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									Free holomorphic functions on noncommutative domains
   Free biholomorphic functions and Cartan type results
Biholomorphic classification of noncommutative domains
      Isomorphisms of noncommutative Hardy algebras




                 Free biholomorphic classification of
                     noncommutative domains

                                            G ELU P OPESCU

                                   University of Texas at San Antonio


                                              August, 2010




                                       G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results
 Biholomorphic classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Plan


          Free holomorphic functions on noncommutative domains
          Free biholomorphic functions and noncommutative Cartan
          type results
          Free biholomorphic classification of noncommutative
          domains
          Isomorphisms of noncommutative Hardy algebras


          Reference : Free biholomorphic classification of
          noncommutative domains, Int. Math. Res. Not., in press.



                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
   Free biholomorphic functions and Cartan type results   Universal models
Biholomorphic classification of noncommutative domains     Free holomorphic functions
      Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Noncommutative Reinhardt domains

         F+ is the unital free semigroup on n generators g1 , . . . , gn
          n
         and the identity g0 .
         |α| stands for the length of the word α ∈ F+ .
                                                    n

         If X := (X1 , . . . , Xn ) ∈ B(H)n , we set Xα := Xi1 · · · Xik if
         α := gi1 · · · gik ∈ F+ , and Xg0 := I.
                                 n
                   ∞
         f :=                    aα Xα is a free holomorphic function on a ball
                  k =1 |α|=k
                                                                                      1/2k
         [B(H)n ]γ , γ > 0, if lim supk →∞                         |α|=k    |aα |2            < ∞.

         f is called positive regular free holomorphic function if
         aα ≥ 0, agi = 0, i = 1, . . . , n.

                                       G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains        Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results     Universal models
 Biholomorphic classification of noncommutative domains       Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras         Composition of free holomorphic functions


Noncommutative Reinhardt domains
          Given m, n ∈ N := {1, 2, . . .} and a positive regular free
          holomorphic function f , define the noncommutative domain
           Dm (H) := X ∈ B(H)n : (id − Φf ,X )k (I) ≥ 0, 1 ≤ k ≤ m ,
            f

          where Φf ,X : B(H) → B(H) is defined by
                                            ∞
                                                                   ∗
                      Φf ,X (Y ) :=                        aα Xα YXα ,             Y ∈ B(H),
                                           k =1 |α|=k

          and the convergence is in the week operator topology.
          Dm (H) can be seen as a noncommutative Reinhardt
           f
          domain, i.e.,
                                     (eiθ1 X1 , . . . , eiθn Xn ) ∈ Dm (H),
                                                                     f
          for X ∈ Dm (H) and θi ∈ R.
                   f
                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Noncommutative Reinhardt domains

          If m = 1, p = X1 + · · · + Xn , then D1 (H) coincides with
                                                p

                                                                  ∗               ∗
              [B(H)n ]1 := {(X1 , . . . , Xn ) :              X1 X1 + · · · + Xn Xn ≤ 1} .

          The study of [B(H)n ]1 has generated a free analogue of
                         s
          Sz.-Nagy–Foia¸ theory.
          Frazho, Bunce, Popescu, Arias-Popescu,
          Davidson-Pitts-Katsoulis, Ball-Vinnikov, and others.
          The domain D1 (H) was studied in
                      f

          G. Popescu, Operator theory on noncommutative domains,
          Mem. Amer. Math. Soc. 205 (2010), No.964, vi+124 pp.

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains          Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results       Universal models
 Biholomorphic classification of noncommutative domains         Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras           Composition of free holomorphic functions


Noncommutative Reinhardt domains
          The domain Dm (H), m ≥ 2, was considered in
                      f

          G. P OPESCU, Noncommutative Berezin transforms and
          multivariable operator model theory, J. Funct. Anal., 254
          (2008), 1003–1057.
          If q = X1 + · · · + Xn and m ≥ 1, then Dm (H) coincides with
                                                        q
          the set of all row contractions (X1 , . . . , Xn ) ∈ [B(H)n ]1
          satisfying the positivity condition
                                     m
                                                           m                ∗
                                           (−1)k                        Xα Xα ≥ 0.
                                                           k
                                    k =0                       |α|=k

          The elements of Dm (H) can be seen as multivariable
                           q
          noncommutative analogues of Agler’s m-hypercontractions
          (when n = 1, m ≥ 2, q = X )
                                        G ELU P OPESCU         Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains        Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results     Universal models
 Biholomorphic classification of noncommutative domains       Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras         Composition of free holomorphic functions


Universal model for Dm
                     f

          Let Hn be an n-dimensional complex Hilbert space with
          orthonormal basis e1 , e2 , . . . , en . The full Fock space of Hn
          is defined by
                                                                                ⊗k
                                        F 2 (Hn ) := C1 ⊕                      Hn .
                                                                        k ≥1

          The weighted left creation operators associated with
          Dm (H) are defined by setting Wi : F 2 (Hn ) → F 2 (Hn ),
           f

                                                           (m)
                                                       bα
                                    Wi eα =                      egi α ,       α ∈ F+ ,
                                                                                    n
                                                        (m)
                                                       bgi α
                         (m)
          where bg0 = 1 and
                                        G ELU P OPESCU           Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains        Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results     Universal models
 Biholomorphic classification of noncommutative domains       Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras         Composition of free holomorphic functions


Universal model for Dm
                     f


                     |α|
        (m)                                                       j +m−1
       bα =                                      aγ1 · · · aγj                             if |α| ≥ 1,
                               γ1 ···γj =α
                                                                    m−1
                    j=1
                           |γ1 |≥1,...,|γj |≥1


  where aα are the coefficients of f .

