Applications of operator algebras to symbolic dynamicaal systems

• Project/Area Number: 23540237
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: Joetsu University of Education
• Principal Investigator: MATSUMOTO Kengo 上越教育大学, 学校教育研究科(研究院), 教授 (40241864)
• Co-Investigator(Renkei-kenkyūsha): TOMIYAMA Jun 東京都立大学, 理学部, 名誉教授 (30006928)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 作用素環 / 記号力学系 / Ｃ＊環 / 位相的マルコフシフト / 軌道同型 / サブシフト / Cuntz-Krieger 環 / Ｃ＊－環 / 軌道同型類 / ＨｉｌｂｅｒｔＣ＊－加群

Outline of Final Research Achievements

I have studied symbolic dynamical systems and its C*-algebras from the following three view points. I have first studied typical examples of symbolic dynamical systems such as Dyck shifts and Markov-Dyck shifts and its C*-algebras. I have computed the K-groups of these C*-algebras. I have second introduced a notion of lambda-synchronization of subshifts with Wolfgang Krieger and proved that the lambda-synchronization is invariant under flow equivalence. Wolfgang Krieger and I have also presented formulae of zeta function and topological entropy for Markov-Dyck shifts. I have third studied continious orbit equivalence of one-sided topological markov shifts. Hiroki Matui and I have succeeded to obtain a complete classification theorem of continuous orbit equivalence of topological Markov shifts.

Research Products

• [Journal Article] Full groups of one-sided topological Markov shifts (2015)
  • Author(s): Kengo Matsumoto
  • Journal Title: Israel Journal of Mathematics Volume: 205 Issue: 1 Pages: 1-33
  • DOI: 10.1007/s11856-014-1134-8
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras. (2014)
  • Author(s): Kengo Matsumoto and Hiroki Matui
  • Journal Title: Ｋｙｏｔｏ Ｊｏｒｎａｌ ｏｆ Ｍａｔｈｅｍａｔｉｃｓ Volume: 54 Issue: 4
  • DOI: 10.1215/21562261-2801849
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Simple C*-algebras arising from certain Markov codes (2014)
  • Author(s): Kengo Matsumoto
  • Journal Title: Acta Sci.Math.(Szeged) Volume: 80
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Topological full groups of C*-algebras arising from beta-expansions (2014)
  • Author(s): Kengo Matsumoto and Hiroki Matui
  • Journal Title: Journal of Australian Mathematical Society Volume: 97 Issue: 2 Pages: 257-287
  • DOI: 10.1017/s1446788714000214
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] C*-algebras associated with Hilbert C*-quad modules of finite type (2014)
  • Author(s): Kengo Matsumoto
  • Journal Title: International Journal of Mathematics and Mathematical Sciences Volume: 952068 Pages: 1-21
  • DOI: 10.1155/2014/952068
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] C*-algebras associated with lambda-synchronizing subshifts and flow equivalence (2013)
  • Author(s): Kengo Matsumoto
  • Journal Title: Journal of the Australian Mathematical Society Volume: 95 Issue: 2 Pages: 241-265
  • DOI: 10.1017/s1446788713000219
  • Peer Reviewed
• [Journal Article] A certain synchronizing property of subshifts and flow equivalence (2013)
  • Author(s): Kengo Matsumoto
  • Journal Title: Israel Journal of Mathematics Volume: 196 Pages: 235-272
  • Peer Reviewed
• [Journal Article] A notion of synchronization of symbolic dynamics and C*-algebras (2013)
  • Author(s): Wolfgang Krieger and Kengo Matsumoto
  • Journal Title: Acta Appl. Math. Volume: 126 Pages: 263-275
• [Journal Article] Cuntz-Krieger algebras and a generalization of Catalan numbers (2013)
  • Author(s): Kengo Matsumoto
  • Journal Title: International Journal of Mathematics Volume: 24
• [Journal Article] Classification of Cuntz-Krieger algebras by orbit equivalence of topological markov shifts (2013)
  • Author(s): Kengo Matsumoto
  • Journal Title: Proceedings of American Mathematical Society Volume: 141
  • Peer Reviewed
• [Journal Article] K-groups of full group action on one-sided topological markov shifts (2013)
  • Author(s): Kengo Matsumoto
  • Journal Title: Duscrete and Continuous Dynamical System Volume: 33
  • Peer Reviewed
• [Journal Article] Some remarks on orbit equivalence of topological markov shifts (2012)
  • Author(s): Kengo matsumoto
  • Journal Title: Yokohama Mathematical Journal Volume: 58
  • Peer Reviewed
• [Journal Article] A class of simple C*-algebras arising from certain non-sofic subshifts (2011)
  • Author(s): Kengo Matsumoto
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 31
  • Peer Reviewed
• [Journal Article] Ergodic properties and KMS conditions on C*-symbolic dynamical systems (2011)
  • Author(s): Kengo Matsumoto
  • Journal Title: Documenta Mtahematica Volume: 16
  • Peer Reviewed
• [Journal Article] C*-algebras arising from Dyck systems of topological Markov chains (2011)
  • Author(s): Kengo Matsumoto
  • Journal Title: Mathematica Scandinavica Volume: 109
  • Peer Reviewed
• [Journal Article] Zeta functions and topological entropy of the Markov Dyck shifts (2011)
  • Author(s): Wolfgang Krieger and Kengo Matsumoto
  • Journal Title: Munster journal of Mathematics Volume: 4 Pages: 171184-171184
  • Peer Reviewed
• [Presentation] Full groups of Cuntz-Krieger algebras and Higman-Thompson groups (2015)
  • Author(s): 松井宏樹、 松本健吾
  • Organizer: 日本数学会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-23
• [Presentation] 位相的マルコフシフトの軌道同型と力学系のゼータ関数 (2014)
  • Author(s): 松井宏樹、 松本健吾
  • Organizer: 日本数学会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-23
• [Presentation] Topological Full groups of C*-algebras arsing from beta-expansions
  • Author(s): 松井 宏樹、松本健吾
  • Organizer: 日本数学会関数解析学分科会
  • Place of Presentation: 愛媛大学
• [Presentation] 位相的マルコフシフトの充足群とCuntz-Krieger 環の分類について
  • Author(s): 松本健吾
  • Organizer: 日本数学会 函数解析分科会
  • Place of Presentation: 九州大学
• [Presentation] Hilbert C*-quad module からできるC*-環について
  • Author(s): 松本健吾
  • Organizer: 日本数学会 函数解析分科会
  • Place of Presentation: 京都大学
• [Presentation] タイルばりの作用からできるＣ＊環とそのＫ－群公式について
  • Author(s): 松本健吾
  • Organizer: 日本数学会 関数解析分科会
  • Place of Presentation: 信州大学（長野県）
• [Presentation] 位相的マルコフシフトの鼓動同型類によるCuntz-Krieger 環の分類について
  • Author(s): 松本健吾
  • Organizer: 日本数学会 関数解析分科会
  • Place of Presentation: 東京理科大学（東京都）