学部・大学院区分
Undergraduate / Graduate

多・博前

時間割コード
Registration Code

3211092

科目区分
Course Category

Ａ類Ⅱ（専門科目）
Category A-2

科目名　【日本語】
Course Title

数理科学特論Ⅵ

科目名　【英語】
Course Title

Topics in Mathematical Science Ⅵ

コースナンバリングコード
Course Numbering Code

担当教員　【日本語】
Instructor

BOURNE Christopher jack ○

担当教員　【英語】
Instructor

BOURNE Christopher jack ○

単位数
Credits

開講期・開講時間帯
Term / Day / Period

秋金曜日２時限
Fall Fri 2

授業形態
Course style

学科・専攻
Department / Program

Graduate School of Mathematics

必修・選択
Required / Selected

Selected

授業の目的　【日本語】
Goals of the Course(JPN)

科目のテーマ：C*環（作用素環）

この科目は英語で行います．
C*環とはヒルベルト空間の有界線形作用素のノルムで閉じた環であり，定義に関数解析の部分と代数学の部分が混ざっています．この科目ではC*環を入門し，基礎の性質（Gelfand-Naimark定理，スペクトル定理，GNS構成法など）を考察します．局所コンパクトHausdorff空間，離散群，離散群作用からC*環が作ることができて，この例を詳しく勉強する予定です．

授業の目的　【英語】
Goals of the Course

Course title: C*-algebras

C*-algebras are algebras that can be naturally considered as a norm-closed subalgebra of the bounded linear operators on a Hilbert space. In this course, we will introduce the theory of C*-algebras, which combines elements of (functional) analysis with abstract algebra, and consider their elementary properties. Topics will include the Gelfand-Naimark Theorem, The Spectral Theorem, continuous functional calculus and the Gelfand-Naimark-Segal (GNS) construction.

Some natural settings where C*-algebras emerge include locally compact Hausdorff spaces, discrete groups and groups acting on spaces. The latter part of the topic will look at these examples in detail, particularly C*-algebras for discrete groups.

到達目標　【日本語】
Objectives of the Course(JPN))

英語版をご覧ください．

到達目標　【英語】
Objectives of the Course

By the end of the course, students will understand the definition and elementary properties of C*-algebras, which bring together elements of analysis and algebra. Students will also be able to construct C*-algebras for discrete groups and, when the group is abelian, its connection to Fourier theory.

授業の内容や構成
Course Content / Plan

A tentative plan for the semester is as follows.
- Banach and C*-algebras
- The Gelfand transform
- The spectral theorem and functional calculus
- Positivity
- The Gelfand-Naimark-Segal (GNS) construction
- C*-algebras for discrete groups
- C*-algebras for group actions on spaces (if time)
- C*-algebras of étale groupoids (if time)

履修条件
Course Prerequisites

Students should be familiar with some basic concepts of functional analysis such as the definition of Banach spaces. Some knowledge of abstract algebra (rings, ideals, group representations) will also be helpful.