          (W1 , . . . , Wn ) is the universal model for Dm .
                                                         f
          The domain algebra An (Dm ) associated with the
                                         f
          noncommutative domain Dm is the norm closure of all
                                           f
          polynomials in W1 , . . . , Wn , and the identity.
                                  ∞
          The Hardy algebra Fn (Dm ) is the SOT-(resp. WOT-, w ∗ -)
                                       f
          closure of all polynomials in W1 , . . . , Wn , and the identity.
                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains        Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results     Universal models
 Biholomorphic classification of noncommutative domains       Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras         Composition of free holomorphic functions


Universal model for Dm
                     f


                     |α|
        (m)                                                       j +m−1
       bα =                                      aγ1 · · · aγj                             if |α| ≥ 1,
                               γ1 ···γj =α
                                                                    m−1
                    j=1
                           |γ1 |≥1,...,|γj |≥1


  where aα are the coefficients of f .

          (W1 , . . . , Wn ) is the universal model for Dm .
                                                         f
          The domain algebra An (Dm ) associated with the
                                         f
          noncommutative domain Dm is the norm closure of all
                                           f
          polynomials in W1 , . . . , Wn , and the identity.
                                  ∞
          The Hardy algebra Fn (Dm ) is the SOT-(resp. WOT-, w ∗ -)
                                       f
          closure of all polynomials in W1 , . . . , Wn , and the identity.
                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Universal model for Dm
                     f



          Assumptions :
              (i) H is a separable infinite dimensional Hilbert space ;
             (ii) Dm (H) is closed in the operator norm topology ;
                   f
            (iii) Dm (H) is starlike domain, i.e.
                   f

                                        r Dm (H) ⊂ Dm (H),
                                           f        f                     r ∈ [0, 1).

          Examples of closed starlike domains :
             (i) D1 (H) ;
                  f
            (ii) Dm (H) if p = a1 X1 + . . . + an Xn , ai > 0.
                  p




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains         Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results      Universal models
 Biholomorphic classification of noncommutative domains        Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras          Composition of free holomorphic functions


Free holomorphic functions
          The radial part of Dm (H) is defined by
                              f

                                       Dm (H) :=
                                        f ,rad                         r Dm (H).
                                                                          f
                                                              0≤r <1

          if q is any positive regular noncommutative polynomial,
          then
          Int(D1 (H)) = D1 (H) and Int(D1 (H)) = D1 (H).
                 q          q,rad                 q       q
          A formal power series G := α∈F+ cα Zα , cα ∈ C, is called
                                                n
          free holomorphic function on Dm if its representation on
                                           f ,rad
          any Hilbert space H, i.e., G : Dm (H) → B(H) given by
                                           f ,rad
                                           ∞
                         G(X ) :=                          cα Xα ,         X ∈ Dm (H),
                                                                                f ,rad
                                         k =0 |α|=k

          is well-defined in the operator norm topology.
                                        G ELU P OPESCU         Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Free holomorphic functions




          The map G is called free holomorphic function on
          Dm (H).
           f ,rad


          Hol(Dm ) denotes the algebra of all free holomorphic
                 f ,rad
          functions on Dm .
                        f ,rad




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Free holomorphic functions

  Theorem
  Let G := α∈F+ cα Zα be a formal power series and let H be a
                 n
  separable infinite dimensional Hilbert space. Then the following
  statements are equivalent :
     (i) G is a free holomorphic function on Dm .
                                              f ,rad
    (ii) For any r ∈ [0, 1), the series
                                                               ∞
                             G(rW1 , . . . , rWn ) :=                        r |α| cα Wα
                                                              k =0 |α|=k

          is convergent in the operator norm topology, where
          (W1 , . . . , Wn ) is the universal model associated with Dm .
                                                                     f


                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Free holomorphic functions

   (iii) The inequality
                                                                            1
                                                                           2k
                                                           1
                                   lim sup               (m)
                                                                |cα |2          ≤ 1,
                                                  |α|=k bα
                                     k →∞


          holds.
                                                               ∞
   (iv) For any r ∈ [0, 1), the series                         k =0        |α|=k    r |α| cα Wα is
        convergent.
                                                                      ∞
    (v) For any X ∈ Dm (H), the series
                     f ,rad                                           k =0         |α|=k    cα Xα is
        convergent.


                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains         Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results      Universal models
 Biholomorphic classification of noncommutative domains        Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras          Composition of free holomorphic functions


Free holomorphic functions

          Connection between the theory of free holomorphic
          functions on noncommutative domains Dm and the
                                                 f ,rad
          theory of holomorphic functions on domains in Cd .

  Remark
                                          ∞
  If p ∈ N and F (X ) :=                                   cα Xα is a free holomorphic
                                        k =0 |α|=k
  function on Dm (H), then its representation on Cp , i.e., the
               f ,rad
  map
               2                                        2
            Cnp ⊃ Dm (Cp ) Λ → F (Λ) ∈ Mp×p ⊂ Cp
                      f ,rad

  is a holomorphic function on the interior of Dm (Cp ).
                                                f


                                        G ELU P OPESCU        Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains         Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results      Universal models
 Biholomorphic classification of noncommutative domains        Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras          Composition of free holomorphic functions


Free holomorphic functions

          Connection between the theory of free holomorphic
          functions on noncommutative domains Dm and the
                                                 f ,rad
          theory of holomorphic functions on domains in Cd .

