C*-structure and K-theory of Boutet de Monvel's algebra
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
We consider the norm closure Θ of the algebra of all operators of order and class zero in Beutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to Θ. If X is connected and ∂X is not empty, we then show that the K-groups of Θ are topologically determined. In case ∂X has torsion free K-theory, we get Ki(Θ/ℜ) ≃ Ki(C(X)) ⊕ K 1-i(C0(T*Ẋ)), i = 0, 1, with ℜ denoting the compact ideal, and T*Ẋ denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis; and show, moreover, that K1 (Θ) ≃ K1 (C(X)) ⊕ ker χ, with χ: K0(T*Ẋ) → Z denoting the topological index. For the case of orientable, two-dimensional X, K 0(Θ) ≃ ℤ2g+m and K1(Θ) ≃ ℤ2g+m-1, where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ ℜ ⊂ script H sign ⊂ Θ, with Θ/script H sign commutative and script H sign/ℜ isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L2(ℝ̄+).
Originalsprog | Engelsk |
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Tidsskrift | Journal fur die Reine und Angewandte Mathematik |
Udgave nummer | 561 |
Sider (fra-til) | 145-175 |
Antal sider | 31 |
ISSN | 0075-4102 |
Status | Udgivet - 1 jan. 2003 |
ID: 237364664