Transactions of the American Mathematical Society

Abstract:

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, Foiaş, 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.