English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Consider a proper, isometric action by a unimodular locally compact group  G  on a Riemannian manifold  M  with boundary, such that  M / G  is compact. Then an equivariant Dirac-type operator  D  on  M  under a suitable boundary condition has an equivariant index    $${{\,\mathrm{index}\,}}_G(D)$$  index G ( D ) in the  K -theory of the reduced group    $$C^*$$  C ∗ -algebra    $$C^*_rG$$  C r ∗ G of  G . This is a common generalisation of the Baum–Connes analytic assembly map and the (equivariant) Atiyah–Patodi–Singer index. In part I of this series, a numerical index    $${{\,\mathrm{index}\,}}_g(D)$$  index g ( D ) was defined for an element    $$g \in G$$  g ∈ G , in terms of a parametrix of  D  and a trace associated to  g . An Atiyah–Patodi–Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions,    $$\begin{aligned} \tau _g({{\,\mathrm{index}\,}}_G(D)) = {{\,\mathrm{index}\,}}_g(D), \end{aligned}$$  τ g ( index G ( D ) ) = index g ( D ) , for a trace    $$\tau _g$$  τ g defined by the orbital integral over the conjugacy class of  g . This implies that the index theorem from part I yields information about the  K -theoretic index    $${{\,\mathrm{index}\,}}_G(D)$$  index G ( D ) . It also shows that    $${{\,\mathrm{index}\,}}_g(D)$$  index g ( D ) is a homotopy-invariant quantity.

An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index