  Remark
                                          ∞
  If p ∈ N and F (X ) :=                                   cα Xα is a free holomorphic
                                        k =0 |α|=k
  function on Dm (H), then its representation on Cp , i.e., the
               f ,rad
  map
               2                                        2
            Cnp ⊃ Dm (Cp ) Λ → F (Λ) ∈ Mp×p ⊂ Cp
                      f ,rad

  is a holomorphic function on the interior of Dm (Cp ).
                                                f


                                        G ELU P OPESCU        Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains        Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results     Universal models
 Biholomorphic classification of noncommutative domains       Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras         Composition of free holomorphic functions


Free holomorphic functions

          When p = 1, the interior Int(Dm (C)) is a Reinhardt domain
                                        f
          in Cn .
                                                                                                         2
          When p ≥ 2, Int(Dm (Cp )) are circular domains in Cnp .
                           f


          A(Dm ) denotes the set of all elements G in Hol(Dm )
             f ,rad                                        f ,rad
          such that the mapping

                                    Dm (H)
                                     f ,rad                X → G(X ) ∈ B(H)

          has a continuous extension to [Dm (H)]− = Dm (H).
                                          f ,rad     f
          A(Dm ) is a Banach algebra under pointwise multiplication
              f ,rad
          and the norm · ∞ and has a unital operator algebra
          structure.

                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains          Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results       Universal models
 Biholomorphic classification of noncommutative domains         Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras           Composition of free holomorphic functions


Free holomorphic functions

  Theorem
  The map Φ : A(Dm ) → An (Dm ) defined by
                 f ,rad     f
                                                          

                                Φ              cα Zα  :=                  cα Wα
                                       α∈F+
                                          n                        α∈F+
                                                                      n


  is a completely isometric isomorphism of operator algebras.
  Moreover, if G := α∈F+ cα Zα is a free holomorphic function on
                          n
  the domain Dm , then the following statements are equivalent :
                f ,rad
     (i) G ∈ A(Dm ) ;
                f ,rad
   (iii) G(rW1 , . . . , rWn ) := ∞
                                  k =0
                                                 |α|
                                       |α|=k cα r Wα is convergent in
         the operator norm topology as r → 1.

                                        G ELU P OPESCU         Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Free holomorphic functions



    (ii) there exists G ∈ An (Dm ) with G = B[G].
                               f

  In this case, Φ(G) = G = limr →1 G(rW1 , . . . , rWn ) and

                          Φ−1 (G) = G = B[G],                    G ∈ An (Dm ),
                                                                          f

  where B is the noncommutative Berezin transform associated
  with Dm .
        f




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains         Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results      Universal models
 Biholomorphic classification of noncommutative domains        Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras          Composition of free holomorphic functions


Free holomorphic functions


  Corollary
                                          ∞
  If p ∈ N and F (X ) :=                                   cα Xα is in A(Dm ), then its
                                                                          f ,rad
                                        k =0 |α|=k
  representation on Cp , i.e., the map
                            2                                                                 2
                     Cnp ⊃ Dm (Cp )
                            f                        Λ → F (Λ) ∈ Mp×p ⊂ Cp

  is a continuous map on Dm (Cp ) and holomorphic on the interior
                          f
  of Dm (Cp ).
       f




                                        G ELU P OPESCU        Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Composition of free holomorphic functions



  Theorem
  Let f and g be positive regular free holomorphic functions with
  n and p indeterminates, respectively, and let m, l ≥ 1.
  If F : Dlg,rad (H) → B(H) and Φ : Dm (H) → Dlg,rad (H) are free
                                      f ,rad
  holomorphic functions, then F ◦ Φ is a free holomorphic
  function on Dm (H).
                   f ,rad
  If, in addition, F is bounded, then F ◦ Φ is bounded and
    F ◦ Φ ∞ ≤ F ∞.




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Composition of free holomorphic functions


  Theorem
  Let f and g be positive regular free holomorphic functions with
  n and p indeterminates, respectively, and let m, l ≥ 1. If
  F : Dlg,rad (H) → B(H) and Φ : Dm (H) → Dlg (H) are bounded
                                   f ,rad
  free holomorphic functions which have continuous extensions
  to the noncommutative domains Dlg (H) and Dm (H),
                                                 f
  respectively, then F ◦ Φ is a bounded free holomorphic function
  on Dm (H) which has continuous extension to Dm (H).
       f ,rad                                        f




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains      Noncommutative Reinhardt domains
    Free biholomorphic functions and Cartan type results   Universal models
 Biholomorphic classification of noncommutative domains     Free holomorphic functions
       Isomorphisms of noncommutative Hardy algebras       Composition of free holomorphic functions


Composition of free holomorphic functions


  Moreover,
                             ˜
    (a) (F ◦ Φ)(X ) = BX BΦ [F ] , X ∈ Dm (H), where BX , BΦ are
                          ˜                                ˜
                                        f
        the noncommutative Berezin transforms ;
    (b) the model boundary function of F ◦ Φ satisfies

                             F ◦ Φ = lim F (r Φ1 , . . . , r Φp ) = BΦ [F ],
                                                                     ˜
                                            r →1

          where the convergence is in the operator norm.




