An index theorem for families of Fredholm operators parametrized by a group
The author studies the problem of realizing real cohomology of some infinite dimensional Lie groups by left invariant differential forms. The stable complex general linear group GL(\(\infty)\) is considered as a subgroup of GL, the group of invertible operators in a separable complex Hilbert space \({\mathcal H}\). For all \(p\geq 1\) the Schatten ideals \(L^ p\) of compact operators in \({\mathcal H}\) are used to construct Banach Lie groups \(GL^ p\). By a theorem of R. Palais all these groups are homotopy equivalent to GL(\(\infty)\). It is proved that every singular cohomology class of GL(\(\infty)\) and \(GL^ 1\) is representable by an invariant form, and that the real cohomology of \(G^ p\), \(p>1\), is represented by invariant forms in sufficiently large degrees depending on p. For smooth principal \(GL^ p\)-bundles \({\mathcal P}\to {\mathcal M}^ a \)Chern-Weil theorem is proved, expressing the Chern character classes \(ch_{\ell}({\mathcal P})\) by a differential form on \({\mathcal M}.\) To study the cohomology of the classifying space BGL(\(\infty)\) explicitly the author constructs a tower of groups \({\mathcal G}^ p=GL/GL^ p\). In absence of a natural good topology \({\mathcal G}^ p\) is considered as an abstract group only. There exists a projection \(\pi\) : \(Fred_ 0\to {\mathcal G}^ p\) from the space of Fredholm operators with index zero to \({\mathcal G}^ p\). Although GL\(\to {\mathcal G}^ p\) is not a bundle, the pullback of GL to \(Fred_ 0\) is a principal \(GL^ p\)-bundle and a model of the universal fibration EGL(\(\infty)\to BGL(\infty)\). The author regards smooth maps T: \(G\to Fred_ 0\) from a Banach Lie group G to \(Fred_ 0\) such that the composition \(G\to^{T}Fred_ 0\to^{\pi}{\mathcal G}^ p\) is a homomorphism of abstract groups. In this case he constructs a left invariant \(L^ p\)-valued 2-form \(\Omega\) on G, which is used to represent \(T^*ch_{\ell}\) as a left invariant form on G for \(\ell \geq p\). This is the index theorem of the title. In the special case of a graded Hilbert space \({\mathcal H}\) it is possible to construct certain groups of operators which are Banach Lie groups and realize BGL(\(\infty)\). From the index theorem the author obtains invariant differential forms representing the Chern character.
scientific article; zbMATH DE number 4102110