K-theory of Boutet de Monvel's algebra
Volume 61 / 2003
Banach Center Publications 61 (2003), 149-156
MSC: Primary 58J40, 46L80; Secondary 58J32, 35S15, 19K56.
DOI: 10.4064/bc61-0-10
Abstract
We consider the norm closure ${\mathfrak A}$ of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact manifold $X$ with boundary $\partial X$. Assuming that all connected components of $X$ have nonempty boundary, we show that $K_1({\mathfrak A})\simeq K_1(C(X))\oplus\ker\chi$, where $\chi:K_0(C_{0}(T^*\dot X))\to{\mathbb Z}$ is the topological index, $T^*\dot X$ denoting the cotangent bundle of the interior. Also $K_0({\mathfrak A})$ is topologically determined. In case $\partial X$ has torsion free K-theory, we get $K_0({\mathfrak A})\simeq K_0(C(X))\oplus K_1(C_{0}(T^*\dot X))$.
