Distributions of functions in noncommuting random variables Serban - PowerPoint PPT Presentation

Distributions of functions in noncommuting random variables Serban T. Belinschi CNRS - Institut de Mathématiques de Toulouse COSy Canadian Operator Symposium 2020 25–29 May 2020, Fields Institute Toronto Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 1 / 19

Contents Noncommutative distributions 1 Noncommutative (joint) distributions Analytic transforms of noncommutative distributions Applications 2 Distributions of polynomials and analytic functions in noncommuting variables Freeness Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 2 / 19

Contents Noncommutative distributions 1 Noncommutative (joint) distributions Analytic transforms of noncommutative distributions Applications 2 Distributions of polynomials and analytic functions in noncommuting variables Freeness Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 3 / 19

Noncommutative probability The process of passing from “commutative” to “noncommutative” [insert object here] is (most) often done by switching the perspective from the [object] to some algebra of functions defined on the [object] , and trying to eliminate the commutativity assumption on that algebra. Noncommutative probability spaces generalize � ( L ∞ ([ 0 , 1 ] , d x ) , E [ · ] = · d x ) . Thus, noncommutative probability space = von Neumann algebra with state. Here we take a slightly (very slightly!) different approach: we assume that (spaces of) noncommutative functions are known, and we define Noncommutative distributions = linear functionals on spaces of noncommutative functions Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 4 / 19

Noncommutative probability The process of passing from “commutative” to “noncommutative” [insert object here] is (most) often done by switching the perspective from the [object] to some algebra of functions defined on the [object] , and trying to eliminate the commutativity assumption on that algebra. Noncommutative probability spaces generalize � ( L ∞ ([ 0 , 1 ] , d x ) , E [ · ] = · d x ) . Thus, noncommutative probability space = von Neumann algebra with state. Here we take a slightly (very slightly!) different approach: we assume that (spaces of) noncommutative functions are known, and we define Noncommutative distributions = linear functionals on spaces of noncommutative functions Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 4 / 19

Noncommutative (joint) distributions Classical distribution on R n = linear functional, continuous on some space of test functions on R n . Our class of noncommutative test functions is C � X 1 , . . . , X n � , the algebra of polynomials in n selfadjoint noncommuting indeterminates 1 (so X 1 , X 2 , . . . , X n satisfy no algebraic relation) A noncommutative distribution is a linear µ : C � X 1 , . . . , X n � → C 1 such that µ ( 1 ) = 1; µ is positive if µ ( P ∗ P ) ≥ 0 for all P ∈ C � X 1 , . . . , X n � ; 2 µ is bounded if for any P ∈ C � X 1 , . . . , X n � there is an R P > 0 such 3 that µ (( P ∗ P ) k ) < R 2 k P for all k ∈ N ; µ is tracial if µ ( PQ ) = µ ( QP ) for any P , Q ∈ C � X 1 , . . . , X n � . 4 The set of positive, bounded tracial distributions is denoted by Σ 0 . 1 For the rest of the talk, think n = 2! Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 5 / 19

Noncommutative (joint) distributions Classical distribution on R n = linear functional, continuous on some space of test functions on R n . Our class of noncommutative test functions is C � X 1 , . . . , X n � , the algebra of polynomials in n selfadjoint noncommuting indeterminates 1 (so X 1 , X 2 , . . . , X n satisfy no algebraic relation) A noncommutative distribution is a linear µ : C � X 1 , . . . , X n � → C 1 such that µ ( 1 ) = 1; µ is positive if µ ( P ∗ P ) ≥ 0 for all P ∈ C � X 1 , . . . , X n � ; 2 µ is bounded if for any P ∈ C � X 1 , . . . , X n � there is an R P > 0 such 3 that µ (( P ∗ P ) k ) < R 2 k P for all k ∈ N ; µ is tracial if µ ( PQ ) = µ ( QP ) for any P , Q ∈ C � X 1 , . . . , X n � . 4 The set of positive, bounded tracial distributions is denoted by Σ 0 . 1 For the rest of the talk, think n = 2! Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 5 / 19

