unless $X$ is $G$-compact. Nonetheless, the map

$g\mapsto g(c)a,g\mapsto g(c)S$

are norm-continuous for any $a\in A$ and for any $S\in\mathfrak{K}(H_{X})$.
Analogous remark applies to some of the arguments in the subsequent proof.

Let $X^{+}$ be the one-point compactification of $X$ and fix any metric on
$X^{+}$ which is compatible with its topology. Let $A$ be the support of a
cut-off function $c$ on $X$. The closed set $A\subset X$ is not compact
(unless $X$ is $G$-compact) but it satisfies that for any compact subset
$X_{0}$ of $X$, $A\cap gX_{0}=\emptyset$ for $g$ outside a compact set of $G$.
We have $GA=X$.

For $n\geq 1$, let $\mathcal{U}_{n}$ be a finite open cover of $X^{+}$ such
that each $U\in\mathcal{U}_{n}$ is contained in a ball of radius $1/n$. Let
$(\phi^{0}_{k,n})_{k\in S_{n}}$ be a finite partition of unity in
$C(X^{+})=C_{0}(X)+\mathbb{C}1_{X^{+}}$ which is sub-ordinate to the cover
$\mathcal{U}_{n}$. We set $\phi_{k,n}=c^{2}\phi^{0}_{k,n}\in C_{b}(X)$. Thus,
we have

$\sum_{k\in S_{n}}\phi_{k,n}=c^{2}$

and the support of $\phi_{k,n}$ is contained in a ball of radius $1/n$ in $A$.
We have

$\sum_{k\in S_{n}}\int_{g\in G}g(\phi_{k,n})d\mu_{G}(g)=1.$

Let

$F_{n}=\sum_{k\in S_{n}}\int_{g\in
G}g(\phi_{k,n})^{1/2}F_{0}g(\phi_{k,n})^{1/2}d\mu_{G}(g).$

We see that $F_{n}$ is an odd, $G$-equivariant, self-adjoint operator on
$H_{X}$ such that

$\lVert F_{n}\rVert\leq\lVert F_{0}\rVert.$

The last one can be seen from $F_{n}=V^{\ast}_{n}\tilde{F}_{0}V_{n}$ where

$V_{n}\colon H_{X}\to\bigoplus_{k\in S_{n}}L^{2}(G)\otimes H_{X}$

is an isometry which sends $v\in H_{X}$ to

$V_{n}(v)(k,g)=g(\phi_{k,n})^{1/2}v\,\,\,(k\in S_{n},g\in G)$

and where $\tilde{F}_{0}$ on $\bigoplus_{k\in S_{n}}L^{2}(G)\otimes H_{X}$ is
the diagonal operator $((k,g)\mapsto F_{0})\in C_{b}(S_{n}\times
G,\mathfrak{L}(H_{X}))$.

We have

$a(F_{n}-F_{0})=\sum_{k\in
S_{n}}\int_{G}ag(\phi_{k,n})^{1/2}[F_{0},g(\phi_{k,n})^{1/2}]d\mu_{G}(g)\in\mathfrak{K}(H_{X})$

for any $a\in A$. The integral is norm convergent in $\mathfrak{K}(H_{X})$ for
$a\in A$ with compact support. For $t\in(n,n+1)$, we set

$F_{t}=(n+1-t)F_{n}+(t-n)F_{n+1}.$

It is clear that $t\mapsto F_{t}\in\mathfrak{L}(H_{X})$ is (uniformly) norm-
continuous, $G$-equivariant, $\lVert F_{t}\rVert\leq\lVert F_{0}\rVert$ and

$a(F-F_{0})\in C_{b}([1,\infty),\mathfrak{K}(H_{X}))$

for any $a\in A$. We show that $F_{t}$ satisfies
$\lim_{t\to\infty}\lVert[F_{t},\phi]\rVert=0$ for any $\phi\in C_{0}(X)$. Let
$\phi\in C_{c}(X)$. We have

$[F_{n},\phi]=\sum_{k\in
S_{n}}\int_{G}g(\phi_{k,n})^{1/2}[F_{0},\phi]g(\phi_{k,n})^{1/2}d\mu_{G}(g).$

Recall that for any $n\geq 1$ and $k\in S_{n}$, the support of $\phi_{k,n}$ is
contained in a ball of radius $1/n$ in $A$. There is a compact subset $K$ of
$G$ such that $g(\phi_{k,n})\phi=0$ for $g\notin K$, and so

