AN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRIES

By U. LAONL

CONTENTS INTRODUCTION NONCOMMUTATIVE SPACES AND ALGEBRAS OF FUNCTIONS ALGEBRAS COMMUTATIVE SPACES NONCOMMUTATIVE SPACES THE JACOBSON (OR HULL-KERNEL) TOPOLOGY COMPACT OPERATORS NONCOMMUTATIVE LATTICES THE TOPOLOGICAL APPROXIMATION ORDER AND TOPOLOGY HOW TO RECOVER THE SPACE BEING APPROXIMATED THE SPACE PRIMA AS A POSET AF-ALGEBRAS FROM BRATTELI DIAGRAMS TO NONCOMMUTATIVE LATTICES FROM NONCOMMUTATIVE LATTICES TO BRATTELI DIAGRAMS HOW TO RECOVER THE ALGEBRA BEING APPROXIMATED OPERATOR VALUED FUNCTIONS ON NONCOMMUTATIVE LATTICES MODULES AS BUNDLES MODULES PROJECTIVE MODULES OF FINITE TYPE HERMITIAN STRUCTURES OVER PROJECTIVE MODULES FEW ELEMENTS OF K-THEORY THE K-THEORY OF THE PENROSE TILING HIGHER ORDER K-GROUPS THE SPECTRAL CALCULUS IN?NITESIMALS THE DIXMIER TRACE WODZICKI RESIDUE AND CONNES’ TRACE THEOREM SPECTRAL TRIPLES THE CANONICAL TRIPLE OVER A MANIFOLD DISTANCE AND INTEGRAL FOR A SPECTRAL TRIPLE REAL SPECTRAL TRIPLES A TWO POINTS SPACE PRODUCTS AND EQUIVALENCE OF SPECTRAL TRIPLES NONCOMMUTATIVE DI?ERENTIAL FORMS UNIVERSAL DI?ERENTIAL FORMS THE UNIVERSAL ALGEBRA OF ORDINARY FUNCTIONS CONNES’ DI?ERENTIAL FORMS THE USUAL EXTERIOR ALGEBRA AGAIN THE TWO POINTS SPACE SCALAR PRODUCT OF FORMS CONNECTIONS ON MODULES ABELIAN GAUGE CONNECTIONS THE USUAL ELECTROMAGNETISM UNIVERSAL CONNECTIONS CONNECTIONS COMPATIBLE WITH HERMITIAN STRUCTURES THE ACTION OF THE GAUGE GROUP CONNECTIONS ON BIMODULES FIELD THEORIES ON MODULES YANG-MILLS MODELS THE USUAL GAUGE THEORY YANG-MILLS ON A TWO POINTS SPACE THE BOSONIC PART OF THE STANDARD MODEL THE BOSONIC SPECTRAL ACTION FERMIONIC MODELS FERMIONIC MODELS ON A TWO POINTS SPACE THE STANDARD MODEL THE FERMIONIC SPECTRAL ACTION GRAVITY MODELS GRAVITY `A LA CONNES-DIXMIER-WODZICKI SPECTRAL GRAVITY LINEAR CONNECTIONS THE USUAL EINSTEIN GRAVITY OTHER GRAVITY MODELS 10 QUANTUM MECHANICAL MODELS ON NONCOMMUTATIVE LATTICES