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E-grāmata: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces

  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1895
  • Izdošanas datums: 15-Nov-2006
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540399469
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Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function SpacesPalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1895
  • Izdošanas datums: 15-Nov-2006
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540399469
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Over the past several decades, the territory of preserver problems has been continuously enlarging within the frame of linear analysis. The aim of this work is to present a sort of cross-section of the modern theory of preservers on infinite dimensional spaces (operator spaces and function spaces) through the author's corresponding results. Special emphasis is put on preserver problems concerning some structures of Hilbert space operators which appear in quantum mechanics. Moreover, local automorphisms and local isometries of operator algebras and function algebras are discussed in details.

The territory of preserver problems has grown continuously within linear analysis. This book presents a cross-section of the modern theory of preservers on infinite dimensional spaces (operator spaces and function spaces) through the author's corresponding results. Special emphasis is placed on preserver problems concerning some structures of Hilbert space operators which appear in quantum mechanics. In addition, local automorphisms and local isometries of operator algebras and function algebras are discussed in detail.

Recenzijas

From the reviews:









"Preserver problems deal with maps on subsets of algebras that preserve certain sets, relations, functions, etc. The book is very well organized. One of its remarkable features is that it links several areas, particularly operator theory and mathematical physics. The audience of this book is therefore potentially wide: operator algebraists, mathematical physicists, linear algebraists, ring theorists, etc. I warmly recommend this book to anyone interested in preserver problems." (Matej Brear, Mathematical Reviews, Issue, 2007 g)



"The monograph under review collects many important and highly nontrivial results and efforts. It is important to recall that the basic material is based on the research done by the author, who belongs to the eminent researchers in this field. The style is very fresh . I recommend to book for students and experts interested in operator algebra, noncommutative measure theory and mathematical foundations of quantum physics. The monograph is welcome in the quantum structures realm." (Anatolij Dvurecenskij, Zentralblatt MATH, Vol. 1119 (21), 2007)

Introduction 1
0.1 Linear Preserver Problems
1
0.2 Survey of the Results of
Chapter 1
9
0.3 Some Structures of Linear Operators in Quantum Mechanics
11
0.4 Survey of the Results of
Chapter 2
16
0.5 Local Derivations, Local Automorphisms and Local Isometries of Operator Algebras and Function Algebras
21
0.6 Survey of the Results of
Chapter 3
25
0.7 Notation
27
1 Some Linear and Multiplicative Preserver Problems on Operator Algebras and Function Algebras 29
1.1 Some Linear Preserver Problems on B(H) Concerning Rank and Corank
29
1.1.1 Summary
29
1.1.2 Formulation of the Results
29
1.1.3 Proofs
32
1.1.4 Remarks
38
1.2 Linear Maps on Factors Which Preserve the Extreme Points of the Unit Ball
39
1.2.1 Summary
39
1.2.2 Formulation of the Results
39
1.2.3 Proofs
41
1.2.4 Remarks
46
1.3 Diameter Preserving Linear Bijections of C(X)
46
1.3.1 Summary
46
1.3.2 Formulation of the Result
47
1.3.3 Proof
48
1.3.4 Remarks
56
1.4 *-Semigroup Endomorphisms of B(H)
57
1.4.1 Summary
57
1.4.2 Formulation of the Results
58
1.4.3 Proof
59
1.4.4 Remarks
64
2 Preservers on Quantum Structures 65
2.1 Transformations on the Set of All n-Dimensional Subspaces of a Hilbert Space Preserving Principal Angles
65
2.1.1 Summary
65
2.1.2 Formulation of the Main Result
65
2.1.3 Proof
67
2.1.4 Remarks
78
2.2 Orthogonality Preserving Transformations on Indefinite Inner Product Spaces: Generalization of Uhlhorn's Version of Wigner's Theorem
79
2.2.1 Summary
79
2.2.2 Formulation of the Results
79
2.2.3 Proofs
84
2.2.4 Remarks
92
2.3 Fidelity Preserving Maps on Quantum States
92
2.3.1 Summary
92
2.3.2 Formulation of the Results
93
2.3.3 Proofs
94
2.3.4 Remarks
97
2.4 Isometries of Quantum States
97
2.4.1 Summary
97
2.4.2 Formulation of the Results
97
2.4.3 Proofs
99
2.4.4 Remarks
104
2.5 Order Automorphisms of the Set of Bounded Observables
104
2.5.1 Summary
104
2.5.2 Formulation of the Results
104
2.5.3 Proofs
106
2.6 Linear Maps on the Set of Bounded Observables Preserving Maximal Deviation
112
2.6.1 Summary
112
2.6.2 Formulation of the Results
112
2.6.3 Proofs
116
2.6.4 Remarks
129
2.7 Some Preservers on Hilbert Space Effects
130
2.7.1 Summary
130
2.7.2 Formulation of the Results
130
2.7.3 Proofs
134
2.7.4 Remarks
141
2.8 Sequential Isomorphisms Between the Sets of von Neumann Algebra Effects
143
2.8.1 Summary
113
2.8.2 Formulation of the Results
143
2.8.3 Proof
145
2.8.4 Remarks
157
3 Local Automorphisms and Local Isometries of Operator Algebras and Function Algebras 159
3.1 The Automorphism and Isometry Groups of B(H) Are Topologically Reflexive
159
3.1.1 Summary
159
3.1.2 Formulation of the Results
159
3.1.3 Proofs
160
3.1.4 Remarks
168
3.2 Reflexivity of the Automorphism and Isometry Groups of C(X)
169
3.2.1 Summary
169
3.2.2 Formulation of the Result
169
3.2.3 Proof
170
3.2.4 Remarks
171
3.3 Reflexivity of the Automorphism and Isometry Groups of the Suspension of B(H)
172
3.3.1 Summary
172
3.3.2 Formulation of the Results
172
3.3.3 Proofs
173
3.3.4 Remarks
185
3.4 2-Local Automorphisms of Operator Algebras on Banach Spaces
186
3.4.1 Summary
186
3.4.2 Formulation of the Results
187
3.4.3 Proofs
188
3.4.4 Remarks
195
3.5 2-Local Automorphisms of Some Quantum Structures
195
3.5.1 Summary
195
3.5.2 Formulation of the Results
196
3.5.3 Proofs
197
3.5.4 Remarks
203
Appendix 205
Recent Results Added in Revision 211
References 217
Index 231


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