On the Quasitrace Problem and a Characterization of W*-algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914239780749312 |
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| author | Gow, Alec |
| author_facet | Gow, Alec |
| contents | We conjecture that a unital C$^*$-algebra is a W$^*$-algebra if and only if each of its maximal abelian self-adjoint subalgebras is a W$^*$-algebra; this is a space-free analogue of a known result due to G.K. Pedersen. Our main result is a proof that this characterization holds for finite C$^*$-algebras if and only if every $2$-quasitrace on a unital C$^*$-algebra is a trace. We also show that the condition in (the spatial version of) Pedersen's Theorem can be substantially weakened in the case of countably decomposable AW$^*$-factors. We conclude with a preliminary result that allows us to relate the question of (quasi)linearity of functionals on AW$^*$-algebras to the question of monotone completeness of AW$^*$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04431 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Quasitrace Problem and a Characterization of W*-algebras Gow, Alec Operator Algebras 46L05, 46L35 We conjecture that a unital C$^*$-algebra is a W$^*$-algebra if and only if each of its maximal abelian self-adjoint subalgebras is a W$^*$-algebra; this is a space-free analogue of a known result due to G.K. Pedersen. Our main result is a proof that this characterization holds for finite C$^*$-algebras if and only if every $2$-quasitrace on a unital C$^*$-algebra is a trace. We also show that the condition in (the spatial version of) Pedersen's Theorem can be substantially weakened in the case of countably decomposable AW$^*$-factors. We conclude with a preliminary result that allows us to relate the question of (quasi)linearity of functionals on AW$^*$-algebras to the question of monotone completeness of AW$^*$-algebras. |
| title | On the Quasitrace Problem and a Characterization of W*-algebras |
| topic | Operator Algebras 46L05, 46L35 |
| url | https://arxiv.org/abs/2601.04431 |