\(K\)-theory of \(C^*\)-algebras of b-pseudodifferential operators. (English) Zbl 0898.46060

Melrose, R.; Nistor, V.

Geom. Funct. Anal. 8, No. 1, 88-122 (1998).

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 R. B. Melrose and 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.

Reviewer: L.Maxim-Răileanu (Iaşi)

MSC:

46L80	\(K\)-theory and operator algebras (including cyclic theory)
55N15	Topological \(K\)-theory
19K56	Index theory

Keywords:

indicial operator; \(K\)-groups; logarithmically divergent measure; Fredholm elements; spectral sequence; equivariant index of operators

Citations:

Zbl 0761.55002