Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball
HTML articles powered by AMS MathViewer
- by Guy Salomon, Orr M. Shalit and Eli Shamovich PDF
- Trans. Amer. Math. Soc. 370 (2018), 8639-8690 Request permission
Abstract:
We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given an nc variety $\mathfrak {V}$ in the nc unit ball $\mathfrak {B}_d$, we identify the algebra of bounded analytic functions on $\mathfrak {V}$—denoted $H^\infty (\mathfrak {V})$—as the multiplier algebra $\mathrm {Mult} \mathcal {H}_{\mathfrak {V}}$ of a certain reproducing kernel Hilbert space $\mathcal {H}_{\mathfrak {V}}$ consisting of nc functions on $\mathfrak {V}$. We find that every such algebra $H^\infty (\mathfrak {V})$ is completely isometrically isomorphic to the quotient $H^\infty (\mathfrak {B}_d)/ \mathcal {J}_{\mathfrak {V}}$ of the algebra of bounded nc holomorphic functions on the ball by the ideal $\mathcal {J}_{\mathfrak {V}}$ of bounded nc holomorphic functions which vanish on $\mathfrak {V}$. In order to demonstrate this isomorphism, we prove that the space $\mathcal {H}_{\mathfrak {V}}$ is an nc complete Pick space (a fact recently proved—by other methods—by Ball, Marx, and Vinnikov).
We investigate the problem of when two algebras $H^\infty (\mathfrak {V})$ and $H^\infty (\mathfrak {W})$ are (completely) isometrically isomorphic. If the variety $\mathfrak {W}$ is the image of $\mathfrak {V}$ under an nc analytic automorphism of $\mathfrak {B}_d$, then $H^\infty (\mathfrak {V})$ and $H^\infty (\mathfrak {W})$ are completely isometrically isomorphic. We prove that the converse holds in the case where the varieties are homogeneous; in general we can only show that if the algebras are completely isometrically isomorphic, then there must be nc holomorphic maps between the varieties (in the case $d = \infty$ we need to assume that the isomorphism is also weak-$*$ continuous).
We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of $\mathfrak {B}_d$ and related norm closed algebras; the results in the norm closed setting are somewhat simpler and work for the case $d = \infty$ without further assumptions.
Along the way, we are led to consider some interesting problems on function theory in the nc unit ball. For example, we study various versions of the Nullstellensatz (that is, the problem of to what extent an ideal is determined by its zero set), and we obtain perfect Nullstellensatz in both the homogeneous as well as the commutative cases.
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
- Jim Agler and John E. McCarthy, Global holomorphic functions in several noncommuting variables, Canad. J. Math. 67 (2015), no. 2, 241–285. MR 3314834, DOI 10.4153/CJM-2014-024-1
- Jim Agler and John E. McCarthy, Non-commutative holomorphic functions on operator domains, Eur. J. Math. 1 (2015), no. 4, 731–745. MR 3426178, DOI 10.1007/s40879-015-0064-2
- Jim Agler and John E. McCarthy, Pick interpolation for free holomorphic functions, Amer. J. Math. 137 (2015), no. 6, 1685–1701. MR 3432271, DOI 10.1353/ajm.2015.0042
- Jim Agler and John E. McCarthy, The implicit function theorem and free algebraic sets, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3157–3175. MR 3451873, DOI 10.1090/tran/6546
- D. Alpay and D. S. Kalyuzhnyĭ-Verbovetzkiĭ, Matrix-$J$-unitary non-commutative rational formal power series, The state space method generalizations and applications, Oper. Theory Adv. Appl., vol. 161, Birkhäuser, Basel, 2006, pp. 49–113. MR 2187742, DOI 10.1007/3-7643-7431-4_{2}
- S. A. Amitsur, A generalization of Hilbert’s Nullstellensatz, Proc. Amer. Math. Soc. 8 (1957), 649–656. MR 87644, DOI 10.1090/S0002-9939-1957-0087644-9
- A. Andersson Berezin quantization of noncommutative projective varieties, arXiv, 1506.01454, 2015.
- T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90. MR 155193
- 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
- J. A. Ball, G. Marx, and V. Vinnikov, Interpolation and transfer-function realization for the noncommutative schur-agler class, arXiv, 1602.00762, 2015.
