Study on Complex dynamical systems and operator algebras

• Project/Area Number: 19340040
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kyushu University
• Principal Investigator: WATATANI Yasuo Kyushu University, 大学院・数理学研究院, 教授 (00175077)
• Co-Investigator(Kenkyū-buntansha): KOSAKI Hideki 九州大学, 大学院・数理学研究院, 教授 (20186612) ; MATSUI Taku 九州大学, 大学院・数理学研究院, 教授 (50199733) ; KAJIWARA Tsuyoshi 岡山大学, 大学院・環境学研究科, 教授 (50169447) ; UEDA Yoshimichi 九州大学, 大学院・数理学研究院, 准教授 (00314724)
• Co-Investigator(Renkei-kenkyūsha): MASUDA Toshihiko 九州大学, 大学院・数理学研究院, 准教授 (60314978)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥11,050,000 (Direct Cost: ¥8,500,000、Indirect Cost: ¥2,550,000); Fiscal Year 2010: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2009: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2008: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2007: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
• Keywords: 作用素環 / 複素力学系 / C*-環 / 有理関数 / 極大可換環 / トレース / 作用素理 / C^*-環 / KMS状態

Research Abstract

In the research we constructed C*-algebras of operators on Hilbert spaces for complex dynamical systems on a complex plane by the iteration of meromorphic functions like rational functions and transcendental entire functions. We found unexpected relations between complex dynamical systems and operator algebras. Self-similarity of Julia sets of complex dynamical systems corresponds to purely infinite simplicity of the associated C*-algebras. The branch structure of complex dynamical systems is related with the structure of KMS states (equilibrium states) for the gauge actions on the associated C*-algebras. We showed that we can recover the degree of the rational function, the number of branched points and the number of exceptional points.

Research Products

• [Journal Article] Fundamental group of simple C*-algebras with unique trace II (2011)
  • Author(s): N.Nawata, Y.Watatani
  • Journal Title: J.Func.Anal. 260 Pages: 428-435
  • NAID: 120004951556
  • Peer Reviewed
• [Journal Article] C*-algebras associated with algebraic correspondneces on the Riemann sphere (2011)
  • Author(s): T.Kajiwara, Y.Watatani
  • Journal Title: J.Operator Theory 65 Pages: 427-449
  • Peer Reviewed
• [Journal Article] Fundamental group of simple C*-algebras with unique trace (2010)
  • Author(s): N.Nawata, Y.Watatani
  • Journal Title: Advances in Math. 225 Pages: 307-318
  • Peer Reviewed
• [Journal Article] Toeplitz-composition C*-algebras for certain finite Blaschke products (2010)
  • Author(s): H.Hamada, Y.Watatani
  • Journal Title: Amer.Math.Soc. 138 Pages: 2113-2123
  • Peer Reviewed
• [Journal Article] Trace Jensen inequality and related weak majorization in semi-finite von Neumann algebras (2010)
  • Author(s): T.Harada, H.Kosaki
  • Journal Title: J.Operator Theory 63 Pages: 129-150
  • Peer Reviewed
• [Journal Article] Spectral gap and split property in quantum spin chains (2010)
  • Author(s): T.Matsui
  • Journal Title: J.Math.Phys 51
  • Peer Reviewed
• [Journal Article] Toeplitz-composition C^*-algebras for certain Blaschke products (2010)
  • Author(s): H.Hamada, Y.Watatani
  • Journal Title: Proc.Amer.Math. Volume: 138 Pages: 2113-2123
  • Peer Reviewed
• [Journal Article] Fundamental group of simple C^*-algebras with unique trace (2010)
  • Author(s): N.Nawata, Y.Watatani
  • Journal Title: Adv.Math. Volume: 225 Pages: 307-318
  • Peer Reviewed
• [Journal Article] Trace Jensen inequality and related weak majorization in semi-finite von Neumann algebras (2010)
  • Author(s): T.Harada, H.Kosaki
  • Journal Title: J.Operator Theory Volume: 63 Pages: 129-150
  • Peer Reviewed
