Atnaujinkite slapukų nuostatas

El. knyga: Invitation to Unbounded Representations of *-Algebras on Hilbert Space

  • Formatas: EPUB+DRM
  • Serija: Graduate Texts in Mathematics 285
  • Išleidimo metai: 28-Jul-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030463663
Kitos knygos pagal šią temą:
Invitation to Unbounded Representations of *-Algebras on Hilbert SpaceDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Graduate Texts in Mathematics 285
  • Išleidimo metai: 28-Jul-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030463663
Kitos knygos pagal šią temą:

DRM apribojimai

  • Kopijuoti:

    neleidžiama

  • Spausdinti:

    neleidžiama

  • El. knygos naudojimas:

    Skaitmeninių teisių valdymas (DRM)
    Leidykla pateikė šią knygą šifruota forma, o tai reiškia, kad norint ją atrakinti ir perskaityti reikia įdiegti nemokamą programinę įrangą. Norint skaityti šią el. knygą, turite susikurti Adobe ID . Daugiau informacijos  čia. El. knygą galima atsisiųsti į 6 įrenginius (vienas vartotojas su tuo pačiu Adobe ID).

    Reikalinga programinė įranga
    Norint skaityti šią el. knygą mobiliajame įrenginyje (telefone ar planšetiniame kompiuteryje), turite įdiegti šią nemokamą programėlę: PocketBook Reader (iOS / Android)

    Norint skaityti šią el. knygą asmeniniame arba „Mac“ kompiuteryje, Jums reikalinga  Adobe Digital Editions “ (tai nemokama programa, specialiai sukurta el. knygoms. Tai nėra tas pats, kas „Adobe Reader“, kurią tikriausiai jau turite savo kompiuteryje.)

    Negalite skaityti šios el. knygos naudodami „Amazon Kindle“.

This textbook provides an introduction to representations of general -algebras by unbounded operators on Hilbert space, a topic that naturally arises in quantum mechanics but has so far only been properly treated in advanced monographs aimed at researchers. The book covers both the general theory of unbounded representation theory on Hilbert space as well as representations of important special classes of -algebra, such as the Weyl algebra and enveloping algebras associated to unitary representations of Lie groups. A broad scope of topics are treated in book form for the first time, including group graded -algebras, the transition probability of states, Archimedean quadratic modules, noncommutative Positivstellensätze, induced representations, well-behaved representations and representations on rigged modules.

Making advanced material accessible to graduate students, this book will appeal to students and researchers interested in advanced functional analysis and mathematical physics, and with many exercises it can be used for courses on the representation theory of Lie groups and its application to quantum physics. A rich selection of material and bibliographic notes also make it a valuable reference.

Recenzijos

This is a fantastic book. The material it covers is wonderful and exciting. The book is well-written, and in fact pleasant to read. It takes a familiar subject --- that is unfortunately not as popular among mathematicians as it should be --- and presents it from a very evocative perspective. Kudos to Schmüdgen. (Michael Berg, MAA Reviews, July 22, 2023)



It is very well written, the style is pleasant and attractive, and the information can be used by beginners and by specialists as well. All chapters are accompanied by exercises and pertinent historical comments. all researchers interested in representation theory may regard this work not only as a reference book but also as a source of inspiration for further development. (Florian-Horia Vasilescu, zbMATH 1458.47002, 2021)

