Studies on structures and isometries of commutative Banach algebras

• Project/Area Number: 23540193
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Shinshu University
• Principal Investigator: TAKAGI Hiroyuki 信州大学, 学術研究院理学系, 教授 (20206725)
• Co-Investigator(Kenkyū-buntansha): HATORI Osamu 新潟大学, 自然科学系, 教授 (70156363)
• Co-Investigator(Renkei-kenkyūsha): MIURA Takeshi 新潟大学, 自然科学系, 教授 (90333989) ; UEKI Sei-Ichiro 茨城大学, 工学部, 准教授 (70512408)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 関数解析学 / バナッハ環 / 保存問題 / 合成作用素

Outline of Final Research Achievements

Inspired by the preserver problem of Banach algebras, we consider the characterization of (not necessarily linear) isometries between continuous function spaces. Here a continuous function space means a linear space consisting of (not necessarily all) continuous functions with the supremum norm. We prove that if continuous function spaces A and B satisfy a certain separation property, then every isometry from A onto B is represented by using a weighted composition operator and its complex conjugate on the Choquet boundary of B.

Research Products

• [Journal Article] Ulam type stability problems for alternative homomorphisms (2014)
  • Author(s): S-E. Takahasi, M. Tsukada, T. Miura, H. Takagi and K. Tanahashi
  • Journal Title: Journal of Inequalities and Applications Volume: 2014 Pages: 1-13
  • DOI: 10.1186/1029-242x-2014-228
  • NAID: 120006650875
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Isometies of the unitary groups in C^*-algebras (2014)
  • Author(s): O. Hatori
  • Journal Title: Studia Math. Volume: 221 Pages: 61-86
  • DOI: 10.4064/sm221-1-4
  • Peer Reviewed
• [Journal Article] Real-linear isometries between subspaces of continuous functions (2014)
  • Author(s): H. Koshimizu, T. Miura, H. Takagi and S.-E. Takahasi
  • Journal Title: J. Math. Anal. Appl. Volume: 413 Pages: 229-241
  • Peer Reviewed
• [Journal Article] Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C^*-algebras (2014)
  • Author(s): O. Hatori and L Molnar
  • Journal Title: J. Math. Anal. Appl. Volume: 409 Pages: 158-167
  • Peer Reviewed
• [Journal Article] Isometries between groups of invertible elements in C*-algebras (2012)
  • Author(s): O. Hatori and K. Watanabe
  • Journal Title: Studia Math. Volume: 209 Pages: 103-106
• [Journal Article] Isometries of the unitary groups (2012)
  • Author(s): O. Hatori and L. Molnar
  • Journal Title: Proc. Amer. Math. Soc. Volume: 140 Pages: 2141-2154
  • Peer Reviewed
• [Journal Article] Backward shifts on function algebras (2011)
  • Author(s): H. Takagi, H. Koshimizu and H. Ariizumi
  • Journal Title: J. Math. Anal. Appl. Volume: 377 Pages: 135-144
  • NAID: 120004966386
  • Peer Reviewed
• [Journal Article] Spectral radii conditions for isomorphisms between unital semisimple commutative Banach algebras (2011)
  • Author(s): G. Hirasawa, T. Miura and H. Takagi
  • Journal Title: Contemp. Math. Volume: 547 Pages: 125-134
  • Peer Reviewed
• [Journal Article] Algebraic properties of isometries between groups of invertible elements in Banach algebras (2011)
  • Author(s): O. Hatori
  • Journal Title: J. Math. Anal. Appl. Volume: 376 Pages: 74-83
  • Peer Reviewed
• [Presentation] 直積空間上のシフト作用素 (2015)
  • Author(s): 高木啓行, 古清水大直
  • Organizer: 第2回山陰基礎論・解析学研究集会
  • Place of Presentation: 鳥取県米子市国際ファミリープラザ
  • Year and Date: 2015-01-24 – 2015-01-25
• [Presentation] SO(4)上のある写像について (2014)
  • Author(s): 羽鳥理
  • Organizer: 関数環研究集会
  • Place of Presentation: 早稲田大学教育学部
  • Year and Date: 2014-09-18 – 2014-09-19
• [Presentation] 直積空間と等長作用素 (2013)
  • Author(s): 高木啓行・古清水大直
  • Organizer: つくばセミナー
  • Place of Presentation: 筑波大学
• [Presentation] Isometries on the unitary groups (2012)
  • Author(s): O. Hatori
  • Organizer: Conference on Mathematics and Statistics
  • Place of Presentation: University of Memphis
• [Presentation] ユニタリー群上の等距離写像 (2012)
  • Author(s): 羽鳥 理
  • Organizer: つくばセミナー
  • Place of Presentation: 筑波大・自然科学系
• [Presentation] Real-linear isometries between subspaces of continuous functions (2011)
  • Author(s): H. Koshimizu, T. Miura, H. Takagi and S.-E. Takahasi
  • Organizer: Research on preserver problems concerning to Banach algebras and its applications
  • Place of Presentation: 京大・数理研
• [Presentation] Criteria for the equivalence of LCA groups (2011)
  • Author(s): O. Hatori
  • Organizer: Workshop on commutative Banach algebra 2011
  • Place of Presentation: 奈良教育大
• [Presentation] Ulam type stability problems for alternative homomorphisms (2011)
  • Author(s): S.-E. Takahasi, T. Miura, H. Takagi and K. Tanahashi
  • Organizer: 第20回関数空間セミナー
  • Place of Presentation: 北大・理
• [Presentation] 保存問題における関数空間上の等距離写像について
  • Author(s): 高木啓行、古清水大直
  • Organizer: 山陰基礎論・解析学セミナー
  • Place of Presentation: 米子高専
• [Presentation] exp A 上の preserver problem について
  • Author(s): 羽鳥理
  • Organizer: 関数環研究集会
  • Place of Presentation: クロスパルにいがた