Spectral lifting in Banach algebras and interpolation in several variables

Abstract

Let A {\mathcal {A}} be a unital Banach algebra and let J J be a closed two-sided ideal of A {\mathcal {A}} . We prove that if any invertible element of A / J {\mathcal {A}}/J has an invertible lifting in A {\mathcal {A}} , then the quotient homomorphism Φ : A → A / J \Phi :{\mathcal {A}}\to {\mathcal {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 ) F_{n}^{\infty }\bar \otimes B({\mathcal {K}}) , the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra F n ∞ F_{n}^{\infty } and B ( K ) B({\mathcal {K}}) , the algebra of bounded operators on a finite dimensional Hilbert space K {\mathcal {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 ) F_{n}^{\infty }\bar \otimes B({\mathcal {K}}) . In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of C n \mathbb {C}^{n} , in which one bounds the spectral radius of the interpolant and not the norm.