The first volume (General Theory) differs from most textbooks as it emphasizes the mathematical structure and mathematical rigor, while being adapted to the teaching the first semester of an advanced course in Quantum Mechanics (the content of the book are the lectures of courses actually delivered.). It differs also from the very few texts in Quantum Mechanics that give emphasis to the mathematical aspects because this book, being written as Lecture Notes, has the structure of lectures delivered in a course, namely introduction of the problem, outline of the relevant points, mathematical tools needed, theorems, proofs. This makes this book particularly useful for self-study and for instructors in the preparation of a second course in Quantum Mechanics (after a first basic course). With some minor additions it can be used also as a basis of a first course in Quantum Mechanics for students in mathematics curricula.

Elements of the history of Quantum Mechanics I.- Elements of the history of Quantum Mechanics II.- Axioms, states, observables, measurement, di¿culties.- Entanglement, decoherence, Bell's inequalities, alternative theories.- Automorphisms; Quantum dynamics; Theorems of Wigner, Kadison, Segal; Continuity andgenerators.- Operators on Hilbert spaces I; Basic elements.- Quadratic forms.- Properties of free motion, AnholonomyGeometric phase.- Elements of C *-algebras, GNS representationautomorphisms and dynamical systems.- Derivations and generators. K.M.S. condition. Elements of modular structure. Standard form.- Semigroups and dissipations. Markov approximation.- Quantum dynamical semigroups I.- Positivity preserving contraction semigroups on C *-algebras.- Conditional expectations.- Complete Dissipations.- Weyl system, Weyl algebra, lifting symplectic maps.- Magnetic Weyl algebra.- A Theorem of Segal.- Representations of Bargmann, Segal, Fock.- Second quantization.- Other quantizations (deformation, geometric).