Essentially normal Hilbert modules and K-homology

Guo, Kun; Wang, Kai

doi:10.1007/s00208-007-0175-2

Cited by 65 publications

(85 citation statements)

References 26 publications

(43 reference statements)

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“…Therefore, the results of this paper extend to show that the closure J and the quotient L 2 a (B 2 )/J are essentially normal A-modules (see also [25] Consider the function f . We prove that f is in I + M j z 0 for every j ≥ 1.…”

Section: 2supporting

confidence: 55%

An analytic Grothendieck Riemann Roch theorem

Douglas

Tang

2016

Advances in Mathematics

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Abstract. We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative K-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let B m be the unit ball in C m , and I an ideal in the polynomial algebra C[z 1 , · · · , z m ]. We prove that when the zero variety Z I is a complete intersection space with only isolated singularities and intersects with the unit sphere S 2m−1 transversely, the representations of C[z 1 , · · · , z m ] on the closure of I in L 2 a (B m ) and also the corresponding quotient space Q I are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on Q I by showing that the representation of C[z 1 , · · · , z m ] on the quotient space Q I gives the fundamental class of the boundary Z I ∩ S 2m−1 . In the appendix, we prove with Kai Wang that if f ∈ L 2 a (B m ) vanishes on Z I ∩ B m , then f is contained inside the closure of the ideal I in L 2 a (B m ).

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Section: 2supporting

confidence: 55%

An analytic Grothendieck Riemann Roch theorem

Douglas

Tang

2016

Advances in Mathematics

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“…Our results are of a nature quite different from other results on this conjecture, e.g., [4,5,15,16,17,18,20,22,26,27,35]; these previous results gave a full verification of the conjecture for limited classes of (typically homogeneous) ideals. Here, we shall present more limited results that hold for all homogeneous ideals, and for a large class of non-homogeneous ideals.…”

Section: Introduction Notation and Preliminariescontrasting

confidence: 99%

Essential normality, essential norms and hyperrigidity

Kennedy

Shalit

2015

Journal of Functional Analysis

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Abstract. Let S = (S 1 , . . . , S d ) denote the compression of the dshift to the complement of a homogeneous ideal I of C[z 1 , . . . , z d ].Arveson conjectured that S is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary.The unital norm closed algebra B I generated by S 1 , . . . , S d modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on V := Z(I) ∩ B d , where Z(I) is the variety corresponding to I. Consequently, the essential norm of an element in B I is equal to the sup norm of its Gelfand transform, and the C*-envelope of B I is identified as the algebra of continuous functions on V ∩ ∂B d , which means it is a complete invariant of the topology of the variety determined by I in the ball.Motivated by this determination of the C*-envelope of B I , we suggest a new, more qualitative approach to the problem of essential normality. We prove the tuple S is essentially normal if and only if it is hyperrigid as the generating set of a C*-algebra, which is a property closely connected to Arveson's notion of a boundary representation.We show that most of our results hold in a much more general setting. In particular, for most of our results, the ideal I can be replaced by an arbitrary (not necessarily homogeneous) invariant subspace of the d-shift.2010 Mathematics Subject Classification. 47A13, 47L30, 46E22.

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“…This conjecture attracted a lot of attention [23,24,53,54,57,58,59,63,67,75,76,77,81,82,109], where the conjecture was proved in particular classes of submodules, but it is still far from being solved. In all cases where the conjecture was verified, the following stronger conjecture due to Douglas was also shown to hold.…”

Section: Essential Normality and The Conjectures Of Arveson And Douglasmentioning

confidence: 99%

Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

Shalit

2015

Operator Theory

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Essentially normal Hilbert modules and K-homology

Cited by 65 publications

References 26 publications

An analytic Grothendieck Riemann Roch theorem

An analytic Grothendieck Riemann Roch theorem

Essential normality, essential norms and hyperrigidity

Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

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