Brent |
| contents | Given a von Neumann algebra $M$ equipped with a faithful normal strictly semifinite weight $φ$, we develop a notion of Murray-von Neumann dimension over $(M,φ)$ that is defined for modules over the basic construction associated to the inclusion $M^φ\subset M$. For $φ=τ$ a faithful normal tracial state, this recovers the usual Murray-von Neumann dimension for finite von Neumann algebras. If $M$ is either a type $\mathrm{III}_λ$ factor with $0<λ<1$ or a full type $\mathrm{III}_1$ factor with $\text{Sd}(M)\neq \mathbb{R}$, then amongst extremal almost periodic weights the dimension function depends on $φ$ only up to scaling. As an application, we show that if an inclusion of diffuse factors with separable preduals $N\subset M$ is with expectation $\mathcal{E}$ and admits a compatible extremal almost periodic state $φ$, then this dimension quantity bounds the index $\text{Ind}{\mathcal{E}}$, and in fact equals it when the modular operators $Δ_φ$ and $Δ_{φ|_N}$ have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_15725 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Murray-von Neumann dimension for strictly semifinite weights Guinto, Aldo Garcia Lorentz, Matthew Nelson, Brent Operator Algebras 46L10, 46L37 Given a von Neumann algebra $M$ equipped with a faithful normal strictly semifinite weight $φ$, we develop a notion of Murray-von Neumann dimension over $(M,φ)$ that is defined for modules over the basic construction associated to the inclusion $M^φ\subset M$. For $φ=τ$ a faithful normal tracial state, this recovers the usual Murray-von Neumann dimension for finite von Neumann algebras. If $M$ is either a type $\mathrm{III}_λ$ factor with $0<λ<1$ or a full type $\mathrm{III}_1$ factor with $\text{Sd}(M)\neq \mathbb{R}$, then amongst extremal almost periodic weights the dimension function depends on $φ$ only up to scaling. As an application, we show that if an inclusion of diffuse factors with separable preduals $N\subset M$ is with expectation $\mathcal{E}$ and admits a compatible extremal almost periodic state $φ$, then this dimension quantity bounds the index $\text{Ind}{\mathcal{E}}$, and in fact equals it when the modular operators $Δ_φ$ and $Δ_{φ|_N}$ have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases. |
| title | Murray-von Neumann dimension for strictly semifinite weights |
| topic | Operator Algebras 46L10, 46L37 |
| url | https://arxiv.org/abs/2405.15725 |