• [Journal Article] Indecomposable representations of quivers on infinite dimensional Hilbert spaces (2009)
  • Author(s): M.Enomoto, Y.Watatani
  • Journal Title: J.funct.Anal. 256 Pages: 959-991
  • NAID: 110002548905
  • Peer Reviewed
• [Journal Article] Complex dynamical systems and associated C*-algebras, Operator structures and dynamical systems (2009)
  • Author(s): Y.Watatani
  • Journal Title: Contemp.Math.Amer.Math. 503 Pages: 291-305
  • Peer Reviewed
• [Journal Article] Countable bases for Hilbert C*-modules and Classification of KMS state, Operator structures and dynamical systems (2009)
  • Author(s): T.Kajiwara
  • Journal Title: Contemp.Math.Amer.Math.Soc. 503
  • Peer Reviewed
• [Journal Article] A log-Sobolev type inequality for free entropy of two projections (2009)
  • Author(s): F.Hiai, Y.Ueda
  • Journal Title: Ann.Inst.Henri Poincare Probab.Stat. 45 Pages: 239-249
  • NAID: 120006714550
  • Peer Reviewed
• [Journal Article] Orbital approach to microstates free entropy, Internat. (2009)
  • Author(s): F.Hiai, T.Miyamoto, Y.Ueda
  • Journal Title: J.Math. 20 Pages: 227-273
  • Peer Reviewed
• [Journal Article] On peak phenomena for non commutative (2009)
  • Author(s): Y.Ueda
  • Journal Title: H^∞ Math.Ann. 343 Pages: 421-429
  • Peer Reviewed
• [Journal Article] Approximate innerness andCentral triviality of endomorphisms (2009)
  • Author(s): T.Masuda, R.Tomatsu
  • Journal Title: Adv.Math. 220 Pages: 1075-1134
  • Peer Reviewed
• [Journal Article] Indecomposable representations of quivers on infinite-dimensional Hilbert spaces (2009)
  • Author(s): M. Enomoto, Y. Watatani
  • Journal Title: J.Funct.Anal. 256 Pages: 959-991
  • NAID: 110002548905
• [Journal Article] Indecomposabe vepresentationo of purven on infimite dimensionel Hiebert spasen (2009)
  • Author(s): M.Enomoto, Y.Watatani
  • Journal Title: J. Funct. Anal. 256 Pages: 959-991
  • Peer Reviewed
• [Journal Article] Indeumr〓os〓 r〓res〓tation〓 of g〓ven on 〓fimite- dimensivnal Hi〓bevt space (2009)
  • Author(s): M. Enomoto, Y. Watatani
  • Journal Title: J. fund. Anal. 256 Pages: 959-991
  • Peer Reviewed
• [Journal Article] C*-algebras associated with Mauldin-Williams graphs (2008)
  • Author(s): M.Ionescu, Y.Watatani
  • Journal Title: Canad.Math.Bull. 51 Pages: 545-560
  • Peer Reviewed
• [Journal Article] On equality condition for trace Jensen Inequality in semi-finite von Neumann algebras (2008)
  • Author(s): T.Harada, H.Kosak
  • Journal Title: Internat.J.Math. 19 Pages: 41-501
  • Peer Reviewed
• [Journal Article] On infinite divisibility of positive definite functions arisingfrom operator means (2008)
  • Author(s): H.Kosaki
  • Journal Title: J.Funct.Anal. 254 Pages: 84-108
  • Peer Reviewed
• [Journal Article] On Haag duality for pure states of quantum spin chains (2008)
  • Author(s): M.Keyl, T.Matsui, D.Schlingemann, R.Werner
  • Journal Title: Rev.Math.Phys. 20 Pages: 707-724
  • Peer Reviewed
• [Journal Article] Remarks on HNN extensions in operator algebras (2008)
  • Author(s): Y.Ueda
  • Journal Title: Illinois J.Math. 52 Pages: 705-725
  • Peer Reviewed
• [Journal Article] Notes on microstate free entropyof projections (2008)
  • Author(s): F.Hiai, Y.Ueda
  • Journal Title: Publ.Res.Inst.Math.Sci. 44 Pages: 49-89
  • Peer Reviewed
• [Journal Article] Classification of actions of duals of Finite groups on the AFD factor of type II_1 (2008)
  • Author(s): T.Masuda
  • Journal Title: J.Operator Theory 60 Pages: 273-300
  • Peer Reviewed
• [Journal Article] C*algebras associated with Mauldin-Williams graphs (2008)
  • Author(s): M. Ionescu, Y. Watatani
  • Journal Title: Canad.Math.Bull. 51 Pages: 545-560
• [Journal Article] C^<〓>lge〓ado associated with Manldin-Wi〓 〓pho (2008)
