Quotients of standard Hilbert modules
HTML articles powered by AMS MathViewer
- by William Arveson PDF
- Trans. Amer. Math. Soc. 359 (2007), 6027-6055 Request permission
Abstract:
We initiate a study of Hilbert modules over the polynomial algebra $\mathcal A=\mathbb C[z_1,\dots ,z_d]$ that are obtained by completing $\mathcal A$ with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity version of one of these. Standard Hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra, and their quotients by submodules give rise to universal solutions of nonlinear relations. Essentially all of the basic Hilbert modules that have received attention over the years are standard, including the Hilbert module of the $d$-shift, the Hardy and Bergman modules of the unit ball, modules associated with more general domains in $\mathbb C^d$, and those associated with projective algebraic varieties. We address the general problem of determining when a quotient $H/M$ of an essentially normal standard Hilbert module $H$ is essentially normal. This problem has been resistant. Our main result is that it can be “linearized” in that the nonlinear relations defining the submodule $M$ can be reduced, appropriately, to linear relations through an iteration procedure, and we give a concrete description of linearized quotients.References
- William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
- William Arveson, The curvature invariant of a Hilbert module over $\textbf {C}[z_1,\cdots ,z_d]$, J. Reine Angew. Math. 522 (2000), 173–236. MR 1758582, DOI 10.1515/crll.2000.037
- William Arveson, The Dirac operator of a commuting $d$-tuple, J. Funct. Anal. 189 (2002), no. 1, 53–79. MR 1887629, DOI 10.1006/jfan.2001.3827
- William B. Arveson, $p$-summable commutators in dimension $d$, J. Operator Theory 54 (2005), no. 1, 101–117. MR 2168861
- L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of $C^*$-algebras and $K$-homology, Ann. of Math. (2) 105 (1977), no. 2, 265–324. MR 458196, DOI 10.2307/1970999
- Raul E. Curto, Fredholm and invertible $n$-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), no. 1, 129–159. MR 613789, DOI 10.1090/S0002-9947-1981-0613789-6
- R. G. Douglas and Roger Howe, On the $C^*$-algebra of Toeplitz operators on the quarterplane, Trans. Amer. Math. Soc. 158 (1971), 203–217. MR 288591, DOI 10.1090/S0002-9947-1971-0288591-1
- Ronald G. Douglas and Gadadhar Misra, Equivalence of quotient Hilbert modules, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), no. 3, 281–291. MR 1999257, DOI 10.1007/BF02829607
- Ronald G. Douglas, Gadadhar Misra, and Cherian Varughese, On quotient modules—the case of arbitrary multiplicity, J. Funct. Anal. 174 (2000), no. 2, 364–398. MR 1768979, DOI 10.1006/jfan.2000.3594
- R. G. Douglas. Essentially reductive Hilbert modules. J. Operator Theory, 2005. arXiv:math.OA/0404167.
- Ronald G. Douglas, Ideals in Toeplitz algebras, Houston J. Math. 31 (2005), no. 2, 529–539. MR 2132850
- S. Gleason, J. Richter and C. Sundberg. On the index of invariant subspaces in spaces of analytic functions in several complex variables. to appear in Crelle’s Journal, 2005.
- Kunyu Guo, Defect operators for submodules of $H_d^2$, J. Reine Angew. Math. 573 (2004), 181–209. MR 2084587, DOI 10.1515/crll.2004.059
- K. Guo and K. Wang. Essentially normal Hilbert modules and $K$-homology. preprint, 2005.
- 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
- 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 joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172–191. MR 0268706, DOI 10.1016/0022-1236(70)90055-8
- Harald Upmeier, Toeplitz $C^{\ast }$-algebras on bounded symmetric domains, Ann. of Math. (2) 119 (1984), no. 3, 549–576. MR 744863, DOI 10.2307/2007085
Additional Information
- William Arveson
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- Email: arveson@math.berkeley.edu
- Received by editor(s): July 19, 2005
- Received by editor(s) in revised form: November 5, 2005
- Published electronically: June 13, 2007
- Additional Notes: The author was supported by NSF grant DMS-0100487
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 6027-6055
- MSC (2000): Primary 46L07, 47A99
- DOI: https://doi.org/10.1090/S0002-9947-07-04209-2
- MathSciNet review: 2336315