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Transactions of the American Mathematical Society

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Spectral lifting in Banach algebras and interpolation in several variables


Author: Gelu Popescu
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
Published electronically: March 12, 2001
MathSciNet review: 1828475
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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, 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.


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  • [AMc] J. Agler and J.E. Mc Carthy, Nevanlinna-Pick kernels and localization, preprint (1997). CMP 2000:17
  • [ArPo1] A. Arias and G. Popescu, Factorization and reflexivity on Fock spaces, Integr. Equat. Oper.Th. 23 (1995), 268-286.
  • [ArPo2] A. Arias and G. Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205-234.
  • [Arv] W.B. Arveson, Subalgebras of $C^{*}$-algebras III: Multivariable operator theory, Acta Math. 181 (1998), 159-228. MR 2000e:47013
  • [BTV] J.A. Ball, T.T.Trent, and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, preprint (1999).
  • [BV] J.A. Ball and V. Vinnikov, Multivariable linear systems, scattering, unitary dilations and operator theory for row contractions, preprint (1999).
  • [B] J.A. Ball, Linear systems, operator model theory and scattering: multivariable generalizations, preprint (1999).
  • [BFT] H. Bercovici, C. Foias, and A. Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991), 741-763. MR 91j:47006
  • [BF] H. Bercovici and C. Foias, On spectral tangential Nevanlinna-Pick interpolation, J.Math. Anal.Appl. 155 (1991), 156-176. MR 92d:47020
  • [DP1] K.R. Davidson and D. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math.Soc., 78 (1999), 401-430.
  • [DP2] K.R. Davidson and D. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275-303. MR 20001e:47082
  • [DP3] K.R. Davidson and D. Pitts, Nevanlinna-Pick interpolation for noncommutative analytic Toeplitz algebras, Integr. Equat. Oper.Th. 31 (1998), 321-337. MR 2000g:47016
  • [DMP] R.G. Douglas, P.S. Muhly and C. Pearcy, Lifting commuting operators, Michigan Math.J. 15 (1968), 385-395. MR 38:5046
  • [F] I.P. Fedcina, A description of of the solutions of the Nevanlinna-Pick tangent problem, Akad. Nauk. Armjan. SSR Dokl. 60 (1975), 37-42. MR 52:5974
  • [N] R. Nevanlinna, Über beschränkte Functionen, die in gegebenen Punkten vorgeschribene Werte annehmen, Ann. Acad. Sci. Fenn. Ser A 13 (1919), 7-23.
  • [P] G. Pick, Über die Beschränkungen analytischer Functionen, welche durch vorgegebene Functionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23.
  • [Po1] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer.Math.Soc. 316 (1989), 523-536. MR 90c:47006
  • [Po2] G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J.Operator Theory 22 (1989), 51-71. MR 91m:47012
  • [Po3] G. Popescu, Von Neumann inequality for $(B(H)^{n})_{1}$, Math.Scand. 68 (1991), 292-304. MR 92k:47073
  • [Po4] G. Popescu, On intertwining dilations for sequences of noncommuting operators, J.Math. Anal.Appl. 167 (1992), 382-402. MR 93e:47012
  • [Po5] G. Popescu, Functional calculus for noncommuting operators, Michigan Math. J. 42 (1995), 345-356. MR 96k:47025
  • [Po6] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31-46. MR 96k:47049
  • [Po7] G. Popescu, Noncommutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), 2137-2148. MR 96k:47077
  • [Po8] G. Popescu, Interpolation problems in several variables, J.Math Anal.Appl. 227 (1998), 227-250. MR 99i:47028
  • [Po9] G. Popescu, Poisson transforms on some $C^{*}$-algebras generated by isometries, J. Funct. Anal. 161 (1999), 27-61. MR 2000m:46117
  • [R] G.C. Rota, On models for linear operators, Comm.Pure Appl.Math. 13 (1960), 469-472. MR 22:2898
  • [S] D. Sarason, Generalized interpolation in $H^{\infty }$, Trans. AMS 127 (1967), 179-203. MR 34:8193
  • [SzF1] B.Sz.-Nagy, C. Foias, Dilation des commutants d'operateurs, C.R.Acad.Sci. Paris, Serie A 266 (1968), 493-495. MR 38:5049
  • [SzF2] B.Sz.-Nagy, C. Foias, Harmonic analysis on operators on Hilbert space, North-Holland, Amsterdam (1970). MR 43:947

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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
Email: gpopescu@math.utsa.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02796-9
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
Article copyright: © Copyright 2001 American Mathematical Society

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