\(K\)-theory of \(C^*\)-algebras of b-pseudodifferential operators
The algebra \(\psi^0_b(M)\) of \(b\)-pseudodifferential (or totally characteristic) operators acting on the compact manifold with corners \(M\), was described in Adv. Math. 92, No. 1, 1-26 (1992; Zbl 0761.55002) by \textit{R. B. Melrose} and \textit{P. Piazza}. In this paper, the authors compute the \(K\)-groups of the norm closure \({\mathcal U}(M)\) of this algebra. Identifying \(\psi^0_b(M)\) with a *-closed subalgebra of the bounded operators on \(L^2_b(M)= L^2(M,\Omega_b)\) (corresponding to logarithmically divergent measure), its Fredholm elements can then be characterized by the invertibility of a joint symbol consisting of the principal symbol, in the ordinary sense, and an ``indicial operator'' at each boundary face, which arises by freezing the coefficients at the boundary face in question. The principal symbol map \(\sigma\) has a continuous extension to \({\mathcal U}(M)\) with values in \(C(^bS^*M)\), where \({^bS^*}M\equiv S^*M\) as manifolds. The algebra \({\mathcal U}(M)\) contains the algebra \(K(L^2_b(M))\) of compact operators on \(L^2_b(M)\); the quotient \({\mathcal D}(M)={\mathcal U}(M)/K(L^2_b(M))\) is called the algebra of joint symbols since it involves both the principal symbol and extra morphisms giving the ``indicial operators''. By using the indicial maps, a composition series for \({\mathcal U}(M)\): \[ {\mathcal U}(M)\supset J_1\supset J_1\supset\cdots\supset J_n,\quad n=\dim M, \] is constructed and the subquotients of this composition series are identified as the sum over the boundary faces of dimension \(\ell\) of the \(C^*\)-algebras of continuous functions vanishing at infinity on \(\mathbb{R}^{n-1}\) and taking values in the compact operators on an associated Hilbert space. The end cases are: \(J_n\cong K(L^2_b(M))\) and \({\mathcal U}(M)/J_0\simeq C(^bS^*M)\). The \(K\)-theory of each of these subquotients is readily computed, and this leads to a spectral sequence for the \(K\)-theory of \({\mathcal U}(M)\). The relation between these results and the \(\eta\)-invariant is discussed. Finally, some results on the equivariant index of operators on manifolds equipped with a proper action of \(\mathbb{R}^k\) are given.
scientific article; zbMATH DE number 1123726