Mathematical quantization

Nik Weaver1584880015, 9781584880011, 9781420036237

With a unique approach and presenting an array of new and intriguing topics, Mathematical Quantization offers a survey of operator algebras and related structures from the point of view that these objects are quantizations of classical mathematical structures. This approach makes possible, with minimal mathematical detail, a unified treatment of a variety of topics.Detailed here for the first time, the fundamental idea of mathematical quantization is that sets are replaced by Hilbert spaces. Building on this idea, and most importantly on the fact that scalar-valued functions on a set correspond to operators on a Hilbert space, one can determine quantum analogs of a variety of classical structures. In particular, because topologies and measure classes on a set can be treated in terms of scalar-valued functions, we can transfer these constructions to the quantum realm, giving rise to C*- and von Neumann algebras.In the first half of the book, the author quickly builds the operator algebra setting. He uses this as a unifying theme in the second half, in which he treats several active research topics, some for the first time in book form. These include the quantum plane and tori, operator spaces, Hilbert modules, Lipschitz algebras, and quantum groups.For graduate students, Mathematical Quantization offers an ideal introduction to a research area of great current interest. For professionals in operator algebras and functional analysis, it provides a readable tour of the current state of the field.

---

Table of contents :
c0015_c000……Page 1
Mathematical Quantization……Page 3
Preface……Page 5
Contents……Page 8
1.1 Classical physics……Page 11
Table of Contents……Page 0
1.2 States and events……Page 12
1.3 Observables……Page 16
1.4 Dynamics……Page 19
1.5 Composite systems……Page 23
1.6 Quantum computation……Page 26
1.7 Notes……Page 27
2.1 Definitions and examples……Page 28
2.2 Subspaces……Page 32
2.3 Orthonormal bases……Page 37
2.4 Duals and direct sums……Page 40
2.5 Tensor products……Page 44
2.6 Quantum logic……Page 49
2.7 Notes……Page 52
3.1 Unitaries and projections……Page 53
3.2 Continuous functional calculus……Page 58
3.3 Borel functional calculus……Page 62
3.4 Spectral measures……Page 65
3.5 The bounded spectral theorem……Page 69
3.6 Unbounded operators……Page 71
3.7 The unbounded spectral theorem……Page 74
3.8 Stone’s theorem……Page 76
3.9 Notes……Page 80
4.1 Position and momentum……Page 81
4.2 The tracial representation……Page 85
4.3 Bargmann-Segal space……Page 87
4.4 Quantum complex analysis……Page 93
4.5 Notes……Page 97
5.1 The algebras C(X)……Page 99
5.2 Topologies from functions……Page 103
5.3 Abelian C*-algebras……Page 107
5.4 The quantum plane……Page 109
5.5 Quantum tori……Page 117
5.6 The GNS construction……Page 124
5.7 Notes……Page 131
6.1 The algebras l1(X)……Page 133
6.2 The algebras L1(X)……Page 136
6.3 Trace class operators……Page 139
6.4 The algebras B(H)……Page 143
6.5 Von Neumann algebras……Page 146
6.6 The quantum plane and tori……Page 151
6.7 Notes……Page 154
7.1 Fock space……Page 155
7.2 CCR algebras……Page 158
7.3 Relativistic particles……Page 163
7.4 Flat spacetime……Page 167
7.5 Curved spacetime……Page 169
7.6 Notes……Page 172
8.1 The spaces V (K)……Page 175
8.2 Matrix norms and convexity……Page 177
8.3 Duality……Page 184
8.4 Matrix-valued functions……Page 188
8.5 Operator systems……Page 192
8.6 Notes……Page 198
9.1 Continuous Hilbert bundles……Page 199
9.2 Hilbert L1-modules……Page 202
9.3 Hilbert C*-modules……Page 205
9.4 Hilbert W*-modules……Page 210
9.5 Crossed products……Page 216
9.6 Hilbert *-bimodules……Page 219
9.7 Notes……Page 225
10.1 The algebras Lip0(X)……Page 227
10.2 Measurable metrics……Page 234
10.3 The derivation theorem……Page 239
10.4 Examples……Page 244
10.5 Quantum Markov semigroups……Page 250
10.6 Notes……Page 256
11.1 Finite dimensional C*-algebras……Page 257
11.2 Finite quantum groups……Page 259
11.3 Compact quantum groups……Page 264
11.4 Haar measure……Page 268
11.5 Notes……Page 271
References……Page 273