- Joseph A. Ball, Gregory Marx, and Victor Vinnikov, Noncommutative reproducing kernel Hilbert spaces, J. Funct. Anal. 271 (2016), no. 7, 1844–1920. MR 3535321, DOI 10.1016/j.jfa.2016.06.010
- S. T. Belinschi, M. Popa, and V. Vinnikov, On the operator-valued analogues of the semicircle, arcsine and Bernoulli laws, J. Operator Theory 70 (2013), no. 1, 239–258. MR 3085826, DOI 10.7900/jot.2011jun24.1963
- David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs. New Series, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR 2111973, DOI 10.1093/acprof:oso/9780198526599.001.0001
- John W. Bunce, Models for $n$-tuples of noncommuting operators, J. Funct. Anal. 57 (1984), no. 1, 21–30. MR 744917, DOI 10.1016/0022-1236(84)90098-3
- Kenneth R. Davidson, Michael Hartz, and Orr Moshe Shalit, Multipliers of embedded discs, Complex Anal. Oper. Theory 9 (2015), no. 2, 287–321. MR 3311940, DOI 10.1007/s11785-014-0360-8
- 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 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, 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
- Kenneth R. Davidson, Christopher Ramsey, and Orr Moshe Shalit, Operator algebras for analytic varieties, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1121–1150. MR 3280039, DOI 10.1090/S0002-9947-2014-05888-1
- Adam Dor-On and Daniel Markiewicz, Operator algebras and subproduct systems arising from stochastic matrices, J. Funct. Anal. 267 (2014), no. 4, 1057–1120. MR 3217058, DOI 10.1016/j.jfa.2014.05.004
- Adam Dor-On and Daniel Markiewicz, $\rm C^*$-envelopes of tensor algebras arising from stochastic matrices, Integral Equations Operator Theory 88 (2017), no. 2, 185–227. MR 3669127, DOI 10.1007/s00020-017-2382-x
- S. W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300–304. MR 480362, DOI 10.1090/S0002-9939-1978-0480362-8
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- David Eisenbud and Melvin Hochster, A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions, J. Algebra 58 (1979), no. 1, 157–161. MR 535850, DOI 10.1016/0021-8693(79)90196-0
- Quanlei Fang and Jingbo Xia, Multipliers and essential norm on the Drury-Arveson space, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2497–2504. MR 2784815, DOI 10.1090/S0002-9939-2010-10680-9
- Arthur E. Frazho, Models for noncommuting operators, J. Functional Analysis 48 (1982), no. 1, 1–11. MR 671311, DOI 10.1016/0022-1236(82)90057-X
- John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
- G. H. Hardy, On the mean modulus of an analytic function, Proc. London Math. Soc. (3), 14:269–277, 1915.
- Lawrence A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, Proceedings on Infinite Dimensional Holomorphy (Internat. Conf., Univ. Kentucky, Lexington, Ky., 1973) Lecture Notes in Math., Vol. 364, Springer, Berlin, 1974, pp. 13–40. MR 0407330
- 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
- Michael Hartz, On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces, Canad. J. Math. 69 (2017), no. 1, 54–106. MR 3589854, DOI 10.4153/CJM-2015-050-6
- J. William Helton, “Positive” noncommutative polynomials are sums of squares, Ann. of Math. (2) 156 (2002), no. 2, 675–694. MR 1933721, DOI 10.2307/3597203
- William Helton, Igor Klep, and Scott McCullough, Free convex algebraic geometry, Semidefinite optimization and convex algebraic geometry, MOS-SIAM Ser. Optim., vol. 13, SIAM, Philadelphia, PA, 2013, pp. 341–405. MR 3050247
- J. William Helton, Igor Klep, and Scott McCullough, The matricial relaxation of a linear matrix inequality, Math. Program. 138 (2013), no. 1-2, Ser. A, 401–445. MR 3034812, DOI 10.1007/s10107-012-0525-z
- J. W. Helton, I. Klep, S. McCullough, and M. Schweighofer, Dilations, linear matrix inequalities, the matrix cube problem and beta distributions, arXiv, 1412.1481, 2015.
- J. William Helton and Scott McCullough, Every convex free basic semi-algebraic set has an LMI representation, Ann. of Math. (2) 176 (2012), no. 2, 979–1013. MR 2950768, DOI 10.4007/annals.2012.176.2.6
- J. William Helton and Scott A. McCullough, A Positivstellensatz for non-commutative polynomials, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3721–3737. MR 2055751, DOI 10.1090/S0002-9947-04-03433-6
- E. T. A. Kakariadis and O. M. Shalit, On operator algebras associated with monomial ideals in noncommuting variables, arXiv, 1501.06495, 2015.
- Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of free noncommutative function theory, Mathematical Surveys and Monographs, vol. 199, American Mathematical Society, Providence, RI, 2014. MR 3244229, DOI 10.1090/surv/199
- Matt Kerr, John E. McCarthy, and Orr Moshe Shalit, On the isomorphism question for complete pick multiplier algebras, Integral Equations Operator Theory 76 (2013), no. 1, 39–53. MR 3041720, DOI 10.1007/s00020-013-2048-2
- A. Ja. Helemskiĭ, Homological methods in the holomorphic calculus of several operators in Banach space, after Taylor, Uspekhi Mat. Nauk 36 (1981), no. 1(217), 127–172, 248 (Russian). MR 608943
- John E. McCarthy and Orr Moshe Shalit, Spaces of Dirichlet series with the complete Pick property, Israel J. Math. 220 (2017), no. 2, 509–530. MR 3666434, DOI 10.1007/s11856-017-1527-6
- John E. McCarthy and Richard M. Timoney, Non-commutative automorphisms of bounded non-commutative domains, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 5, 1037–1045. MR 3569149, DOI 10.1017/S0308210515000748
- Meghna Mittal and Vern I. Paulsen, Operator algebras of functions, J. Funct. Anal. 258 (2010), no. 9, 3195–3225. MR 2595740, DOI 10.1016/j.jfa.2010.01.006
- Paul S. Muhly and Baruch Solel, Hardy algebras, $W^\ast$-correspondences and interpolation theory, Math. Ann. 330 (2004), no. 2, 353–415. MR 2089431, DOI 10.1007/s00208-004-0554-x
- Paul S. Muhly and Baruch Solel, Schur class operator functions and automorphisms of Hardy algebras, Doc. Math. 13 (2008), 365–411. MR 2520475
- Paul S. Muhly and Baruch Solel, Representations of Hardy algebras: absolute continuity, intertwiners, and superharmonic operators, Integral Equations Operator Theory 70 (2011), no. 2, 151–203. MR 2794388, DOI 10.1007/s00020-011-1869-0
- Paul S. Muhly and Baruch Solel, Tensorial function theory: from Berezin transforms to Taylor’s Taylor series and back, Integral Equations Operator Theory 76 (2013), no. 4, 463–508. MR 3073943, DOI 10.1007/s00020-013-2062-4
- Paul S. Muhly and Baruch Solel, Matricial function theory and weighted shifts, Integral Equations Operator Theory 84 (2016), no. 4, 501–553. MR 3483873, DOI 10.1007/s00020-016-2281-6
- V. Müller and F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), no. 4, 979–989. MR 1112498, DOI 10.1090/S0002-9939-1993-1112498-0
- R. Nevanlinna, Über beschränkte Funktionen, die gegenbenen Funkten vorgeschreiben Werte annehmen, Ann. Acad. Sci. Fenn. Ser. A 13 (1), 1919.
- R. Nevanlinna, Über beschränkte Funktionen, Ann. Acad. Sci. Fenn. Ser. A 32 (7), 1929.
- Stephen Parrott, Unitary dilations for commuting contractions, Pacific J. Math. 34 (1970), 481–490. MR 268710
- Georg Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1915), no. 1, 7–23 (German). MR 1511844, DOI 10.1007/BF01456817
- Mihai Popa and Victor Vinnikov, Non-commutative functions and the non-commutative free Lévy-Hinčin formula, Adv. Math. 236 (2013), 131–157. MR 3019719, DOI 10.1016/j.aim.2012.12.013
- Gelu Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), no. 2, 523–536. MR 972704, DOI 10.1090/S0002-9947-1989-0972704-3
- Gelu Popescu, von Neumann inequality for $(B({\scr H})^n)_1$, Math. Scand. 68 (1991), no. 2, 292–304. MR 1129595, DOI 10.7146/math.scand.a-12363
- Gelu Popescu, On intertwining dilations for sequences of noncommuting operators, J. Math. Anal. Appl. 167 (1992), no. 2, 382–402. MR 1168596, DOI 10.1016/0022-247X(92)90214-X
- Gelu Popescu, Functional calculus for noncommuting operators, Michigan Math. J. 42 (1995), no. 2, 345–356. MR 1342494, DOI 10.1307/mmj/1029005232
- Gelu Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), no. 1, 31–46. MR 1348353, DOI 10.1007/BF01460977
- Gelu Popescu, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2137–2148. MR 1343719, DOI 10.1090/S0002-9939-96-03514-9
- Gelu Popescu, Free holomorphic functions on the unit ball of $B(\scr H)^n$, J. Funct. Anal. 241 (2006), no. 1, 268–333. MR 2264252, DOI 10.1016/j.jfa.2006.07.004
- 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
- C. Ramsey. Maximal ideal space techniques in non-selfadjoint operator algebras. PhD thesis, University of Waterloo, 2013.
