Projective free algebras of bounded holomorphic functions on infinitely connected domains

Brudnyi, A.

doi:10.1090/spmj/1718

Projective free algebras of bounded holomorphic functions on infinitely connected domains
HTML articles powered by AMS MathViewer

by A. Brudnyi

St. Petersburg Math. J. 33 (2022), 619-631

DOI: https://doi.org/10.1090/spmj/1718

Published electronically: June 27, 2022

PDF | Request permission

Abstract:

The algebra $H^\infty (D)$ of bounded holomorphic functions on $D\subset \mathbb {C}$ is projective free for a wide class of infinitely connected domains. In particular, for such $D$ every rectangular left-invertible matrix with entries in $H^\infty (D)$ can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of $H^\infty (D)$ asserting that its covering dimension is $2$ and the second Čech cohomology group is trivial.

References

Michael Frederick Behrens, The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains, Trans. Amer. Math. Soc. 161 (1971), 359–379. MR 435420, DOI 10.1090/S0002-9947-1971-0435420-9
Alexander Brudnyi, Projections in the space $H^\infty$ and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces, Ark. Mat. 42 (2004), no. 1, 31–59. MR 2056544, DOI 10.1007/BF02432909
Alexander Brudnyi, Grauert- and Lax-Halmos-type theorems and extension of matrices with entries in $H^\infty$, J. Funct. Anal. 206 (2004), no. 1, 87–108. MR 2024347, DOI 10.1016/S0022-1236(03)00097-1
Alexander Brudnyi, Holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds, Compos. Math. 142 (2006), no. 4, 1018–1038. MR 2249540, DOI 10.1112/S0010437X06002156
A. Brudnyi, Extension of matrices with entries in $H^\infty$ on coverings of Riemann surfaces of finite type, Algebra i Analiz 21 (2009), no. 3, 79–92; English transl., St. Petersburg Math. J. 21 (2010), no. 3, 423–432. MR 2588763, DOI 10.1090/S1061-0022-10-01101-5
Alexander Brudnyi, Topology of the maximal ideal space of $H^\infty$ revisited, Adv. Math. 299 (2016), 931–939. MR 3519483, DOI 10.1016/j.aim.2016.06.001
A. Brudnyi, Structure of the maximal ideal space of $H^\infty$ on the countable disjoint union of open disks, Algebra i Analiz 32 (2020), no. 6, 58–71; English transl., St. Petersburg Math. J. 32 (2021), no. 6, 999–1009. MR 4219491, DOI 10.1090/spmj/1681
Alexander Brudnyi, On homotopy invariants of tensor products of Banach algebras, Integral Equations Operator Theory 92 (2020), no. 2, Paper No. 19, 14. MR 4088578, DOI 10.1007/s00020-020-02575-8
Lutz Bungart, On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers, Topology 7 (1967), 55–68. MR 222338, DOI 10.1016/0040-9383(86)90015-7
Alexander Brudnyi and Amol Sasane, Sufficient conditions for the projective freeness of Banach algebras, J. Funct. Anal. 257 (2009), no. 12, 4003–4014. MR 2557732, DOI 10.1016/j.jfa.2009.05.003
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
Ronald G. Douglas, Steven G. Krantz, Eric T. Sawyer, Sergei Treil, and Brett D. Wicks, A history of the corona problem, The corona problem, Fields Inst. Commun., vol. 72, Springer, New York, 2014, pp. 1–29. MR 3329539, DOI 10.1007/978-1-4939-1255-1_{1}
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
Keiô Nagami, Dimension theory, Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. With an appendix by Yukihiro Kodama. MR 0271918
N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
Fernando Daniel Suárez, Čech cohomology and covering dimension for the $H^\infty$ maximal ideal space, J. Funct. Anal. 123 (1994), no. 2, 233–263. MR 1283028, DOI 10.1006/jfan.1994.1088
Vadim Tolokonnikov, Extension problem to an invertible matrix, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1023–1030. MR 1123668, DOI 10.1090/S0002-9939-1993-1123668-X
V. Tolokonnikov, Stable rank of $H^\infty$ in multiply connected domains, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3151–3156. MR 1273527, DOI 10.1090/S0002-9939-1995-1273527-9