The eta invariant and families of pseudodifferential operators (Q1910654)

Let \(Y\) be a compact manifold without boundary and denote by \(\Psi^*(Y)\) the algebra of 1-step polyhomogeneous (classical) pseudodifferential operators. The author constructs what he calls the suspended algebra \(\Psi^*_{\text{sus}}(Y)\). This is a subalgebra of \(\Psi^*(Y\times {\mathbf R})\) consisting of operators which are translation-invariant in the second variable, i.e. morally they act by convolution. Via the Fourier transform on \({R}\), one obtains one-parameter families of pseudodifferential operators, i.e. \(\Psi^*_{\text{sus}}(Y)\) is a sort of suspension of the algebra \(\Psi^*(Y)\). The suspension, \(C^\infty_0({R}, \Psi^{-\infty}(Y))\), of the algebra of smoothing operators is isomorphic to a subalgebra of \(\Psi^*_{\text{sus}}(Y)\). The author constructs a trace functional \(\overline{\text{Tr}}\colon \Psi^*_{\text{sus}}(Y)\longrightarrow C\). This trace functional is a regularization of the standard trace on the suspension of \(\Psi^{-\infty}(Y)\). More precisely, if \(A\in\Psi_{\text{sus}}^m(Y), m<-\dim Y-1\), then \(\overline{\text{Tr}}(A)=(2\pi)^{-1}\int_{-\infty}^\infty \widehat A(t)dt\), where \(\widehat A\) is the Fourier transform of \(A\) in the second variable. The author constructs a homomorphism \(\eta\) from the group of invertible elements in \(\Psi^*_{\text{sus}}(Y)\) into the real numbers. If \(A=\text{Id}+S\) with \(S\in \Psi_{\text{sus}}^m, m<-\dim Y\), then \(\eta(A)\) is just the winding number of the Fredhom determinant \(\det(\text{Id}+\widehat S)\). Thus, \(\eta\) is a regularized version of this winding number. A surprising feature of this \(\eta\)-homomorphism is the fact that for a Dirac operator \(D\), the (spectrally defined) \(\eta\)-invariant coincides with the \(\eta\)-homomorphism evaluated on the operator \(D+\partial/\partial t\). Although this is stated only for Dirac operators, this result seems to hold for arbitrary first-order elliptic self-adjoint differential operators. Furthermore, the author gives a variation formula for \(\eta\) and he shows that \(\eta\) leads to an invariant which distinguishes the connected components of \(\text{Inv}_m\cap\text{Ell}_m(A_0)\). Here \(\text{Inv}_m\) denotes the group of invertible pseudodifferential operators of order \(m\) and \(\text{Ell}_m(A_0)\) denotes the path component of \(A_0\) within elliptic pseudodifferential operators of order \(m\).