VIEWS: 1 PAGES: 59 CATEGORY: Business Letters POSTED ON: 4/17/2012 Public Domain
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