TY - JOUR

T1 - C*-structure and K-theory of Boutet de Monvel's algebra

AU - Melo, S. T.

AU - Nest, R.

AU - Schrohe, E.

N1 - Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2003

Y1 - 2003

N2 - 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+).

AB - 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+).

UR - http://www.scopus.com/inward/record.url?scp=0042353687&partnerID=8YFLogxK

U2 - 10.1515/crll.2003.064

DO - 10.1515/crll.2003.064

M3 - Article

AN - SCOPUS:0042353687

SP - 145

EP - 175

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 561

ER -