Construction of the notions of independence in non-commutative probability theory

• Project/Area Number: 23654039
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Iwate Prefectural University
• Principal Investigator: MURAKI Naofumi 岩手県立大学, 総合政策学部, 教授 (60229979)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• 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 / 普遍積 / 自然積 / q-畳み込み / ブール独立性 / 証明の単純化 / 独立性概念 / q-変形フォック空間 / 普遍計算規則 / q-変形ブラウン運動 / q-変形正準交換関係

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).

Research Products

• [Journal Article] A simple proof of the classification theorem for positive natural products (2013)
  • Author(s): N. Muraki
  • Journal Title: Prob. Math. Statistics Volume: 33, No.2 Pages: 315-326
  • Peer Reviewed
• [Journal Article] A simple proof of the classification theorem for positive natural products (2013)
  • Author(s): Naofumi Muraki
  • Journal Title: Probability and Mathematical Statistics Volume: 33 Pages: 315-326
  • Peer Reviewed
• [Presentation] q-Deformation of free independence (2013)
  • Author(s): N. Muraki
  • Organizer: Seminaire d'Analyse Functionnelle
  • Place of Presentation: Besancon, フランス
  • Year and Date: 2013-10-01
• [Presentation] q-Deformation of free independence (2013)
  • Author(s): N. Muraki
  • Organizer: Algebra und functional analytische Anwendungen
  • Place of Presentation: Greifswald, ドイツ
  • Year and Date: 2013-09-25
• [Presentation] q-Deformation of free independence (2013)
  • Author(s): N. Muraki
  • Organizer: Free Probability Seminar
  • Place of Presentation: Saskatoon, カナダ
  • Year and Date: 2013-07-11
• [Presentation] On a q-deformation of free independence (2013)
  • Author(s): N. Muraki
  • Organizer: Workshop on Combinatorial and Random Matrix Aspects of Noncommutative Distributions and Free Probability
  • Place of Presentation: Toronto, カナダ
  • Year and Date: 2013-07-05
• [Presentation] Twisted independence from twisted CAR (2013)
  • Author(s): N. Muraki
  • Organizer: Recent Advances in Operator Theory and Operator Algebras
  • Place of Presentation: Bangalore, インド
  • Year and Date: 2013-01-08
• [Presentation] On a q-deformation of free independence (2013)
  • Author(s): Naofumi Muraki
  • Organizer: Workshop on Combinatorial and Random Matrix Aspects of Noncommutative Distributions and Free Probability
  • Place of Presentation: The Fields Institute for Research in Mathematical Sciences, Toronto, Canada
  • Invited
• [Presentation] q-Deformation of free independence (2013)
  • Author(s): Naofumi Muraki
  • Organizer: Free Probability Seminar
  • Place of Presentation: Department of Mathematics & Statistics, University of Saskatchewan, Saskatoon, Canada
• [Presentation] q-Deformation of free independence (2013)
  • Author(s): Naofumi Muraki
  • Organizer: Algebra und funktionalanalytische Anwendungen
  • Place of Presentation: Institut fur Mathematik und Informatik, Ernst Moritz Arndt Universitat Greifswald, Greifswald, Germany
• [Presentation] q-Deformation of free independence (2013)
  • Author(s): Naofumi Muraki
  • Organizer: Seminaire d’Analyse Fonctionnelle
  • Place of Presentation: Departement de mathematiques de Besancon, Universite de Franche-Comte, Besancon, France
• [Presentation] Twisted independence from twisted CAR (2013)
  • Author(s): Naofumi MUraki
  • Organizer: Recent Advances in Operator Theory and Operator Algebras
  • Place of Presentation: Indian Statistical Institute, Bangalore, India
  • Invited
• [Presentation] q-Deformation of free independence (2012)
  • Author(s): N. Muraki
  • Organizer: International Conference on Operator Algebras in Non-equilibrium Statistical Mechanics
  • Place of Presentation: Goa, インド
  • Year and Date: 2012-12-21
• [Presentation] Twisted independence for noncommutative random variables (2012)
  • Author(s): N. Muraki
  • Organizer: 15th Workshop on Non-Commutative Harmonic Analysis
  • Place of Presentation: Bedlewo, ポーランド
  • Year and Date: 2012-09-28
• [Presentation] 非可換確率論における独立性概念 (2012)
  • Author(s): 村木尚文
  • Organizer: CMRU 研究集会「量子確率論と量子ウォーク」
  • Place of Presentation: 東北大学情報科学研究科
  • Year and Date: 2012-07-04
• [Presentation] q-Deformation of stochastic independence in non-commutative probability theory (2012)
  • Author(s): N. Muraki
  • Organizer: International Workshop on Anomalous Statistics, Generalized Entropies, and Entropies, and Information Geometry
  • Place of Presentation: 奈良教育大学
  • Year and Date: 2012-03-08
• [Presentation] 非可換確率論における独立性概念 (2012)
  • Author(s): 村木尚文
  • Organizer: CMRU研究集会「量子確率論と量子ウォーク」
  • Place of Presentation: 東北大学情報科学研究科
• [Presentation] Twisted independence for noncommutative random variables (2012)
  • Author(s): Naofumi Muraki
  • Organizer: 15th Workshop: Non-Commutative Harmonic Analysis
  • Place of Presentation: Mathematics Reserach and Conference Center, Bedlewo, Poland
• [Presentation] q-Deformation of free independence (2012)
  • Author(s): Naofumi MUraki
  • Organizer: International Conference on Operator Algebras in Non-equilibrium Statistical Mechanics
  • Place of Presentation: Goa, India
  • Invited
• [Presentation] q-Deformation of stochastic independence in non-commutative probability theory (2012)
  • Author(s): Naofumi Muraki
  • Organizer: International Workshop on Anomalous Statistics, Generalized Entropies, and Information Geometry
  • Place of Presentation: Nara Women's University, Nara, Japan
• [Presentation] On a q-interpolation between classical independence and free independence (2011)
  • Author(s): N. Muraki
  • Organizer: Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-12-19
• [Presentation] Noncommutative independence (2011)
  • Author(s): N. Muraki
  • Organizer: 32nd International Conference on Quantum Probability and Related Topics
  • Place of Presentation: Levico Terme, イタリア
  • Year and Date: 2011-06-02
• [Presentation] Non-commutative independence (2011)
  • Author(s): Naofumi Muraki
  • Organizer: 32nd International Conference on Quantum Probability and Related Topics
  • Place of Presentation: Levico Terme, Italy
• [Presentation] On a q-interpolation beteen classical independence and free independence (2011)
  • Author(s): Naofumi Muraki
  • Organizer: RIMS Workshop: Mathematical Studies on Independence and Dependence Structure - Algebra meets Probability
  • Place of Presentation: RIMS, Kyoto University, Kyoto, Japan