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Pseudodifferential subspaces and their applications in elliptic theory. (English) Zbl 1113.58010
Bojarski, Bogdan (ed.) et al., \(C^*\)-algebras and elliptic theory. Basically formed by contributions from the international conference, Bȩdlewo, Poland, February 2004. In cooperation with Dan Burghelea, Richard Melrose and Victor Nistor. Basel: Birkhäuser (ISBN 3-7643-7686-4/hbk). Trends in Mathematics, 247-289 (2006).
Summary: The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah-Patodi-Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological \(K\)-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on the theory of elliptic operators on closed manifolds acting in subspaces.
For the entire collection see [Zbl 1097.58001].

58J20 Index theory and related fixed-point theorems on manifolds
58J28 Eta-invariants, Chern-Simons invariants
58J32 Boundary value problems on manifolds
19K56 Index theory