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.