Operator-Valued Chordal Loewner Chains and Non-Commutative Probability David A. Jekel University of California, Los Angeles Extended Probabilistic Operator Algebras Seminar, November 2017 David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 1 / 43
Introduction David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 2 / 43
Chordal Loewner Chains Definition A normalized chordal Loewner chain on [ 0 , T ] is a family of analytic functions F t ∶ H → H such that F 0 ( z ) = z . The F t ’s are analytic in a neighborhood of ∞ . If F t ( z ) = z + t / z + O ( 1 / z 2 ) . For s < t , we have F t = F s ◦ F s , t for some F s , t ∶ H → H . Fact The F t ’s are conformal maps from H onto H \ K t , where K t is a growing compact region touching the real line, e.g. a growing slit. David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 3 / 43
Chordal Loewner Chains Theorem (Bauer 2005) Every normalized Loewner chain satisfies the generalized Loewner equation ∂ t F t ( z ) = D z F t ( z ) ⋅ V ( z , t ) where V ( z , t ) is some vector field of the form V ( z , t ) = − G ν t ( z ) . Conversely, given such a vector field, the Loewner equation has a unique solution. History Loewner chains in the disk were studied by Loewner in 1923 in the case F t maps D onto D minus a slit. Kufarev and Pommerenke considered more general Loewner chains in the disk. Loewner chains in the half-plane were studied by Schramm in the case V ( z , t ) = − 1 /( z − B t ) where B t is a Brownian motion (SLE). David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 4 / 43
Loewner Chains and Free Probability Theorem (Voiculescu, Biane) If X and Y are freely independent, then G X + Y = G X ◦ F for some analytic F ∶ H → H . Observation (Bauer 2004, Schleißinger 2017) If X t is a process with freely independent increments, and if E ( X t ) = 0 and E ( X 2 t ) = t , then F X t ( z ) = 1 / G X t ( z ) is a normalized chordal Loewner chain. Remark The converse is not true. In fact, if σ is the semicircle law and if F µ = F σ ◦ F σ , then µ cannot be written as σ ⊞ ν . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 5 / 43
Loewner Chains and Monotone Probability Theorem (Muraki 2000-2001) If X and Y are monotone independent, then F X + Y = F X ◦ F Y . Observation (Schleißinger 2017) If X t is a process with monotone independent increments, and if E ( X t ) = 0 and E ( X 2 t ) = t , then F t ( z ) = 1 / G X t ( z ) is a normalized chordal Loewner chain. Every normalized Loewner chain arises in this way. History The differential equation ∂ t F t ( z ) = DF t ( z )[ V ( z )] was studied earlier by Muraki and Hasebe, and Schleißinger connected it with the Loewner equation. David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 6 / 43
Overview Goal Adapt the theory of Loewner chains to the non-commutative upper half-plane H ( A ) for a C ∗ algebra A . Overview: 1 Background on operator-valued laws. 2 Loewner chains F t = F µ t and the Loewner equation. 3 Combinatorial computation of moments for µ t . 4 Central limit theorem describing behavior for large t . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 7 / 43
Operator-valued Laws and Cauchy Transforms David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 8 / 43
A -valued Probability Spaces Definition Let A be a C ∗ -algebra. An A -valued probability space ( B , E ) is a C ∗ algebra B ⊇ A together with a bounded, completely positive, unital, A -bimodule map E ∶ B → A , called the expectation . Definition A ⟨ X ⟩ denotes the ∗ -algebra generated by A and a non-commutating self-adjoint indeterminate X . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 9 / 43
A -valued Laws Definition A linear map µ ∶ A ⟨ X ⟩ → A is called a (bounded) law if 1 µ is a unital A -bimodule map. 2 µ is completely positive. 3 There exist C > 0 and M > 0 such that ∥ µ ( a 0 Xa 1 X . . . a n − 1 Xa n )∥ ≤ CM n ∥ a 0 ∥ . . . ∥ a n ∥ . Definition We call µ a (bounded) generalized law if it satisfies (2) and (3) but not necessarily (1). David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 10 / 43
A -valued Laws Definition For a generalized law µ , we define rad ( µ ) = inf { M > 0 ∶ ∃ C > 0 s.t. condition (3) is satisfied } . Theorem (Popa-Vinnikov 2013, Williams 2013) For a generalized law µ , there exists a C ∗ -algebra B , a ∗ -homomorphism π ∶ A ⟨ X ⟩ → B which is bounded on A , and a completely positive µ ◦ π and ∥ π ( X )∥ = rad ( µ ) . µ ∶ B → A such that µ = ˜ ˜ In particular, every law µ is realized as the law of a self-adjoint π ( X ) in a probability space ( B , ˜ µ ) . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 11 / 43
