CONTENTS xi 5.2.1 Algebraic noncommutative probability spaces and laws 328 5.2.2 C*-probability spaces and the weak-*topology 332 5.2.3 W*-probability spaces 341 5.3 Free independence 351 5.3.1 Independence and free independence 351 5.3.2 Free independence and combinatorics 356 5.3.3 Consequence of free independence:free convolution 362 5.3.4 Free central limit theorem 371 5.3.5 Freeness for unbounded variables 372 5.4 Link with random matrices 377 5.5 Convergence of the operator norm of polynomials of inde- pendent GUE matrices 396 5.6 Bibliographical Notes 412 Appendices 417 A Linear algebra preliminaries 417 A.1 Identities and bounds 417 A.2 Perturbations for normal and Hermitian matrices 418 A.3 Noncommutative Matrix LP-norms 419 A.4 Brief review of resultants and discriminants 420 B Topological Preliminaries 421 B.1 Generalities 421 B.2 Topological Vector Spaces and Weak Topologies 424 B.3 Banach and Polish Spaces 425 B.4 Some elements of analysis 426 Probability measures on Polish spaces 427 C.1 Generalities 427 C.2 Weak Topology 429 D Basic notions of large deviations 431 E The skew field H of quaternions,and matrix theory over F 434 E.1 Matrix terminology over F,and factorization theorems435CONTENTS xi 5.2.1 Algebraic noncommutative probability spaces and laws 328 5.2.2 C ∗ - probability spaces and the weak-* topology 332 5.2.3 W∗ - probability spaces 341 5.3 Free independence 351 5.3.1 Independence and free independence 351 5.3.2 Free independence and combinatorics 356 5.3.3 Consequence of free independence: free convolution 362 5.3.4 Free central limit theorem 371 5.3.5 Freeness for unbounded variables 372 5.4 Link with random matrices 377 5.5 Convergence of the operator norm of polynomials of inde￾pendent GUE matrices 396 5.6 Bibliographical Notes 412 Appendices 417 A Linear algebra preliminaries 417 A.1 Identities and bounds 417 A.2 Perturbations for normal and Hermitian matrices 418 A.3 Noncommutative Matrix L p -norms 419 A.4 Brief review of resultants and discriminants 420 B Topological Preliminaries 421 B.1 Generalities 421 B.2 Topological Vector Spaces and Weak Topologies 424 B.3 Banach and Polish Spaces 425 B.4 Some elements of analysis 426 C Probability measures on Polish spaces 427 C.1 Generalities 427 C.2 Weak Topology 429 D Basic notions of large deviations 431 E The skew field H of quaternions, and matrix theory over F 434 E.1 Matrix terminology over F, and factorization theorems 435