Subspaces defined by pseudodifferential projections, and some of their applications. (Q1432240)

This nicely written short paper reviews the results of the authors from [Sb. Math. 191, No. 8, 1191--1213 (2000; Zbl 0981.58018)] and [Sb. Math. 190, No. 8, 1195--1228 (1999; Zbl 0963.58008)]. The subspaces from the title are images in \(C^\infty(M,E)\) of pseudodifferential projections of order \(0\) over a closed manifold \(M\) which are ``admissible'', in the sense that the parity of their symbol with respect to the antipodal map is opposite to the parity of \(\dim M\). There exists a map \(d\) from the semigroup of homotopy classes of such projections into \(\mathbb Z[1/2]\) with the following property: If \(A\) is an elliptic pseudodifferential operator with ``parity'' opposite to \(\dim M\), and \(L_+(A)\) is the image of the spectral projection coming from the nonnegative eigenvalues of \(A\), then \(d(L_+(A))\) coincides with the eta invariant of \(A\). This implies that \(\eta(A)\) belongs to \(\mathbb Z[1/2]\), which answers positively a conjecture of \textit{P. Gilkey} [Adv. Math. 58, 243--284 (1985; Zbl 0602.58041)]. The functional \(d\) appears also in an index formula for elliptic operators acting on subspaces as above, as well as for the index of certain boundary-value problems.