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Free biholomorphic functions

          Let f and g be positive regular free holomorphic functions
          with n and q indeterminates, respectively, and let m, l ≥ 1.
          A map F : Dm (H) → Dlg (H) is called free biholomorphic
                        f
          function if F is a homeomorphism in the operator norm
          topology and F |Dm (H) and F −1 |Dl (H) are free
                             f ,rad                                g,rad

          holomorphic functions on Dm (H) and Dlg,rad (H),
                                    f ,rad
          respectively.
          Dm (H) and Dlg (H) are called free biholomorphic equivalent
           f
          and denote Dm Dlg .
                       f
          Bih(Dm , Dlg ) denotes the set of all the free biholomorphic
                f
          functions F : Dm (H) → Dlg (H).
                           f


                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Free biholomorphic functions
          Two domains Ω1 , Ω2 in Cd are called biholomorphic
          equivalent if there are holomorphic maps ϕ : Ω1 → Ω2 and
          ψ : Ω2 → Ω1 be such that ϕ ◦ ψ = idΩ2 and ψ ◦ ϕ = idΩ1 .

  Theorem
  Let f and g be positive regular free holomorphic functions with
  n and q indeterminates, respectively, and let m, l, p ≥ 1. If
  F : Dm (H) → Dlg (H) is a free biholomorphic function, then
        f
  n = q and its representation on Cp , i.e., the map
                         2                                                                  2
                   Cnp ⊃ Dm (Cp )
                          f                       Λ → F (Λ) ∈ Dlg (Cp ) ⊂ Cqp

  is a homeomorphism from Dm (Cp ) onto Dlg (Cp ) and a
                              f
  biholomorphic function from Int(Dm (Cp )) onto Int(Dlg (Cp )).
                                   f

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Free biholomorphic functions

          The theory of functions in several complex variables =⇒
          results on the classification of the noncommutative
          domains Dm (H).
                      f


  Corollary
  Let f and g be positive regular free holomorphic functions with
  n and q indeterminates, respectively, and let m, l ≥ 1. If n = q
  or there is p ∈ {1, 2, . . .} such that Int(Dm (Cp )) is not
                                               f
  biholomorphic equivalent to Int(Dlg (Cp )), then the
  noncommutative domains Dm (H) and Dlg (H) are not free
                                   f
  biholomorphic equivalent.


                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Free biholomorphic functions


          Since Int(Dm (C)) ⊂ Cn and Int(Dlg (C)) ⊂ Cq are Reinhardt
                        f
          domains which contain 0, Sunada’s result implies that
          there exists a permutation σ of the set {1, . . . , n} and
          scalars µ1 , . . . , µn > 0 such that the map

                               (z1 , . . . , zn ) → (µ1 zσ(1) , . . . , µn zσ(n) )

          is a biholomorphic map from Int(Dm (C)) onto Int(Dlg (C)).
                                           f

          Open question : Is there an analogue of Sunada’s result for
          the noncommutative domains Dm .f




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results



          N := (N1 , . . . , Nn ) ∈ B(H)n is called nilpotent if there is
          p ∈ N := {1, 2, . . .} such that Nα = 0 for any α ∈ F+ with
                                                                    n
          |α| = p.

          The nilpotent part of the noncommutative domain Dm (H) is
                                                           f
          defined by

                         Dm (H) := {N ∈ Dm (H) : N is nilpotent}.
                          f ,nil         f




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results

  Theorem
  Let f be a positive regular free holomorphic function with n
  indeterminates and let m ≥ 1. Let H1 , . . . , Hn be formal power
  series in n noncommuting indeterminates Z = (Z1 , . . . , Zn ) of
  the form
                                                       (i)
                      Hi (Z ) :=                      aα Zα ,          i = 1, . . . , n.
                                       k =2 |α|=k

  If F (Z ) := (Z1 + H1 (Z ), . . . , Zn + Hn (Z )) has the property that

                                      F (Dm (H)) ⊆ Dm (H)
                                          f ,nil    f ,nil

  for any Hilbert space H, then F (Z ) = Z .

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results
  Theorem
  Let f and g be positive regular free holomorphic functions with
  n indeterminates and let m, l ≥ 1. Let F = (F1 , . . . , Fn ) and
  G = (G1 , . . . , Gn ) be n-tuples of formal power series in n
  noncommuting indeterminates such that

                  F (0) = G(0) = 0                   and     F ◦ G = G ◦ F = id.

  If F (Dm (H)) = Dlg,nil (H) for any Hilbert space H, then F has
         f ,nil
  the form
                  F (Z1 , . . . , Zn ) = [Z1 , . . . , Zn ]U,
  where U is an invertible bounded linear operator on Cn .
                (f )             (f )
          (W1 , . . . , Wn ) is the universal model associated with Dm .
                                                                     f
                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results
  Theorem
  Let f and g be positive regular free holomorphic functions with
  n indeterminates and let m, l ≥ 1. Let F = (F1 , . . . , Fn ) and
  G = (G1 , . . . , Gn ) be n-tuples of formal power series in n
  noncommuting indeterminates such that

                  F (0) = G(0) = 0                   and     F ◦ G = G ◦ F = id.