- Freidrich Riesz, Über die Randwerte einer analytischen Funktion, Math. Z. 18 (1923), no. 1, 87–95 (German). MR 1544621, DOI 10.1007/BF01192397
- Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
- 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
- Guy Salomon and Orr Moshe Shalit, The isomorphism problem for complete Pick algebras: a survey, Operator theory, function spaces, and applications, Oper. Theory Adv. Appl., vol. 255, Birkhäuser/Springer, Cham, 2016, pp. 167–198. MR 3617006
- Donald Sarason, Operator-theoretic aspects of the Nevanlinna-Pick interpolation problem, Operators and function theory (Lancaster, 1984) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 153, Reidel, Dordrecht, 1985, pp. 279–314. MR 810449
- Ichirô Satake, Algebraic structures of symmetric domains, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980. MR 591460
- O. M. Shalit, Operator theory and function theory in Drury-Arveson space and its quotients, in D. Alpay, editor, Handbook of Operator Theory, pages 1125–1180. Springer, Basel, 2015.
- Orr Moshe Shalit and Baruch Solel, Subproduct systems, Doc. Math. 14 (2009), 801–868. MR 2608451
- M. Skeide and O. M. Shalit, CP-semigroups and dilations, subproduct systems and superproduct systems: The multi-parameter case and beyond, in preparation.
- Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy, Harmonic analysis of operators on Hilbert space, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR 2760647, DOI 10.1007/978-1-4419-6094-8
- Joseph L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38. MR 271741, DOI 10.1007/BF02392329
- Joseph L. Taylor, A general framework for a multi-operator functional calculus, Advances in Math. 9 (1972), 183–252. MR 328625, DOI 10.1016/0001-8708(72)90017-5
- Joseph L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973), 1–34. MR 315446, DOI 10.1090/S0002-9904-1973-13077-0
- Ami Viselter, Covariant representations of subproduct systems, Proc. Lond. Math. Soc. (3) 102 (2011), no. 4, 767–800. MR 2793449, DOI 10.1112/plms/pdq047
- Ami Viselter, Cuntz-Pimsner algebras for subproduct systems, Internat. J. Math. 23 (2012), no. 8, 1250081, 32. MR 2949219, DOI 10.1142/S0129167X12500814
- Dan Voiculescu, Symmetries of some reduced free product $C^\ast$-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588. MR 799593, DOI 10.1007/BFb0074909
- Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. MR 839105, DOI 10.1016/0022-1236(86)90062-5
- Dan Voiculescu, Free analysis questions. I. Duality transform for the coalgebra of $\partial _{X\colon B}$, Int. Math. Res. Not. 16 (2004), 793–822. MR 2036956, DOI 10.1155/S1073792804132443
- Dan-Virgil Voiculescu, Free analysis questions II: the Grassmannian completion and the series expansions at the origin, J. Reine Angew. Math. 645 (2010), 155–236. MR 2673426, DOI 10.1515/CRELLE.2010.063
- Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124
Additional Information
- Guy Salomon
- Affiliation: Department of Mathematics, Technion - Israel Institute of Mathematics, Haifa, 3200003, Israel
- MR Author ID: 993215
- Email: guy.salomon@technion.ac.il
- Orr M. Shalit
- Affiliation: Department of Mathematics, Technion - Israel Institute of Mathematics, Haifa, 3200003, Israel
- MR Author ID: 829657
- Email: oshalit@technion.ac.il
- Eli Shamovich
- Affiliation: Department of Mathematics, Technion - Israel Institute of Mathematics, Haifa, 3200003, Israel
- MR Author ID: 1197796
- ORCID: setImmediate$0.6024528153333779$6
- Email: eshamovich@uwaterloo.ca
- Received by editor(s): February 13, 2017
- Received by editor(s) in revised form: May 12, 2017, and June 13, 2017
- Published electronically: August 21, 2018
- Additional Notes: The first author was partially supported by the Clore Foundation. The second author was partially supported by Israel Science Foundation Grants no. 474/12 and 195/16, and by EU FP7/2007-2013 Grant no. 321749
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 8639-8690
- MSC (2010): Primary 47LXX, 46L07, 47L25
- DOI: https://doi.org/10.1090/tran/7308
- MathSciNet review: 3864390