Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers
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arXiv
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| Natura: | Preprint |
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2024
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| author | Nagisa, Masaru Watatani, Yasuo |
| author_facet | Nagisa, Masaru Watatani, Yasuo |
| contents | We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor $\mathcal{M}$ as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need weighted dimension function $p \mapsto α(τ(p))$ for projections $p \in \mathcal{M}$, which is an analog of a monotone measure. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and Sugeno type respectively. Based on the notion of generalized eigenvalues and singular values, we show that non-linear traces of the Choquet type are closely related to the Lorentz function spaces and the Lorentz operator spaces if the weight functions $α$ are concave. For the algebras of compact operators and factors of type ${\rm II}$, we completely determine the condition that the associated weighted $L^p$-spaces for the non-linear traces become quasi-normed spaces in terms of the weight functions $α$ for any $0 < p < \infty$. We also show that any non-linear trace of the Sugeno type gives a certain metric on the factor. This is an attempt at non-linear and non-commutative integration theory on semifinite factors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18339 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers Nagisa, Masaru Watatani, Yasuo Operator Algebras Functional Analysis Primary 47B10. Secondary 46L51, 47B06 We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor $\mathcal{M}$ as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need weighted dimension function $p \mapsto α(τ(p))$ for projections $p \in \mathcal{M}$, which is an analog of a monotone measure. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and Sugeno type respectively. Based on the notion of generalized eigenvalues and singular values, we show that non-linear traces of the Choquet type are closely related to the Lorentz function spaces and the Lorentz operator spaces if the weight functions $α$ are concave. For the algebras of compact operators and factors of type ${\rm II}$, we completely determine the condition that the associated weighted $L^p$-spaces for the non-linear traces become quasi-normed spaces in terms of the weight functions $α$ for any $0 < p < \infty$. We also show that any non-linear trace of the Sugeno type gives a certain metric on the factor. This is an attempt at non-linear and non-commutative integration theory on semifinite factors. |
| title | Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers |
| topic | Operator Algebras Functional Analysis Primary 47B10. Secondary 46L51, 47B06 |
| url | https://arxiv.org/abs/2404.18339 |