  • Author(s): M, Ionesscu, Y, watatani
  • Journal Title: Canada, Math, Bull. 51 Pages: 545-560
  • Peer Reviewed
• [Journal Article] On infimte divisabil〓t〓 of positin〓 defim〓te fun〓ion avising from operatn mea〓o (2008)
  • Author(s): H, Kosaki
  • Journal Title: J. Fud, Anal. 254 Pages: 84-108
  • Peer Reviewed
• [Journal Article] KMS states and branched points (2007)
  • Author(s): M.Izumi, T.Kajiwara, Y.Watatani
  • Journal Title: Ergodic Theory Dynam. Systems 27 Pages: 1887-1918
  • Peer Reviewed
• [Journal Article] Representations of Cuntz algebras on fractal sets (2007)
  • Author(s): M.Mori, O.Suzuki, Y.Watatani
  • Journal Title: Kyushu J.Math. 61 Pages: 443-456
  • NAID: 110006377556
  • Peer Reviewed
• [Journal Article] Exotic indecomposable systems of four subspaces in a Hilbert Spaces (2007)
  • Author(s): M.Enomoto, Y.Watatani
  • Journal Title: Integral Equations Operator Theory 59 Pages: 149-164
  • Peer Reviewed
• [Journal Article] Mean matrices and infinite divisibility (2007)
  • Author(s): R.Bhatia, H.Kosaki
  • Journal Title: Linear Algebras Appl.Linear Algebas Appl. 424 Pages: 36-54
  • Peer Reviewed
• [Journal Article] Evans-Kishimoto type argument for actions of discrete amenable groups on McDuff factors (2007)
  • Author(s): T.Masuda
  • Journal Title: Math.Scand. 101 Pages: 48-64
  • Peer Reviewed
• [Journal Article] Classification of minimal Actions of a compact Kac algebra with amenable dual (2007)
  • Author(s): T.Masuda, R.Tomatsu
  • Journal Title: Comm.Math.Phys. 274 Pages: 487-551
  • Peer Reviewed
• [Journal Article] KMS states and branched points (2007)
  • Author(s): M. Izumi, T. Kajiwara, Y. Watatani
  • Journal Title: Ergodic Th. & Dynam.Sys. 27 Pages: 1887-1918
• [Journal Article] Exotic indecomposable systems of four subspaces in a Hilbert space (2007)
  • Author(s): M. Enomoto, Y. Watatani
  • Journal Title: Integr.Equ.and Oper.Theory 59 Pages: 149-164
• [Journal Article] KMS stale and branched point (2007)
  • Author(s): M. Izumi, T. kajimara, Y. Watatahi
  • Journal Title: Ergodic Th.&.Dynam.sys. 27 Pages: 1887-1918
  • Peer Reviewed
• [Journal Article] Representation of Cunt algetna on fractalset (2007)
  • Author(s): M. Mori, O. Suzuli and Y. Watatani
  • Journal Title: kyushe J.Math 61 Pages: 443-456
  • Peer Reviewed
• [Journal Article] Exotie indecomnosclle system offour subspace in a Hilhatspae (2007)
  • Author(s): M. Enomoto, Y. watatani
  • Journal Title: Integn.egn.and oper.Theary 59 Pages: 149-164
  • Peer Reviewed
• [Journal Article] Fundamental group of simple C*-algebras with unique trace
  • Author(s): N. Nawata, Y. Watatani
  • Journal Title: Adv.Math. (to appear)
• [Journal Article] Toeplitz composition C*-algebras for certain finite Blaschke products
  • Author(s): H. Hamada, Y. Watatani
  • Journal Title: Proc.Amer.Math.Soc. (to appear)
• [Journal Article] Toeplitg-compwition Ctlgelnes for centain finite Blasche products
  • Author(s): H.Hamad, Y.Watatani
  • Journal Title: Proc. Amer, Math, Soc. (to appean)
  • Peer Reviewed
• [Journal Article] Trace Jensen Inefuality and related weah majorization in seml-fimite oon Neumeas lgehas
  • Author(s): T.Haradn, H.Kosaki
  • Journal Title: J. Openatu Thevry (to appean)
  • Peer Reviewed
• [Presentation] Self-similarity and fundamental group of simple C*-algebras (2010)
  • Author(s): Y.Watatani
  • Organizer: KOTAC International Conference
  • Place of Presentation: Inchon, Korea
  • Year and Date: 2010-06-18
• [Presentation] Self-similarity and fundamental group of simple C^*-algebras (2010)
  • Author(s): Y.Watatani
  • Organizer: KOTAC
  • Place of Presentation: Inchon, Korea