1 Prologue: The Algebraic Approach to Quantum Theories
1(6)
2 *-Algebras
7(32)
2.1 *-Algebras: Definitions and Examples
7(5)
2.2 Constructions with *-Algebras
12(5)
2.3 Quadratic Modules
17(3)
2.4 Positive Functionals and States on Complex *-Algebras
20(3)
2.5 Positive Functionals on Real *-Algebras
23(2)
2.6 Characters of Unital Algebras
25(3)
2.7 Hermitian Characters of Unital *-Algebras
28(5)
2.8 Hermitian and Symmetric *-Algebras
33(2)
2.9 Exercises
35(3)
2.10 Notes
38(1)
3 O* -Algebras
39(20)
3.1 O*-Algebras and Their Graph Topologies
39(5)
3.2 Bounded Commutants
44(5)
3.3 Trace Functionals on O*-Algebras
49(5)
3.4 The Mittag-Leffler Lemma
54(3)
3.5 Exercises
57(1)
3.6 Notes
57(2)
4 * -Representations
59(34)
4.1 Basic Concepts on *-Representations
59(8)
4.2 Domains of Representations in Terms of Generators
67(6)
4.3 Invariant Subspaces and Reducing Subspaces
73(4)
4.4 The GNS Construction
77(5)
4.5 Examples of GNS Representations
82(3)
4.6 Positive Semi-definite Functions on Groups
85(3)
4.7 Pathologies with Unbounded Representations
88(1)
4.8 Exercises
89(2)
4.9 Notes
91(2)
5 Positive Linear Functionals
93(28)
5.1 Ordering of Positive Functionals
93(4)
5.2 Orthogonal Positive Functionals
97(2)
5.3 The Transition Probability of Positive Functionals
99(5)
5.4 Examples of Transition Probabilities
104(6)
5.5 A Radon-Nikodym Theorem for Positive Functionals
110(3)
5.6 Extremal Decomposition of Positive Functionals
113(4)
5.7 Quadratic Modules and *-Representations
117(2)
5.8 Exercises
119(1)
5.9 Notes
119(2)
6 Representations of Tensor Algebras
121(16)
6.1 Tensor Algebras
121(2)
6.2 Positive Functionals on Tensor Algebras
123(2)
6.3 Operations with Positive Functionals
125(2)
6.4 Representations of Free Field Type
127(2)
6.5 Topological Tensor Algebras
129(5)
6.6 Exercises
134(1)
6.7 Notes
135(2)
7 Intert able Representations of Commutative *-Algebras
137(16)
7.1 Some Auxiliary Operator-Theoretic Results
137(2)
7.2 "Bad" Representations of the Polynomial Algebra C[ xi,*2]
139(3)
7.3 Integrable Representations of Commutative *-Algebras
142(6)
7.4 Spectral Measures of Integrable Representations
148(3)
7.5 Exercises
151(1)
7.6 Notes
152(1)
8 The Weyl Algebra and the Canonical Commutation Relation
153(34)
8.1 The Weyl Algebra
154(3)
8.2 The Operator Equation AA* =A*A + I
157(5)
8.3 The Bargmann-Fock Representation of the Weyl Algebra
162(2)
8.4 The Schrodinger Representation of the Weyl Algebra
164(5)
8.5 The Stone-von Neumann Theorem
169(4)
8.6 A Resolvent Approach to Schrodinger Pairs
173(3)
8.7 The Uncertainty Principle
176(2)
8.8 The Groenewold-van Hove Theorem
178(5)
8.9 Exercises
183(1)
8.10 Notes
184(3)
9 Integrable Representations of Enveloping Algebras
187(38)
9.1 Preliminaries on Lie Groups and Enveloping Algebras
188(1)
9.2 Infinitesimal Representations of Unitary Representations
189(7)
9.3 The Graph Topology of the Infinitesimal Representation
196(3)
9.4 Elliptic Elements
199(11)
9.4.1 Preliminaries on Elliptic Operators
199(5)
9.4.2 Main Results on Elliptic Elements
204(1)
9.4.3 Applications of Elliptic Elements
205(5)
9.5 Two Examples
210(2)
9.6 Analytic Vectors
212(5)
9.6.1 Analytic Vectors for Single Operators
213(2)
9.6.2 Analytic Vectors for Unitary Representations
215(1)
9.6.3 Exponentiation of Representations of Enveloping Algebras
216(1)
9.7 Analytic Vectors and Unitary Representations of SL(2,R)
217(4)
9.8 Exercises
221(1)
9.9 Notes
222(3)
10 Archimedean Quadratic Modules and Positivstellensatze
225(26)
10.1 Archimedean Quadratic Modules and Bounded Elements
225(6)
10.2 Representations of *-Algebras with Archimedean Quadratic Modules
231(2)
10.3 Stellensatze for Archimedean Quadratic Modules
233(2)
10.4 Application to Matrix Algebras of Polynomials
235(2)
10.5 A Bounded *-Algebra Related to the Weyl Algebra
237(4)
10.6 A Positivstellensatz for the Weyl Algebra
241(3)
10.7 A Theorem About the Closedness of the Cone A2
244(4)
10.8 Exercises
248(1)
10.9 Notes
249(2)
11 The Operator Relation XX* = F(X*X)
251(32)
11.1 A Prelude: Power Partial Isometries
252(2)
11.2 The Operator Relation AB = BF(A)
254(4)
11.3 Strong Solutions of the Relation XX* = F(X*X)
258(3)
11.4 Finite-Dimensional Representations
261(4)
11.5 Infinite-Dimensional Representations
265(5)
11.6 The Hermitian Quantum Plane
270(3)
11.7 The q-Oscillator Algebra
273(3)
11.8 The Real Quantum Plane
276(4)
11.9 Exercises
280(1)
11.10 Notes
281(2)
12 Induced *-Representations
283(18)
12.1 Conditional Expectations
283(5)
12.2 Induced *-Representations
288(4)
12.3 Induced Representations of Group Graded *-Algebras from Hermitian Characters
292(5)
12.4 Exercises
297(2)
12.5 Notes
299(2)
13 Well-Behaved Representations
301(18)
13.1 Well-Behaved Representations of Some Group Graded *-Algebras
302(2)
13.2 Representations Associated with *-Algebras of Fractions
304(6)
13.3 Application to the Weyl Algebra
310(2)
13.4 Compatible Pairs of *-Algebras
312(2)
13.5 Application to Enveloping Algebras
314(2)
13.6 Exercises
316(1)
13.7 Notes
317(2)
14 Representations on Rigged Spaces and Hilbert C* -Modules
319(28)
14.1 Rigged Spaces
319(5)
14.2 Weak Imprimitivity Bimodules
324(4)
14.3 Positive Semi-definite Riggings
328(3)
14.4 Imprimitivity Bimodules
331(3)
14.5 Hilbert C*-modules
334(4)
14.6 Representations on Hilbert C*-modules
338(6)
14.7 Exercises
344(1)
14.8 Notes
344(3)
Appendix A Unbounded Operators on Hilbert Space 347(6)
Appendix B C*-Algebras and Representations 353(4)
Appendix C Locally Convex Spaces and Separation of Convex Sets 357(6)
References 363(10)
Index 373(6)
Symbol Index 379
Konrad Schmüdgen is Emeritus Professor at the Mathematical Institute of the University of Leipzig. He has worked for decades on unbounded representations and made important contributions. Among these are trace representation theorems for linear functionals, noncommutative Positivstellensätze, results on the transition probability, the theory of induced and well-behaved representations and classifications results of representations of special classes of algebras. He is the author of several books, including the Graduate Texts in Mathematics Unbounded Self-adjoint Operators on Hilbert Space (2012) and The Moment Problem (2017).
Daugiau apie elektronines knygas Daugiau apie elektronines knygas
Pastovi nuoroda: https://www.kriso.lt/db/97830304636636e.html
Raktažodžiai: