Operator algebras for analytic varieties

Davidson, Kenneth; Ramsey, Christopher; Shalit, Orr

doi:10.1090/S0002-9947-2014-05888-1

Operator algebras for analytic varieties
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by Kenneth R. Davidson, Christopher Ramsey and Orr Moshe Shalit PDF

Trans. Amer. Math. Soc. 367 (2015), 1121-1150 Request permission

Abstract:

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions $\mathcal {M}_V$ of the multiplier algebra $\mathcal {M}$ of Drury-Arveson space to a holomorphic subvariety $V$ of the unit ball $\mathbb {B}_d$.

We find that $\mathcal {M}_V$ is completely isometrically isomorphic to $\mathcal {M}_W$ if and only if $W$ is the image of $V$ under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when $d<\infty$ every isometric isomorphism is completely isometric.

The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When $V$ and $W$ are each a finite union of irreducible varieties and a discrete variety, when $d<\infty$, an isomorphism between $\mathcal {M}_V$ and $\mathcal {M}_W$ determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-$*$ continuous.

We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold—particularly, smooth curves and Blaschke sequences.

We also discuss the norm closed algebras associated to a variety, and point out some of the differences.

References

Jim Agler and John E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal. 175 (2000), no. 1, 111–124. MR 1774853, DOI 10.1006/jfan.2000.3599
Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259, DOI 10.1090/gsm/044
Daniel Alpay, Mihai Putinar, and Victor Vinnikov, A Hilbert space approach to bounded analytic extension in the ball, Commun. Pure Appl. Anal. 2 (2003), no. 2, 139–145. MR 1975056, DOI 10.3934/cpaa.2003.2.139
N. Arcozzi, R. Rochberg, and E. Sawyer, Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls, Adv. Math. 218 (2008), no. 4, 1107–1180. MR 2419381, DOI 10.1016/j.aim.2008.03.001
Alvaro Arias and Frédéric Latrémolière, Ergodic actions of convergent Fuchsian groups on quotients of the noncommutative Hardy algebras, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2485–2496. MR 2784814, DOI 10.1090/S0002-9939-2011-10794-9
Alvaro Arias and Gelu Popescu, Factorization and reflexivity on Fock spaces, Integral Equations Operator Theory 23 (1995), no. 3, 268–286. MR 1356335, DOI 10.1007/BF01198485
Alvaro Arias and Gelu Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205–234. MR 1749679, DOI 10.1007/BF02810587
William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
Şerban Costea, Eric T. Sawyer, and Brett D. Wick, The corona theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in $\Bbb C^n$, Anal. PDE 4 (2011), no. 4, 499–550. MR 3077143, DOI 10.2140/apde.2011.4.499
H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), no. 2, 129–183. MR 500923, DOI 10.1112/blms/10.2.129
Kenneth R. Davidson and Ryan Hamilton, Nevanlinna-Pick interpolation and factorization of linear functionals, Integral Equations Operator Theory 70 (2011), no. 1, 125–149. MR 2786738, DOI 10.1007/s00020-011-1862-7
Kenneth R. Davidson and David R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. (3) 78 (1999), no. 2, 401–430. MR 1665248, DOI 10.1112/S002461159900180X
Kenneth R. Davidson and David R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), no. 2, 275–303. MR 1625750, DOI 10.1007/s002080050188
Kenneth R. Davidson and David R. Pitts, Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), no. 3, 321–337. MR 1627901, DOI 10.1007/BF01195123
Kenneth R. Davidson, Christopher Ramsey, and Orr Moshe Shalit, The isomorphism problem for some universal operator algebras, Adv. Math. 228 (2011), no. 1, 167–218. MR 2822231, DOI 10.1016/j.aim.2011.05.015
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
Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. II, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990. Local theory. MR 1057177
Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
Michael Hartz, Topological isomorphisms for some universal operator algebras, J. Funct. Anal. 263 (2012), no. 11, 3564–3587. MR 2984075, DOI 10.1016/j.jfa.2012.08.028
Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
Michael T. Jury, Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3669–3675. MR 2336583, DOI 10.1090/S0002-9939-07-08931-9
Gelu Popescu, Poisson transforms on some $C^*$-algebras generated by isometries, J. Funct. Anal. 161 (1999), no. 1, 27–61. MR 1670202, DOI 10.1006/jfan.1998.3346
Gelu Popescu, Operator theory on noncommutative varieties, Indiana Univ. Math. J. 55 (2006), no. 2, 389–442. MR 2225440, DOI 10.1512/iumj.2006.55.2771
Gelu Popescu, Free holomorphic automorphisms of the unit ball of $B(\scr H)^n$, J. Reine Angew. Math. 638 (2010), 119–168. MR 2595338, DOI 10.1515/CRELLE.2010.005
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0387634