An Invitation to Unbounded Representations of ∗-Algebras on Hilbert Space

Preface and Overview
Contents
General Notation
1 Prologue: The Algebraic Approach  to Quantum Theories
2 ast-Algebras
	2.1 ast-Algebras: Definitions and Examples
	2.2 Constructions with ast-Algebras
	2.3 Quadratic Modules
	2.4 Positive Functionals and States on Complex ast-Algebras
	2.5 Positive Functionals on Real ast-Algebras
	2.6 Characters of Unital Algebras
	2.7 Hermitian Characters of Unital ast-Algebras
	2.8 Hermitian and Symmetric ast-Algebras
	2.9 Exercises
	2.10 Notes
3 O*-Algebras
	3.1 Oast-Algebras and Their Graph Topologies
	3.2 Bounded Commutants
	3.3 Trace Functionals on O*-Algebras
	3.4 The Mittag-Leffler Lemma
	3.5 Exercises
	3.6 Notes
4 ast-Representations
	4.1 Basic Concepts on ast-Representations
	4.2 Domains of Representations in Terms of Generators
	4.3 Invariant Subspaces and Reducing Subspaces
	4.4 The GNS Construction
	4.5 Examples of GNS Representations
	4.6 Positive Semi-definite Functions on Groups
	4.7 Pathologies with Unbounded Representations
	4.8 Exercises
	4.9 Notes
5 Positive Linear Functionals
	5.1 Ordering of Positive Functionals
	5.2 Orthogonal Positive Functionals
	5.3 The Transition Probability of Positive Functionals
	5.4 Examples of Transition Probabilities
	5.5 A Radon–Nikodym Theorem for Positive Functionals
	5.6 Extremal Decomposition of Positive Functionals
	5.7 Quadratic Modules and *-Representations
	5.8 Exercises
	5.9 Notes
6 Representations of Tensor Algebras
	6.1 Tensor Algebras
	6.2 Positive Functionals on Tensor Algebras
	6.3 Operations with Positive Functionals
	6.4 Representations of Free Field Type
	6.5 Topological Tensor Algebras
	6.6 Exercises
	6.7 Notes
7 Integrable Representations  of Commutative ast-Algebras
	7.1 Some Auxiliary Operator-Theoretic Results
	7.2 ``Bad'' Representations of the Polynomial Algebra mathbbC[x1,x2]
	7.3 Integrable Representations of Commutative ast-Algebras
	7.4 Spectral Measures of Integrable Representations
	7.5 Exercises
	7.6 Notes
8 The Weyl Algebra and the Canonical Commutation Relation
	8.1 The Weyl Algebra
	8.2 The Operator Equation AA*=A*A+ I
	8.3 The Bargmann–Fock Representation of the Weyl Algebra
	8.4 The Schrödinger Representation of the Weyl Algebra
	8.5 The Stone–von Neumann Theorem
	8.6 A Resolvent Approach to Schrödinger Pairs
	8.7 The Uncertainty Principle
	8.8 The Groenewold–van Hove Theorem
	8.9 Exercises
	8.10 Notes
9 Integrable Representations of Enveloping Algebras
	9.1 Preliminaries on Lie Groups and Enveloping Algebras
	9.2 Infinitesimal Representations of Unitary Representations
	9.3 The Graph Topology of the Infinitesimal Representation
	9.4 Elliptic Elements
		9.4.1 Preliminaries on Elliptic Operators
		9.4.2 Main Results on Elliptic Elements
		9.4.3 Applications of Elliptic Elements
	9.5 Two Examples
	9.6 Analytic Vectors
		9.6.1 Analytic Vectors for Single Operators
		9.6.2 Analytic Vectors for Unitary Representations
		9.6.3 Exponentiation of Representations of Enveloping Algebras
	9.7 Analytic Vectors and Unitary Representations  of SL(2,mathbbR)
	9.8 Exercises
	9.9 Notes
10 Archimedean Quadratic Modules and Positivstellensätze
	10.1 Archimedean Quadratic Modules and Bounded Elements
	10.2 Representations of *-Algebras with Archimedean Quadratic Modules
	10.3 Stellensätze for Archimedean Quadratic Modules
	10.4 Application to Matrix Algebras of Polynomials
	10.5 A Bounded *-Algebra Related to the Weyl Algebra
	10.6 A Positivstellensatz for the Weyl Algebra
	10.7 A Theorem About the Closedness of the Cone sumA2
	10.8 Exercises
	10.9 Notes
11 The Operator Relation XX*=F(X*X)
	11.1 A Prelude: Power Partial Isometries
	11.2 The Operator Relation AB=BF(A)
	11.3 Strong Solutions of the Relation XX*=F(X*X)
	11.4 Finite-Dimensional Representations
	11.5 Infinite-Dimensional Representations
	11.6 The Hermitian Quantum Plane
	11.7 The q-Oscillator Algebra
	11.8 The Real Quantum Plane
	11.9 Exercises
	11.10 Notes
12 Induced ast-Representations
	12.1 Conditional Expectations
	12.2 Induced ast-Representations
	12.3 Induced Representations of Group Graded *-Algebras from Hermitian Characters
	12.4 Exercises
	12.5 Notes
13 Well-Behaved Representations
	13.1 Well-Behaved Representations of Some Group Graded ast-Algebras
	13.2 Representations Associated with *-Algebras of Fractions
	13.3 Application to the Weyl Algebra
	13.4 Compatible Pairs of *-Algebras
	13.5 Application to Enveloping Algebras
	13.6 Exercises
	13.7 Notes
14 Representations on Rigged Spaces  and Hilbert C*-Modules
	14.1 Rigged Spaces
	14.2 Weak Imprimitivity Bimodules
	14.3 Positive Semi-definite Riggings
	14.4 Imprimitivity Bimodules
	14.5 Hilbert C*-modules
	14.6 Representations on Hilbert C*-modules
	14.7 Exercises
	14.8 Notes
Appendix A Unbounded Operators on Hilbert Space
Appendix B C*-Algebras and Representations
Appendix C Locally Convex Spaces and Separation  of Convex Sets
Appendix  References
Index
Symbol Index