• [Presentation] Fumdamental group of simple Cdgethes with umigue tace (2009)
  • Author(s): Y Watatani
  • Organizer: KMS state and Nonwmmatatine geomety
  • Place of Presentation: Univensity of Vatoria
  • Year and Date: 2009-07-07
• [Presentation] Norm comparison for operator means : general theory and new results (2009)
  • Author(s): H.Kosaki
  • Organizer: ICTP
  • Place of Presentation: Trieste, Italy
  • Year and Date: 2009-07-06
• [Presentation] Composition operators and C*-algebras associated with complex dynamical systems (2009)
  • Author(s): Y.Watatani
  • Organizer: Asian Mathematical Conference 2009
  • Place of Presentation: Kuala Lumpur, Malaysia
  • Year and Date: 2009-06-24
• [Presentation] Fundamental group of simple C*-algebras with unique trace (2009)
  • Author(s): Y.Watatani
  • Organizer: Conference on KMS states and non-commutative geometry
  • Place of Presentation: Victoria, Canada
• [Presentation] Fundamental group of simple C*-algebras with unique trace (2009)
  • Author(s): Y. Watatani
  • Organizer: University of Victoria
  • Place of Presentation: Victoria, Canada
• [Presentation] Composition operators and C*-algebras associated with complex dynamical systems (2009)
  • Author(s): Y. Watatani
  • Organizer: Asia Mathematical Conference 2009
  • Place of Presentation: Kuala Lumpur, Malaysia
• [Presentation] Complex dynamical systems and associated C*-algebras (2008)
  • Author(s): Y.Watatani
  • Organizer: Operator structures and dynamical systems
  • Place of Presentation: Leiden, Netherlands
  • Year and Date: 2008-07-21
• [Presentation] Complex dynami〓ay sy〓m and associated C^<〓>d〓g〓 (2008)
  • Author(s): 綿谷安男
  • Organizer: Lorenty center のワークショッブ
  • Place of Presentation: オランダのライデン市
  • Year and Date: 2008-07-21
• [Presentation] C*-algebras associated with algebraic correspondencs (2008)
  • Author(s): Y.Watatani
  • Organizer: C*-algebras associated to discrete and dynamical systems, BIRS
  • Place of Presentation: Bunff, Canada
• [Presentation] Trace Jensen type Inequality in von Neumann algebras and related Topics (2008)
  • Author(s): H.Kosaki
  • Organizer: Operator algebras and Quantum fields theory
  • Place of Presentation: Vienna, Austria
• [Presentation] Complex dynamical systems and associated C*-algebras (2008)
  • Author(s): Y. Watatani
  • Organizer: Lorentz Center
  • Place of Presentation: Leiden, Netherlands
• [Presentation] C*-algebras associated with algebraic correspondences (2008)
  • Author(s): Y. Watatani
  • Organizer: Bunff International Research Station
  • Place of Presentation: Bunff, Canada
• [Presentation] Branched points and C*-algebras associated with complex dynamica systems (2007)
  • Author(s): Y.Watatani
  • Organizer: KOTAC
  • Place of Presentation: Seoul, Korea
• [Presentation] On Jensen-type majorizations in semi-finite von Neumann algebras, Noncommutative L_p spaces, Operator spaces and Applications (2007)
  • Author(s): H.Kosaki
  • Organizer: CIRM
  • Place of Presentation: Luminy, France
• [Presentation] Branched points and C*-algebras associated with complex dynamical systems (2007)
  • Author(s): Y. Watatani
  • Organizer: Hanyang University
  • Place of Presentation: Seoul, Korea
• [Presentation] Branchedpoint and C^*_lgelna associates wich cagler dynamicelsystem (2007)
  • Author(s): Y. Watatani
  • Organizer: KOTAC
  • Place of Presentation: ソウル
• [Book] Positive definiteness of functions with applications to operator norm inequalities
  • Author(s): H.Kosaki
  • Publisher: appear in Memoirs, AMS
• [Remarks] ホームページ
  • URL: http://hyoka.ofc.kyushu-u.ac.jp/search/details/K000506/index.html
• [Remarks]
  • URL: http://hyoka.ofc.kyushu-u.ac.jp/search/details/K000506/index.html