Inoue A. — Tomita-Takesaki Theory in Algebras of Unbounded Operators (Lecture Notes in Mathematics) :: Электронная библиотека попечительского совета мехмата МГУ

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Inoue A. — Tomita-Takesaki Theory in Algebras of Unbounded Operators (Lecture Notes in Mathematics)

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Название: Tomita-Takesaki Theory in Algebras of Unbounded Operators (Lecture Notes in Mathematics)

Автор: Inoue A.

Аннотация:

These notes are devoted to a systematic study of developing the Tomita-Takesaki theory for von Neumann algebras in unbounded operator algebras called O*-algebras and to its applications to quantum physics. The notions of standard generalized vectors and standard weights for an O*-algebra are introduced and they lead to a Tomita-Takesaki theory of modular automorphisms. The Tomita-Takesaki theory in O*-algebras is applied to quantum moment problem, quantum statistical mechanics and the Wightman quantum field theory. This will be of interest to graduate students and researchers in the field of (unbounded) operator algebras and mathematical physics.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1998

Количество страниц: 248

Добавлена в каталог: 30.03.2014

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Предметный указатель

$\mathcal{M}$ -invariant subspace      15
(Quasi-)weight on $\mathcal{M}_{+}$       115
(Quasi-)weight on $\mathcal{P(M)}$       114
(Quasi-)weight on $\mathcal{P}(\pi_{\varphi}(\mathcal{M})^{''}_{w\sigma})$ induced by a quasi-weight on $\mathcal{P(M)}$       127
(Quasi-)weight, $\sigma$ -weakly continuous      130
(Quasi-)weight, $\varphi$ -absolutely continuous      143
(Quasi-)weight, $\varphi$ -dominated      143
(Quasi-)weight, $\varphi$ -singular      143
(Quasi-)weight, $\{\alpha_{t}\}$ -KMS      128
(Quasi-)weight, $\{\sigma^{\varphi}_{t}\}$ -invariant      148
(Quasi-)weight, $\{\sigma^{\varphi}_{t}\}$ -KMS      127
(Quasi-)weight, defined by a vector in $\mathcal{H}$       116
(Quasi-)weight, essentially standard      126
(Quasi-)weight, faithful      125
(Quasi-)weight, quasi-regular      120
(Quasi-)weight, quasi-standard      126
(Quasi-)weight, regular      120
(Quasi-)weight, semifinite      125
(Quasi-)weight, singular      120
(Quasi-)weight, standard      126
(Quasi-)weight, strongly faithful      141
*-representation      27
*-representation, algebraically self-adjoint      28
*-representation, closed      27
*-representation, essentially self-adjoint      28
*-representation, self-adjoint      28
*-representation, strongly positive      199
Absolutely convergent series      32
BCS-Bogolubov model      209
Canonical algebra for s-degree of freedom      196
Canonical representation of a trace class operator      31
Canonical TCP-operator      214
Causal complements      214
Closed double cones      214
Closed forward light cone      213
Closure of O-algebra      9
Commutants of generalized vectors      43
Commutants of O*-algebras, strong      12
Commutants of O*-algebras, unbounded      26
Commutants of O*-algebras, weak      12
Essentially self-adjoint $\mathcal{M}$ -invariant subspace      16
Extended W*-algebra (EW*-algebra)      26
Extension of (*)-representation      27
Extension of (quasi-)weights      143
Extension of generalized vectors      42
Extension of O-algebra      8
Generalized Connes cocycle theorem for (quasi-)weights      136
Generalized Connes cocycle theorem for generalized vector      92
Generalized H-bounded      218
Generalized Pedersen — Takesaki Radon — Nikodym theorem for (quasi-)weights      145
Generalized Pedersen — Takesaki Radon — Nikodym theorem for generalized vector      103
Generalized vector      42
Generalized vector, cyclic      42
Generalized vector, essentially standard      57
Generalized vector, full      48
Generalized vector, induced by a quasi-weight      124 125
Generalized vector, modular      58
Generalized vector, quasi-standard      57
Generalized vector, regular      44
Generalized vector, relative modular      94
Generalized vector, semifinite      88
Generalized vector, standard      57
Generalized vector, strongly cyclic      42
Generalized vector, strongly regular      44
Generalized vector, tracial      61
Generalized von Neumann algebra      27
GNS-construction for (quasi-)wights      115
GNS-construction for positive linear functionals      30
Graph topology      9 27
Heisenberg commutation relation      196
Hereditary positive subcone      115
Induced extension of O*-algebras      19
Induced topology      9
Induction of O*-algebras      15
Interacting Boson Model      206

Intertwing space      28
KMS-condition      54
Lebesgue decomposition theorem for weights      144
Minkowski space      212
Momentum operator      18
Net property      119
O*-algebra      8
O*-algebra, algebraically self-adjoint      10
O*-algebra, closed      9
O*-algebra, defined by a generalized vector      42
O*-algebra, essentially self-adjoint      10
O*-algebra, having the von Neumann density type property      153
O*-algebra, integrable      10
O*-algebra, QMP-solvable      177
O*-algebra, self-adjoint      10
O*-algebra, semifinite      70
O*-algebra, spatially semifinite      104
O*-algebra, standard      10
O*-algebra, symmetric      26
O-algebra      8
Poincare group      213
Poincare-equivalent      213
Position operator      18
Positive linear functional on *-algebras      29
Positive linear functional on O*-algebras      30
Positive linear functional, $\sigma$ -weakly continuous      130
Positive linear functional, centrally positive      219
Quantum moment problem      171
Radon — Nikodym theorem for (quasi-)weights      143
Radon — Nikodym theorem for positive linear functionals      153
Regular part of a vector      165
Representation      27
Schroedinger representation      4 9 197
Schwartz space      9
Singular part of a vector      165
Space-like separated      213
Strongly commuting self-adjoint operator      20
Strongly positive linear functional on CCR-algebras      199
Strongly positive linear functional on O*-algebra      30
System, cyclic and separating      44
System, essentially standard      56
System, full      48
System, generalized modular      109
System, generalized standard      107 109
System, modular      58
System, quasi-standard      57
System, regular      43
System, standard      52 56
System, strongly cyclic and separating      45
System, strongly regular      44
Topology on O*-algebras, $\lambda$ -topology      25
Topology on O*-algebras, $\rho$ -topology      25
Topology on O*-algebras, precompact      36
Topology on O*-algebras, quasi-uniform      24
Topology on O*-algebras, strong*, $\sigma$ -strong*      23
Topology on O*-algebras, strong, $\sigma$ -strong      23
Topology on O*-algebras, uniform      24
Topology on O*-algebras, weak, $\sigma$ -weak      23
Trio-commutants for a quasi-weight      120
Unbounded bicommutants of O*-algebras      26
Unbounded CCR-algebra      4 82 196
Unbounded Tomita algebra      204
Unbounded vector      177
Unitary equivalence      29
Vacuum state      213
Vector, cyclic      17
Vector, modular      61
Vector, quasi-standard      61
Vector, standard      61
Vector, strongly cyclic      17
Vector, ultra-cyclic      17
Wedge-regions      213
Weyl formulation of the commutation relations      198
Weyl representation      198
Wightman field      212
Wightman field, associated to a local net of von Neumann algebras      213

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