  If F (Dm (H)) = Dlg,nil (H) for any Hilbert space H, then F has
         f ,nil
  the form
                  F (Z1 , . . . , Zn ) = [Z1 , . . . , Zn ]U,
  where U is an invertible bounded linear operator on Cn .
                (f )             (f )
          (W1 , . . . , Wn ) is the universal model associated with Dm .
                                                                     f
                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results     Free biholomorphic functions
 Biholomorphic classification of noncommutative domains       Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results
  Theorem
  Let f and g be positive regular free holomorphic functions with
  n and q indeterminates, respectively, and let m, l ≥ 1. A map
  F : Dm (H) → Dlg (H) is a free biholomorphic function with
        f
  F (0) = 0 if and only if n = q and F has the form

           F (X ) = [X1 , . . . , Xn ]U,                   X = (X1 , . . . , Xn ) ∈ Dm (H),
                                                                                     f

  where U is an invertible bounded linear operator on Cn such
  that
                    (f )         (f )
                  [W1 , . . . , Wn ]U ∈ Dlg (F 2 (Hn ))
  and
                                (g)               (g)
                           [W1 , . . . , Wn ]U −1 ∈ Dm (F 2 (Hn )).
                                                     f


                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results
          Characterization the unit ball of B(H)n among the
          noncommutative domains Dm (H), up to free
                                       f
          biholomorphisms.
  Corollary
  Let g be a positive regular free holomorphic function with q
  indeterminates and let l ≥ 1. Then the noncommutative domain
  Dlg (H) is biholomorphic equivalent to the unit ball [B(H)n ]1 if
  and only if q = n and there is an invertible operator U ∈ B(Cn )
  such that
                     [S1 , . . . , Sn ]U ∈ Dlg (F 2 (Hn ))
  and
                                   (g)               (g)
                              [W1 , . . . , Wn ]U −1 ∈ [B(H)n ]1 .

                                         G ELU P OPESCU    Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results    Free biholomorphic functions
 Biholomorphic classification of noncommutative domains      Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results
          Aut0 (Dm ) denotes the subgroup of all free holomorphic
                 f
          automorphisms of Dm (H) that fix the origin.
                               f

  Corollary
  A map Ψ : Dm (H) → Dm (H) is in the subgroup Aut0 (Dm ) if and
             f        f                               f
  only if

           Ψ(X ) = [X1 , . . . , Xn ]U,                    X = (X1 , . . . , Xn ) ∈ Dm (H),
                                                                                     f

  for some invertible operator U on Cn such that

                      [W1 , . . . , Wn ]U            and       [W1 , . . . , Wn ]U −1

  are in Dm (F 2 (Hn )).
          f

                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results       Free biholomorphic functions
 Biholomorphic classification of noncommutative domains         Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results
          The theory of functions in several complex variables =⇒
          results on the classification of the domains Dm (H).
                                                       f

  Theorem
  Let f and g be positive regular free holomorphic functions with
  n indeterminates and let m, l ≥ 1. Assume that there is
  p ∈ {1, 2, . . .} such that the domains Int(Dm (Cp )) and
                                                 f
  Int(Dlg (Cp )) are linearly equivalent and all the automorphisms
  of Int(Dm (Cp )) fix the origin.
            f
  Then Dm (H) and Dlg (H) are free biholomorphic equivalent if
           f
  and only if there is an invertible operator U ∈ B(Cn ) such that
                                      (f )              (f )
                                [W1 , . . . , Wn ]U ∈ Dlg (F 2 (Hn ))
                                (g)               (g)
                           [W1 , . . . , Wn ]U −1 ∈ Dm (F 2 (Hn )).
                                                     f
                                        G ELU P OPESCU         Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Free biholomorphic functions
 Biholomorphic classification of noncommutative domains     Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results




          Thullen’s theorem. If a bounded Reinhardt domain in C2
          has a biholomorphic map that does not fix the origin, then
          the domain is linearly equivalent to one of the following :
          polydisc, unit ball, or the so-called Thullen domain.




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results     Free biholomorphic functions
 Biholomorphic classification of noncommutative domains       Cartan type results
       Isomorphisms of noncommutative Hardy algebras


Cartan type results

  Corollary
  Let f and g be positive regular free holomorphic functions with
  2 indeterminates and let m, l ≥ 1. Assume that the Reinhardt
  domains Int(Dm (C)) and Int(Dlg (C)) are linearly equivalent but
                  f
  they are not linearly equivalent to either the polydisc, the unit
  ball, or any Thullen domain in C2 .
  Then the noncommutative domains Dm (H) and Dlg (H) are free
                                          f
  biholomorphic equivalent if and only if there is an invertible
  bounded linear operator U ∈ B(C2 ) such that
         (f )      (f )                                       (g)        (g)
   [W1 , W2 ]U ∈ Dlg (F 2 (H2 )),                          [W1 , W2 ]U −1 ∈ Dm (F 2 (H2 )).
                                                                             f




                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results    Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains      Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Noncommutative domain algebras
          If Φ : A(Dm ) → A(Dlg,rad ) is a unital algebra
                      f ,rad
          homomorphism, it induces a unique unital homomorphism
          Φ : An (Dm ) → Aq (Dlg ) such that the diagram
                    f
                                                            Φ
                                       An (Dm )
                                          f               −−
                                                           −−→        Aq (Dlg )
                                                                         
                                                                        
                                          B                             B
                                                            Φ
                                       A(Dm ) − − → A(Dlg,rad )
                                          f ,rad −−

          is commutative, i.e., ΦB = BΦ. The homomorphisms Φ
          and Φ uniquely determine each other by the formulas :
                          [Φ(χ)](X ) = BX [Φ(χ)],                      X ∈ Dlg,rad (H),
          and
                                   Φ(χ) = Φ(χ),                  χ ∈ An (Dm ).
                                                                          f
                                        G ELU P OPESCU      Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Noncommutative domain algebras


