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博碩士論文 etd-0723112-130530 詳細資訊
Title page for etd-0723112-130530
論文名稱 Title |
實數與複數西爾伯特丙星模間的等距算子 Isometries of real and complex Hilbert C*-modules |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
50 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2012-07-10 |
繳交日期 Date of Submission |
2012-07-23 |
關鍵字 Keywords |
實丙星代數、完備等距算子、實西爾伯特丙星模、丙星代數、西爾伯特丙星模 ternary rings of operators, complete isometries, JB*-triples, real JB*-triples, Hilbert C*-modules, real Hilbert C*-modules, real C*-algebras, C*-algebras |
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統計 Statistics |
本論文已被瀏覽 5763 次，被下載 991 次 The thesis/dissertation has been browsed 5763 times, has been downloaded 991 times. |
中文摘要 |
令A和B是丙星代數，V和W分別為佈於A和B的西爾伯特丙星模。假設T是從V到W的線性一對一滿射。我們將證明下列敘述是等價的。 (a) T是酉算子，也就是存在一同構映射α:A→B使得 <Tx,Ty>=α(<x,y>), ∀ x,y∈ V; (b) T保持TRO結構，也就是 T(x<y,z>)=Tx<Ty,Tz>, ∀ x,y,z in V ; (c) T是2-等距算子； (d) T是完備等距算子。 如果A和B是可交換的，則上列敘述也等價於 (e) T是等距算子。 假設A和B是複的，則T是酉算子若且唯若T是模映射，也就是 T(xa)=(Tx)α(a), ∀ x ∈ V,a ∈ A. |
Abstract |
Let A and B be real or complex C*-algebras. Let V and W be real or complex (right) full Hilbert C*-modules over A and B, respectively. Let T be a linear bijective map from V onto W. We show the following four statements are equivalent. (a) T is a unitary operator, i.e., there is a ∗-isomorphism α : A → B such that <Tx,Ty> = α(<x,y>), ∀ x,y∈ V ; (b) T preserves TRO products, i.e., T(x<y,z>) =Tx<Ty,Tz>, ∀ x,y,z in V ; (c) T is a 2-isometry; (d) T is a complete isometry. Moreover, if A and B are commutative, the four statements are also equivalent to (e) T is a isometry. On the other hand, if V and W are complex Hilbert C*-modules over complex C*-algebras, then T is unitary if and only if it is a module map, i.e., T(xa) = (Tx)α(a), ∀ x ∈ V,a ∈ A. |
目次 Table of Contents |
Chapter 1: Introduction 1 Chapter 2: C∗-algebras, Hilbert C∗-modules and JB∗-triples 4 2.1 C∗-algebras . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Complexification of real Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Real C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Hilbert bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Hilbert C∗-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Real Hilbert C∗-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 JB∗-triples and JB-triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Chapter 3: Isometries of real and complex Hilbert C∗-modules 22 3.1 Isometries of Hilbert C∗-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Isometries of Hilbert C∗-modules over commutative complex C∗-algebras . . . . . . . . . . . . . . 27 3.3 Isometries of real Hilbert C∗-modules over commutative real C∗-algebras . . . . . . . . . . . . . . . . . . . 32 3.4 Isometries of Hilbert C∗-modules over complex C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Isometries of real Hilbert C∗-modules over real C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |
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