Spectral Lifting in Banach Algebras and Interpolation in Several Variables
نویسنده
چکیده
Let A be a unital Banach algebra and let J be a closed two-sided ideal of A. We prove that if any invertible element of A/J has an invertible lifting in A, then the quotient homomorphism Φ : A → A/J is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foiaş, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna–Pick, and Carathéodory type interpolation for F∞ n ⊗̄B(K), the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra F∞ n and B(K), the algebra of bounded operators on a finite dimensional Hilbert space K. A spectral tangential commutant lifting theorem in several variables is considered and used to obtain a spectral tangential version of the Nevanlinna-Pick interpolation for F∞ n ⊗̄B(K). In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of Cn, in which one bounds the spectral radius of the interpolant and not the norm.
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تاریخ انتشار 2001