          Consider the closed operator systems
                                                              (f ) ∗
                                               (f
                                   Sf := span{Wα ) Wβ                  ; α, β ∈ F+ }
                                                                                 n

          and
                                            (g)              (g) ∗
                                Sg := span{Wα Wβ                       ; α, β ∈ F+ },
                                                                                 q
                            (f )             (f )             (g)              (g)
          where (W1 , . . . , Wn ) and (W1 , . . . , Wq ) are the
          universal models of Dm and Dlg , respectively.
                                 f




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results     Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains       Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Noncommutative domain algebras
          Let Φ : A(Dm ) → A(Dlg,rad ) be a unital completely
                      f ,rad
          isometric isomorphism. We say that Φ has completely
          contractive hereditary extension if the linear maps
          Φ∗ : Sf → Sg defined by
                                                (f ) ∗                                 (f ) ∗
                                (f
                            Φ∗ Wα ) Wβ                           (f
                                                           := Φ Wα ) Φ Wβ

          and (Φ−1 )∗ : Sg → Sf defined by
                                               (g) ∗                                            (g) ∗
                   (Φ−1 )∗ Wα Wβ
                            (g)
                                                           := Φ−1 Wα Φ−1 Wβ
                                                                   (g)


          are completely contractive.
          Consequently, the map Φ∗ : Sf → Sg is a unital completely
          isometric linear isomorphism which extends Φ.
                                        G ELU P OPESCU       Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains

  Theorem
  Let f and g be positive regular free holomorphic functions with
  n and q indeterminates, respectively, and let m, l ≥ 1. Then the
  following statements are equivalent :
     (i) Ψ : A(Dm ) → A(Dlg,rad ) is a unital completely isometric
                f ,rad
         isomorphism with completely contractive hereditary
         extension ;
    (ii) there is ϕ ∈ Bih(Dlg , Dm ) such that
                                 f

                                   Ψ(χ) = χ ◦ ϕ,                χ ∈ A(Dm ).
                                                                       f ,rad


  In this case, Ψ(χ) = Bϕ [χ], χ ∈ An (Dm ), where Bϕ is the
                  ˜     ˜ ˜ ˜           f           ˜
  noncommutative Berezin transform at ϕ.  ˜

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains

  In the particular case when m = l = 1, any unital completely
  isometric isomorphism has c.c. hereditary extension.

  Remark
  Let Ψ : A(Dm ) → A(Dlg,rad ) be a unital algebra
              f ,rad
  homomorphism. Then Ψ is a unital completely isometric
  isomorphism having completely contractive hereditary
  extension if and only if Ψ is a continuous homeomorphism such
  that
                   ˆ (f )         ˆ  (f )
                 (Ψ(W1 ), . . . , Ψ(Wn )) ∈ Dm (F 2 (Hn ))
                                             f

  and
                                   (g)                      (g)
                    (Ψ−1 (W1 ), . . . , Ψ−1 (Wq )) ∈ Dlg (F 2 (Hq )).
                     ˆ                  ˆ


                                         G ELU P OPESCU    Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains

  In the particular case when m = l = 1, any unital completely
  isometric isomorphism has c.c. hereditary extension.

  Remark
  Let Ψ : A(Dm ) → A(Dlg,rad ) be a unital algebra
              f ,rad
  homomorphism. Then Ψ is a unital completely isometric
  isomorphism having completely contractive hereditary
  extension if and only if Ψ is a continuous homeomorphism such
  that
                   ˆ (f )         ˆ  (f )
                 (Ψ(W1 ), . . . , Ψ(Wn )) ∈ Dm (F 2 (Hn ))
                                             f

  and
                                   (g)                      (g)
                    (Ψ−1 (W1 ), . . . , Ψ−1 (Wq )) ∈ Dlg (F 2 (Hq )).
                     ˆ                  ˆ


                                         G ELU P OPESCU    Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains


  Corollary
  Let f be a positive regular free holomorphic function with n
  indeterminates, and let m ≥ 1. Then the following statements
  are equivalent :
                ∗
     (i) Ψ ∈ Autci (A(Dm )) ;
                       f ,rad
    (ii) there is ϕ ∈ Aut(Dm ) such that
                           f

                                     Ψ(χ) = χ ◦ ϕ,            χ ∈ A(Dm ).
                                                                     f ,rad

                   ∗
  Consequently, Autci (A(Dm ))                             Aut(Dm ).
                          f ,rad                                f




                                        G ELU P OPESCU      Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains
  Corollary
  The noncommutative domains D1 (H) and D1 (H) are free
                                   f           g
  biholomorphic equivalent if and only if the domain algebras
  An (D1 ) and Aq (D1 ) are completely isometrically isomorphic.
       f            g
  Moreover,
                      ∗
                   Autci (A(D1,rad )) = Autci (A(D1,rad ))
                             f                    f                           Aut(D1 ).
                                                                                   f