Noncommutative (joint) distributions Classical distribution on R n = linear functional, continuous on some space of test functions on R n . Our class of noncommutative test functions is C � X 1 , . . . , X n � , the algebra of polynomials in n selfadjoint noncommuting indeterminates 1 (so X 1 , X 2 , . . . , X n satisfy no algebraic relation) A noncommutative distribution is a linear µ : C � X 1 , . . . , X n � → C 1 such that µ ( 1 ) = 1; µ is positive if µ ( P ∗ P ) ≥ 0 for all P ∈ C � X 1 , . . . , X n � ; 2 µ is bounded if for any P ∈ C � X 1 , . . . , X n � there is an R P > 0 such 3 that µ (( P ∗ P ) k ) < R 2 k P for all k ∈ N ; µ is tracial if µ ( PQ ) = µ ( QP ) for any P , Q ∈ C � X 1 , . . . , X n � . 4 The set of positive, bounded tracial distributions is denoted by Σ 0 . 1 For the rest of the talk, think n = 2! Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 5 / 19

Noncommutative (joint) distributions Classical distribution on R n = linear functional, continuous on some space of test functions on R n . Our class of noncommutative test functions is C � X 1 , . . . , X n � , the algebra of polynomials in n selfadjoint noncommuting indeterminates 1 (so X 1 , X 2 , . . . , X n satisfy no algebraic relation) A noncommutative distribution is a linear µ : C � X 1 , . . . , X n � → C 1 such that µ ( 1 ) = 1; µ is positive if µ ( P ∗ P ) ≥ 0 for all P ∈ C � X 1 , . . . , X n � ; 2 µ is bounded if for any P ∈ C � X 1 , . . . , X n � there is an R P > 0 such 3 that µ (( P ∗ P ) k ) < R 2 k P for all k ∈ N ; µ is tracial if µ ( PQ ) = µ ( QP ) for any P , Q ∈ C � X 1 , . . . , X n � . 4 The set of positive, bounded tracial distributions is denoted by Σ 0 . 1 For the rest of the talk, think n = 2! Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 5 / 19

Noncommutative (joint) distributions Classical distribution on R n = linear functional, continuous on some space of test functions on R n . Our class of noncommutative test functions is C � X 1 , . . . , X n � , the algebra of polynomials in n selfadjoint noncommuting indeterminates 1 (so X 1 , X 2 , . . . , X n satisfy no algebraic relation) A noncommutative distribution is a linear µ : C � X 1 , . . . , X n � → C 1 such that µ ( 1 ) = 1; µ is positive if µ ( P ∗ P ) ≥ 0 for all P ∈ C � X 1 , . . . , X n � ; 2 µ is bounded if for any P ∈ C � X 1 , . . . , X n � there is an R P > 0 such 3 that µ (( P ∗ P ) k ) < R 2 k P for all k ∈ N ; µ is tracial if µ ( PQ ) = µ ( QP ) for any P , Q ∈ C � X 1 , . . . , X n � . 4 The set of positive, bounded tracial distributions is denoted by Σ 0 . 1 For the rest of the talk, think n = 2! Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 5 / 19

Realizing and encoding nc distributions As in classical probability: one can realize a given distribution µ ∈ Σ 0 as the distribution of a tuple of selfadjoint elements (“random variables”) x = ( x 1 , . . . , x n ) in a tracial C ∗ -algebra, here via the GNS construction with respect to � P , Q � µ = µ ( Q ∗ P ) . We write µ x when we view µ as the distribution of the variables x = ( x 1 , . . . , x n ) Convention: Upper case X j denote indeterminates, lower case x j denote random variables in a tracial C ∗ - or W ∗ -algebra. By linearity, the matrix of moments (or moment matrix) M ( µ ) given by ( X w ) ∗ X v � , v , w ∈ F + � M ( µ ) v , w = µ n , the free semigroup in n generators , encodes µ . (Note the similarity with the classical problem of moments.) Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 6 / 19

Realizing and encoding nc distributions As in classical probability: one can realize a given distribution µ ∈ Σ 0 as the distribution of a tuple of selfadjoint elements (“random variables”) x = ( x 1 , . . . , x n ) in a tracial C ∗ -algebra, here via the GNS construction with respect to � P , Q � µ = µ ( Q ∗ P ) . We write µ x when we view µ as the distribution of the variables x = ( x 1 , . . . , x n ) Convention: Upper case X j denote indeterminates, lower case x j denote random variables in a tracial C ∗ - or W ∗ -algebra. By linearity, the matrix of moments (or moment matrix) M ( µ ) given by ( X w ) ∗ X v � , v , w ∈ F + � M ( µ ) v , w = µ n , the free semigroup in n generators , encodes µ . (Note the similarity with the classical problem of moments.) Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 6 / 19

Contents Noncommutative distributions 1 Noncommutative (joint) distributions Analytic transforms of noncommutative distributions Applications 2 Distributions of polynomials and analytic functions in noncommuting variables Freeness Serban T. Belinschi (CNRS-IMT) Nc distributions Toronto 05/29/2020 7 / 19