General Notation.- 1 Prologue: The Algebraic Approach to Quantum Theories.- 2 ∗-Algebras.- 3 O*-Algebras.- 4 ∗-Representations.- 5 Positive Linear Functionals.- 6 Representations of Tensor Algebras.- 7 Integrable Representations of Commutative ∗-Algebras.- 8 The Weyl Algebra and the Canonical Commutation Relation.- 9 Integrable Representations of Enveloping Algebras.- 10 Archimedean Quadratic Modules and Positivstellens?tze.-...