          Remarks. The case f = g = X1 + · · · + Xn .
          Autci (An ) Aut(B(H)n ).
                                1
          (Davidson-Pitts ’98) Autci (An ) Aut(Bn ).
          (P.’10) Aut(B(H)n ) Aut(Bn ) Autci (An ).
                           1
                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains
  Corollary
  The noncommutative domains D1 (H) and D1 (H) are free
                                   f           g
  biholomorphic equivalent if and only if the domain algebras
  An (D1 ) and Aq (D1 ) are completely isometrically isomorphic.
       f            g
  Moreover,
                      ∗
                   Autci (A(D1,rad )) = Autci (A(D1,rad ))
                             f                    f                           Aut(D1 ).
                                                                                   f




          Remarks. The case f = g = X1 + · · · + Xn .
          Autci (An ) Aut(B(H)n ).
                                1
          (Davidson-Pitts ’98) Autci (An ) Aut(Bn ).
          (P.’10) Aut(B(H)n ) Aut(Bn ) Autci (An ).
                           1
                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains




  Corollary
  Let g be a positive regular free holomorphic function with q
  indeterminates. Then D1 (H) is biholomorphic equivalent to the
                             g
  unit ball [B(H)n ]1 if and only if q = n and g = c1 X1 + · · · + cn Xn
  for some ci > 0.




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains
  Theorem
  A map Ψ : A(Dm ) → A(Dlg,rad ) is a unital completely isometric
                 f ,rad
  isomorphism having completely contractive hereditary
  extension and such that its symbol ϕ fixes the origin if and only
  if n = q and Ψ is given by

                               Ψ(χ) = χ ◦ ϕ,                χ ∈ A(Dm ),
                                                                   f ,rad

  for some ϕ ∈ Bih(Dlg , Dm ) of the form ϕ(X ) = [X1 , . . . , Xn ]U ,
                          f
  X ∈ Dlg,rad (H), where U is an invertible operator on Cn such that

                                   (g)              (g)
                             [W1 , . . . , Wn ]U ∈ Dm (F 2 (Hn )),
                                                    f

  and
                                         G ELU P OPESCU    Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains

                                 (f )             (f )
                            [W1 , . . . , Wn ]U −1 ∈ Dlg (F 2 (Hn )).
  In this case, we have
                          (f )                   (f )                 (g)              (g)
                [Ψ(W1 ), . . . , Ψ(Wn )] = ϕ = [W1 , . . . , Wn ]U.


          When m = l = 1, Arias and Latrémolière proved that if
          there is a completely isometric isomorphism between
          An (D1 ) and An (D1 ), whose dual map fixes the origin, then
                f           g
          the algebras are related by a linear relation of their
          generators. Our Theorem implies and strengthens their
          result and also provides a converse.

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative domain algebras
 Biholomorphic classification of noncommutative domains     Classification of noncommutative domains
       Isomorphisms of noncommutative Hardy algebras


Classification of noncommutative domains

                                 (f )             (f )
                            [W1 , . . . , Wn ]U −1 ∈ Dlg (F 2 (Hn )).
  In this case, we have
                          (f )                   (f )                 (g)              (g)
                [Ψ(W1 ), . . . , Ψ(Wn )] = ϕ = [W1 , . . . , Wn ]U.


          When m = l = 1, Arias and Latrémolière proved that if
          there is a completely isometric isomorphism between
          An (D1 ) and An (D1 ), whose dual map fixes the origin, then
                f           g
          the algebras are related by a linear relation of their
          generators. Our Theorem implies and strengthens their
          result and also provides a converse.

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains     Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Noncommutative Hardy algebras


          H ∞ (Dm ) denotes the set of all elements ϕ in Hol(Dm )
                f ,rad                                        f ,rad
          such that
                          ϕ ∞ := sup ϕ(X ) < ∞,
          where the sup is taken over all n-tuples X ∈ Dm (H).
                                                        f ,rad


          H ∞ (Dm ) is a Banach algebra under pointwise
                f ,rad
          multiplication and the norm · ∞ , and has a unital
          operator algebra structure.




                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains     Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Noncommutative Hardy algebras
  Theorem
                           ∞
  The map Φ : H ∞ (Dm ) → Fn (Dm ) defined by
                    f ,rad     f

  Φ               cα Zα        :=            cα Wα is a completely isometric
         α∈F+
            n                       α∈F+
                                       n
  isomorphism of operator algebras. Moreover, if G :=                                              cα Zα
                                                                                           α∈F+
                                                                                              n
  is a free holomorphic function on Dm , then the following
                                     f ,rad
  statements are equivalent :
     (i) G ∈ H ∞ (Dm ) ;
                   f ,rad
    (ii) sup           G(rW1 , . . . , rWn ) < ∞, where
          0≤r <1

                                                              ∞
                             G(rW1 , . . . , rWn ) :=                       cα r |α| Wα ;
                                        G ELU P OPESCU       k =0 |α|=k
                                                           Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results      Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains        Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Noncommutative Hardy algebras
                           ∞
   (iii) there exists G ∈ Fn (Dm ) with G = B[G].
                               f

  In this case, Φ(G) = G = SOT- limr →1 G(rW1 , . . . , rWn ) and

                        Φ−1 (ϕ) = G = B[G],                               ∞
                                                                     G ∈ Fn (Dm ),
                                                                              f

  where B is the noncommutative Berezin transform.

