Spectral lifting in Banach algebras and interpolation in several variables

Let ${\mathcal{A}}$ be a unital Banach algebra and let $J$ be a closed two-sided ideal of ${\mathcal{A}}$. We prove that if any invertible element of ${\mathcal{A}}/J$ has an invertible lifting in ${\mathcal{A}}$, then the quotient homomorphism $\Phi :{\mathcal{A}}\to {\mathcal{A}}/J$ is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foias, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna-Pick, and Carathéodory type interpolation for $F_{n}^{\infty }\bar \otimes B({\mathcal{K}})$, the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra $F_{n}^{\infty }$ and $B({\mathcal{K}})$, the algebra of bounded operators on a finite dimensional Hilbert space ${\mathcal{K}}$. 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 $F_{n}^{\infty }\bar \otimes B({\mathcal{K}})$.

In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of $\mathbb{C} ^{n}$, in which one bounds the spectral radius of the interpolant and not the norm.