کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4591684 1335046 2009 41 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Noncommutative hyperbolic geometry on the unit ball of Bn(H)B(H)n
کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Noncommutative hyperbolic geometry on the unit ball of Bn(H)B(H)n
چکیده انگلیسی

In this paper we introduce a hyperbolic (Poincaré–Bergman type) distance δ on the noncommutative open ball[B(H)n]1:={(X1,…,Xn)∈B(H)n:‖X1X1∗+⋯+XnXn∗‖1/2<1}, where B(H)B(H) is the algebra of all bounded linear operators on a Hilbert space HH. It is proved that δ   is invariant under the action of the free holomorphic automorphism group of [Bn(H)]1[B(H)n]1, i.e.,δ(Ψ(X),Ψ(Y))=δ(X,Y),X,Y∈[B(H)n]1, for all Ψ∈Aut([Bn(H)]1)Ψ∈Aut([B(H)n]1). Moreover, we show that the δ  -topology and the usual operator norm topology coincide on [Bn(H)]1[B(H)n]1. While the open ball [Bn(H)]1[B(H)n]1 is not a complete metric space with respect to the operator norm topology, we prove that [Bn(H)]1[B(H)n]1 is a complete metric space with respect to the hyperbolic metric δ. We obtain an explicit formula for δ in terms of the reconstruction operatorRX:=X1∗⊗R1+⋯+Xn∗⊗Rn,X:=(X1,…,Xn)∈[B(H)n]1, associated with the right creation operators R1,…,RnR1,…,Rn on the full Fock space with n   generators. In the particular case when H=CH=C, we show that the hyperbolic distance δ coincides with the Poincaré–Bergman distance on the open unit ballBn:={z=(z1,…,zn)∈Cn:‖z‖2<1}. We obtain a Schwarz–Pick lemma for free holomorphic functions on [Bn(H)]1[B(H)n]1 with respect to the hyperbolic metric, i.e., if F:=(F1,…,Fm)F:=(F1,…,Fm) is a contractive (‖F‖∞⩽1‖F‖∞⩽1) free holomorphic function, thenδ(F(X),F(Y))⩽δ(X,Y),X,Y∈[B(H)n]1. As consequences, we show that the Carathéodory and the Kobayashi distances, with respect to δ, coincide with δ   on [Bn(H)]1[B(H)n]1. The results of this paper are presented in the more general context of Harnack parts of the closed ball [B(H)n]1−, which are noncommutative analogues of the Gleason parts of the Gelfand spectrum of a function algebra.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Functional Analysis - Volume 256, Issue 12, 15 June 2009, Pages 4030–4070
نویسندگان
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