xii CONTENTS E.2 The spectral theorem and key corollaries 437 E.3 A specialized result on projectors 438 E.4 Algebra for curvature computations 439 Manifolds 441 F1 Manifolds embedded in Euclidean space 442 F.2 Proof of the coarea formula 446 F.3 Metrics,connections,curvature,hessians,and the Laplace-Beltrami operator 449 G Appendix on Operator Algebras 454 G.1 Basic definitions 454 G.2 Spectral properties 456 G.3 States and positivity 458 G.4 von Neumann algebras 459 G.5 Noncommutative functional calculus 461 H Stochastic calculus notions 463 References 468 General Conventions 484xii CONTENTS E.2 The spectral theorem and key corollaries 437 E.3 A specialized result on projectors 438 E.4 Algebra for curvature computations 439 F Manifolds 441 F.1 Manifolds embedded in Euclidean space 442 F.2 Proof of the coarea formula 446 F.3 Metrics, connections, curvature, hessians, and the Laplace-Beltrami operator 449 G Appendix on Operator Algebras 454 G.1 Basic definitions 454 G.2 Spectral properties 456 G.3 States and positivity 458 G.4 von Neumann algebras 459 G.5 Noncommutative functional calculus 461 H Stochastic calculus notions 463 References 468 General Conventions 484