Spectral lifting in Banach algebras and interpolation in several variables
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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.
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Additional Information
- Gelu Popescu
- Affiliation: Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, Texas 78249
- MR Author ID: 234950
- Email: gpopescu@math.utsa.edu
- Received by editor(s): December 22, 1998
- Received by editor(s) in revised form: October 4, 1999
- Published electronically: March 12, 2001
- Additional Notes: Partially supported by NSF Grant DMS-9531954
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2843-2857
- MSC (2000): Primary 47L25, 47A57, 47A20; Secondary 30E05
- DOI: https://doi.org/10.1090/S0002-9947-01-02796-9
- MathSciNet review: 1828475