          T := (T1 , . . . , Tn ) ∈ Dm (H) is pure if
                                     f

                                            SOT- lim Φk,T (I) = 0,
                                                      f
                                                     k →∞
                                              ∞                            ∗
          where Φf ,T (X ) =                  k =1         |α|=k   aα Tα XTα .
          Define
                            Dm (H) := {X ∈ Dm (H) : X is pure}.
                             f ,pure        f
                                        G ELU P OPESCU        Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results      Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains        Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Noncommutative Hardy algebras
                           ∞
   (iii) there exists G ∈ Fn (Dm ) with G = B[G].
                               f

  In this case, Φ(G) = G = SOT- limr →1 G(rW1 , . . . , rWn ) and

                        Φ−1 (ϕ) = G = B[G],                               ∞
                                                                     G ∈ Fn (Dm ),
                                                                              f

  where B is the noncommutative Berezin transform.

          T := (T1 , . . . , Tn ) ∈ Dm (H) is pure if
                                     f

                                            SOT- lim Φk,T (I) = 0,
                                                      f
                                                     k →∞
                                              ∞                            ∗
          where Φf ,T (X ) =                  k =1         |α|=k   aα Tα XTα .
          Define
                            Dm (H) := {X ∈ Dm (H) : X is pure}.
                             f ,pure        f
                                        G ELU P OPESCU        Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains     Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Noncommutative Hardy algebras
          Note that : Dm (H) ⊂ Dm (H) ⊂ Dm (H).
                       f ,rad   f ,pure  f
          Bih(Dlg,pure , Dm ) is the set of all bijections
                          f ,pure

                                       ϕ : Dlg,pure (H) → Dm (H)
                                                           f ,pure

          such that ϕ|Dl (H) and ϕ−1 |Dm (H) are free holomorphic
                                        f ,rad
                        g,rad
          functions with their model boundary functions pure, and ϕ
          and ϕ−1 are their radial extensions in the strong operator
          topology, respectively, i.e.,
                        ϕ(X ) = SOT- lim ϕ(rX ),                       X ∈ Dlg,pure (H),
                                                r →1

          and
                    ϕ−1 (X ) = SOT- lim ϕ−1 (rX ),                         X ∈ Dm (H).
                                                                                f ,pure
                                                r →1

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains     Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Unitarily implemented isomorphisms
  Theorem
  A map Ψ : H ∞ (Dm ) → H ∞ (Dlg,rad ) is a unitarily implemented
                  f ,rad
  isomorphism if and only if it has the form

                            Ψ(χ) := χ ◦ ϕ,                 χ ∈ H ∞ (Dm ),
                                                                     f ,rad

  for some ϕ ∈ Bih(Dlg,pure , Dm ) such that ϕ is unitarily
                               f ,pure
                                                                (f )             (f )
  equivalent to the universal model (W1 , . . . , Wn ) associated
  with Dm . In this case,
        f

                                         (m) ∗                         (m)              ∞
       Ψ(χ) = Bϕ [χ] := Kf ,ϕ (χ ⊗ IDf ,m,ϕ )Kf ,ϕ ,
         ˜     ˜ ˜          ˜  ˜          ˜      ˜                                 χ ∈ Fn (Dm ),
                                                                                   ˜        f

                                                                             (m)
  where the noncommutative Berezin kernel Kf ,ϕ is a unitary
                                              ˜
  operator and dim Df ,m,ϕ = 1.
                         ˜
                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results   Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains     Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Unitarily implemented isomorphisms
  Remark
  If m = 1, then ϕ is unitarily equivalent to the universal model
                     ˜
      (f )       (f )
  (W1 , . . . , Wn ) if and only if
      i) SOT- lim Φk,ϕ (I) = 0
                   f ˜
                   k →∞
    (ii) rank [I − Φf ,ϕ (I)] = 1
                       ˜
   (iii) the characteristic function Θϕ = 0.
                                      ˜


          Autw (Dm ) is the group of all ϕ ∈ Bih(Dm , Dm ) such
                 f ,pure                          f ,pure f ,pure
                                                                   (f )            (f )
          that ϕ is unitarily equivalent to (W1 , . . . , Wn ).

  Corollary
                                       ∞
                                Autu (Fn (Df ))            Autw (Dm ).
                                                                  f ,pure

                                        G ELU P OPESCU     Free biholomorphic classification of noncommutative domains
 Free holomorphic functions on noncommutative domains
    Free biholomorphic functions and Cartan type results      Noncommutative Hardy algebras
 Biholomorphic classification of noncommutative domains        Unitarily implemented isomorphisms
       Isomorphisms of noncommutative Hardy algebras


Unitarily implemented isomorphisms

          The case m = 1 and f = X1 + · · · + Xn .
                                      ∞
          (Davidson-Pitts ’98) Autu (Fn )                            Aut(Bn ).
          (P.’10)      Aut(B(H)n )               Aut(Bn )                  ∞
                                                                    Autu (Fn ).
                               1


  Theorem
  Let Autw (Dm )) be the group of all
             f
  ψ ∈ Bih(Dlg , Dm ) such that ψ is unitarily equivalent to the
                 f
                                      (f )             (f )
  universal model (W1 , . . . , Wn ). Then

                                  Autu (An (Dm ))
                                             f                  Autw (Dm ).
                                                                       f




                                        G ELU P OPESCU        Free biholomorphic classification of noncommutative domains
Free holomorphic functions on noncommutative domains
   Free biholomorphic functions and Cartan type results   Noncommutative Hardy algebras
Biholomorphic classification of noncommutative domains     Unitarily implemented isomorphisms
      Isomorphisms of noncommutative Hardy algebras




         THANK YOU




                                       G ELU P OPESCU     Free biholomorphic classification of noncommutative domains

								
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