Lecture Notes on C Algebras and Quantum Mechanics Draft April NP Landsman Kortewegde Vries Institute for Mathematics University of Amsterdam Plantage Muidergracht TV AMSTERDAM THE NETHERLANDS email nplwinsuvanl homepage httpturingwinsuvanl npl telephone oce Euclides a CONTENTS Contents Historical notes Origins in functional analysis and quantum mechanics Rings of op erators von Neumann algebras Reduction of unitary group representations The classication of factors C algebras Elementary theory of C algebras Basic denitions Banach algebra basics CommutativeBanach algebras Commutative C algebras Sp ectrum and functional calculus Positivityin C algebras Ideals in C algebras States Representations and the GNSconstruction The GelfandNeumark theorem Complete p ositivity Pure states and irreducible representations The C algebra of compact op erators The double commutant theorem Hilb ert C mo dules and induced representations Vector bundles Hilb ert C mo dules The C algebra of a Hilb ert C mo dule Morita equivalence Rieel induction The imprimitivity theorem Group C algebras C dynamical systems and crossed pro ducts Transformation group C algebras The abstract transitive imprimitivit ytheorem Induced group representations Mackeys transitive imprimitivity theorem Applications to quantum mechanics The mathematical structure of classical and quantum mechanics Quantization Stinesprings theorem and coherent states Covariant lo calization in conguration space Covariantquantization on phase space Literature CONTENTS Historical notes Origins in functional analysis and quantum mechanics The emergence of the theory of op erator algebras may b e traced back to at least three develop ments The work of Hilb ert and his pupils in Gottingen on integral equations sp ectral theory and innitedimensional quadratic forms The discovery of quantum mechanics by Heisenb erg in Gottingen and indep endently bySchrodinger in Zuric h The arrival of John von Neumann in Gottingen to b ecome Hilb erts assistant Hilb erts memoirs on integral equations app eared b etween and In his studentE Schmidt dened the space in the mo dern sense F Riesz studied the space of all continuous linear maps on and various examples of L spaces emerged around the same time However the abstract concept of a Hilb ert space was still missing Heisenb erg discovered a form of quantum mechanics which at the time was called matrix hrodinger was led to a dierent formulation of the theory which he called wave mechanics Sc mechanics The relationship and p ossible equivalence b etween these alternativeformulations of quantum mechanics which at rst sight lo oked completely dierent was much discussed at the time It was clear from either approach that the b o dy of work mentioned in the previous paragraph was relevanttoquantum mechanics Heisenb ergs pap er initiating matrix mechanics was followed by the Dreimannerarb eit of Born Heisenb erg and Jordan all three were in Gottingen at that time Born was one of the few physicists of his time to be familiar with the concept of a matrix in previous research he had even used innite matrices Heisenbergs fundamental equations could only be satised by innitedimensional matrices Born turned to his former teacher Hilb ert for mathematical advice Hilb ert had been interested in the mathematical structure of physical theories for a long time his Sixth Problem called for the mathematical axiomatization of physics Aided by his assistants Nordheim and von Neumann Hilb ert thus ran a seminar on the mathematical structure of quantum mechanics and the three wrote a joint pap er on the sub ject now obsolete It was von Neumann alone who at the age of saw his way through all structures and mathematical diculties In a series of pap ers written b etween culminating in his b o ok Mathematische Grund lagen der Quantenmechanik he formulated the abstract concept of a Hilb ert space develop ed the sp ectral theory of b ounded as wellasunb ounded normal op erators on a Hilb ert space and proved the mathematical equivalence b etween matrix mechanics and wave mechanics Initiating and largely completing the theory of selfadjoint op erators on a Hilb ert space and intro ducing notions such as density matrices and quantum entropy this b o ok remains the denitive account of the mathematical structure of elementary quantum mechanics von Neumanns book was preceded by Diracs The Principles of Quantum Mechanics which contains a heuristic and mathematically unsatisfactory accountofquantum mechanics in terms of linear spaces and op erators Rings of op erators von Neumann algebras In one of his pap ers on Hilb ert space theory von Neumann denes a ring of op erators M nowadays called a von Neumann algebra as a subalgebra of the algebra BH of all b ounded op erators on a Hilb ert space H ie a subalgebra which is closed under the involution A A that is closed ie sequentially complete in the weak op erator top ology The latter may b e dened by its notion of convergence a sequence fA g of b ounded op erators weakly converges n to A when A A for all H This typeofconvergence is partly motivated by n quantum mechanics in whichA is the exp ectation value of the observable A in the state t and has unit norm provided that A is selfadjoin HISTORICAL NOTES For example BH is itself a von Neumann algebra Since the weak top ology is weaker than the uniform or norm top ology on BH avon Neumann algebra is automatically normclosed as well so that in terminology to b e intro duced later on a von Neumann algebra b ecomes a C algebra when one changes the top ology from the weak to the uniform one However the natural top ology on a von Neumann algebra is neither the weak nor the uniform one In the same pap er von Neumann proves what is still the basic theorem of the sub ject a subalgebra M of BH containing the unit op erator Iis weakly closed i M M Here the commutant M of a collection M of b ounded op erators consists of all b ounded op erators which commute with all elements of M and the bicommutant M is simply M This theorem is remarkable in relating a top ological condition to an algebraic one one is reminded of the much simpler fact that a linear subspace K of H is closed i K where K is the orthogonal complement of K in H Von Neumanns motivation in studying rings of op erators was plurifold His primary motivation probably came from quantum mechanics unlike many physicists then and even now he knew that all Hilb ert spaces of a given dimension are isomorphic so that one cannot characterize a physical system bysaying that its Hilb ert space of pure states is L R Instead von Neumann quantummechanical systems by algebraic conditions on the observables hop ed to characterize This programme has to some extent b een realized in algebraic quantum eld theory Haag and followers Among von Neumanns interest in quantum mechanics was the notion of entropy he wished n to dene states of minimal information When H C for n such a state is given by the density matrix Inbut for innitedimensional Hilb ert spaces this state may no longer be dened Density matrices maybe regarded as states on the von Neumann algebraBH in the sense of p ositive linear functionals which map I to As we shall see there are von Neumann algebras on innitedimensional Hilb ert spaces which do admit states of minimal information that generalize In viz the factors of typ e I I see b elow Furthermore von Neumann hop ed that the divergences in quantum eld theory mightbere moved by considering algebras of observables dierentfromBH This hop e has not materialized although in algebraic quantum eld theory the basic algebras of lo cal observables are indeed not of the form BH but are all isomorphic to the unique hyp ernite factor of typ e I I I see b elow Motivation from a dierent direction came from the structure theory of algebras In the present context a theorem of Wedderburn says that a von Neumann algebra on a nitedimensional Hilb ert space is isomorphic to a direct sum of matrix algebras Von Neumann wondered if this or a similar result in which direct sums are replaced by direct integrals see below still holds when the dimension of H is innite As we shall see it do es not ation came from group representations Von Neumanns bicom Finallyvon Neumanns motiv mutant theorem implies a useful alternativecharacterization of von Neumann algebras from now on we add to the