Matricial Upper Half-Plane Definition The matricial upper half-plane is defined by H ( n ) ( A ) = ⋃ { z ∈ M n ( A ) ∶ Im z ≥ ǫ } ǫ > 0 H ( A ) = { H ( n ) ( A )} n ≥ 1 . Definition A matricial analytic function on H ( A ) is a sequence of analytic functions F ( n ) ( z ) defined on H ( n ) ( A ) such that F preserves direct sums of matrices and conjugation by scalar matrices. David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 12 / 43
Cauchy Transforms Definition (Voiculescu) The Cauchy transform of a generalized law µ is defined by G ( n ) µ ( z ) = µ ⊗ id M n ( C ) [( z − X ⊗ 1 M n ( C ) ) − 1 ] . Theorem (Williams 2013, Williams-Anshelevich 2015) A matricial analytic function G ∶ H ( A ) → − H ( A ) is the Cauchy transform of a generalized law µ with rad ( µ ) ≤ M if and only if 1 G is matricial analytic. G ( z ) ∶ = G ( z − 1 ) extends to be matricial analytic on {∥ z ∥ < 1 / M } . ˜ 2 3 ∥ G ( n ) ( z )∥ ≤ C ǫ for ∥ z ∥ < 1 /( M + ǫ ) , where C ǫ is independent of n. G ( z ∗ ) = ˜ ˜ G ( z ) ∗ . 4 ˜ G ( 0 ) = 0 . 5 Also, µ is a generalized law if and only if lim z → 0 z − 1 ˜ G ( n ) ( z ) = 1 for each n. David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 13 / 43
A -valued Chordal Loewner Chains David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 14 / 43
Definition Definition An A -valued chordal Loewner chain on [ 0 , T ] is a family of matricial analytic functions F t ( z ) = F ( z , t ) on H ( A ) such that F 0 = id F t is the recriprocal Cauchy transform of an A -valued law µ t . If s < t , then F t = F s ◦ F s , t for some matricial analytic F s , t ∶ H ( A ) → H ( A ) . µ t ( X ) and µ t ( X 2 ) are continuous functions of t . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 15 / 43
Basic Properties Remark Loewner chains relate to free and monotone independence over A just as in the scalar case. Lemma F s , t is unique. F 0 , t = F t . F s , t ◦ F t , u = F s , u . F s , t is the F-transform of a law µ s , t . sup s , t rad ( µ s , t ) ≤ C rad ( µ T ) + C sup t ∥ µ t ( X )∥ . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 16 / 43
Basic Properties Lemma There exists a generalized law σ s , t such that F s , t ( z ) = z − µ s , t ( X ) − G σ s , t ( z ) . We have rad ( σ s , t ) ≤ 2 rad ( µ s , t ) and σ s , t ( 1 ) = µ s , t ( X 2 ) = µ t ( X 2 ) − µ s ( X 2 ) . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 17 / 43
Biholomorphicity Theorem Each F s , t is a biholomorphic map onto a matricial domain and the inverse is matricial analytic. Moreover, given ǫ > 0 , there exists δ > 0 depending only on ǫ and the modulus of continuity of t ↦ µ t ( X 2 ) , such that 1 Im z ≥ ǫ ⟹ ∥ DF s , t ( z ) − 1 ∥ ≤ 1 / δ . 2 Im z , Im z ′ ≥ ǫ ⟹ ∥ F s , t ( z ) − F s , t ( z ′ )∥ ≥ δ ∥ z − z ′ ∥ . Proof: By the inverse function theorem, it suffices to prove the estimates (1) and (2). Renormalize so that µ t has mean zero. David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 18 / 43
Biholomorphicity Fix ǫ > 0. If t − s is small, then F s , t ( z ) ≈ z because F s , t ( z ) − z = G σ s , t ( z ) = O ( γ ) , where γ = ǫ − 1 ∥ σ s , t ( 1 )∥ = ǫ − 1 ∥ µ t ( X 2 ) − µ s ( X 2 )∥ , which goes to zero as t − s → 0. Similar estimates show that DF s , t ( z ) = id + O ( γ ) and F s , t ( z ) − F s , t ( z ′ ) = z − z ′ + O ( γ ∥ z − z ′ ∥) . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 19 / 43
Biholomorphicity Hence, the claims hold when t − s is sufficiently small. The claims hold for arbitrary s < t using iteration: F s , t = F s , t 1 ◦ F t 1 , t 2 ◦ ⋅ ⋅ ⋅ ◦ F t n − 1 , t and each function maps { Im z ≥ ǫ } into { Im z ≥ ǫ } . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 20 / 43
The Loewner Equation David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 21 / 43
The Loewner Equation The operator-valued version of the Loewner equation is ∂ t F ( z , t ) = DF ( z , t )[ V ( z , t )] , where DF ( z , t ) is the Fr´ echet derivative with respect to z , and V ( z , t ) is a vector field of the form V ( z , t ) = − G ν t ( z ) for a generalized law ν t . We want to show that the Loewner equation defines a bijection between Loewner chains F ( z , t ) and Herglotz vector fields V ( z , t ) on [ 0 , T ] . David A. Jekel (UCLA) Operator-valued Loewner Chains EPOAS 2017 22 / 43
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