2013 Fiscal Year Final Research Report
Construction of the notions of independence in non-commutative probability theory
| Project/Area Number |
23654039
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Iwate Prefectural University |
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Principal Investigator |
MURAKI Naofumi 岩手県立大学, 総合政策学部, 教授 (60229979)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 非可換確率論 / 量子確率論 / 自由独立性 / 単調独立性 / q-独立性 / q-キュムラント / 捩れ独立性 / 独立性の分類定理 |
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| Research Abstract |
Notions of independence are important in non-commutative probability theory (= quantum probability theory). We constructed a new notion of independence (= q-independence) which is a one-parameter family of independence interpolating classical independence and Voiculescu's free independence. The construction is based on the q-product operation for a family of non-commutative probability spaces. It is connected with the q-Fock space of Bozejko-Speicher. We constructed q-analogues of central limit theorem, law of small numbers, convolution and cumulants. We also discovered a new notion of independence which we call twisted independence. It arises from the twisted canonical anti-commutation relations of W. Pusz. We also gave a simple proof for the classification theorem for positive natural products (= positive universal independences).
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