$g(\phi_{k,n})^{1/2}[F_{0},\phi]g(\phi_{k,n})^{1/2}=0$

for any $n\geq 1$, $k\in S_{n}$ and for any $g\notin K$. Now, consider a
family of functions $g^{-1}(\phi)\in C_{0}(X)$ for $g\in K$. This family is
compact in $C_{0}(X)\subset C(X^{+})$ so $g^{-1}(\phi)$ $(g\in K)$ are equi-
uniformly-continuous on $X^{+}$. It follows that for any $\epsilon$, there is
$N>0$ so that

$|g^{-1}(\phi)(x)-g^{-1}(\phi)(x_{0})|<\epsilon$

holds for any $n\geq N$, for any $g\in K$ and for any
$x,x_{0}\in\mathrm{supp}(\phi_{k,n})$ for any $k\in S_{n}$. In other words,

$|\phi(x)-\phi(x_{0})|<\epsilon$

holds for any $n\geq N$, for any $x,x_{0}\in\mathrm{supp}(g(\phi_{k,n}))$ for
any $g\in K$ and $k\in S_{n}$. Using this, we see that

$\lVert[F_{n},\phi]\rVert=||\sum_{k\in
S_{n}}\int_{K}g(\phi_{k,n})^{1/2}[F_{0},\phi-\phi(gx_{k,n})]g(\phi_{k,n})^{1/2}d\mu_{G}(g)||<2\epsilon\lVert
F_{0}\rVert$

for $n\geq N$ where $x_{k,n}$ is any point in $\mathrm{supp}(\phi_{k,n})$.
This shows $\lim_{t\to\infty}\lVert[F_{t},\phi]\rVert=0$.

It follows that $(t\mapsto F_{t})\in\mathfrak{L}(H_{X}\otimes[1,\infty))$
defines an odd, $G$-equivariant, self-adjoint element $F$ in
$M(RL^{*}_{c}(H_{X}))$ such that

$a(F-F_{0})\in C_{b}([1,\infty),\mathfrak{K}(H_{X}))$

for any $a\in A$. It follow from this and from the properties of $F_{0}$,

$a(1-F^{2}),[a,F],a(g(F)-F)\in C_{b}([1,\infty),\mathfrak{K}(H_{X}))$

for any $a\in A$ and $g\in G$. Moreover, if $x$ is any of these elements
$a(1-F^{2}),[a,F],a(g(F)-F)$, we can see that for any $\phi\in C_{0}(X)$,

$\lVert[x,\phi]\rVert\in C_{0}([1,\infty),\mathfrak{K}(H_{X}))$

because $[F,\phi]\in C_{0}([1,\infty),\mathfrak{L}(H_{X}))$. We can also see
that for any $\epsilon>0$, there is $\phi\in C_{0}(X)$

$\lVert(1-\phi)x\rVert<\epsilon.$

These imply $x\in RL^{*}_{c}(H_{X})$ as desired. For example, to see the last
assertion holds for $x=[a,F]$, we can assume $a$ has compact support so that
$a=\chi a$ for $\chi\in C_{c}(X)$. Then, we have

$(1-\phi)[a,F]=(1-\phi)aF-(1-\phi)F\chi
a=(1-\phi)aF-(1-\phi)[F,\chi]a-(1-\phi)\chi Fa$

which can be made arbitrarily small for some $\phi\in C_{0}(X)$ since
$[F,\chi]a\in C_{0}([1,\infty),\mathfrak{K}(H_{X}))$. This ends our proof. ∎

######  Lemma 14.8.

Let $X$ be a proper $G$-space and $A$ be a separable, graded
$G$-$C_{0}(X)$-algebra which is non-degenerately represented on a graded
$G$-Hilbert space $H$. We view $H$ naturally as an $X$-$G$-module through this
representation and set $H_{X}=H$. Let $D_{1}\subset M(A)$ and $D_{2}\subset
M(RL^{*}_{c}(H_{X}))$ be separable $G$-$C^{*}$-subalgebras such that
$[A,D_{2}]\subset RL^{*}_{c}(H_{X})$. Let $D_{3}$ be the subalgebra of $D_{2}$
consisting of $d\in D_{2}$ such that $ad\in RL^{*}_{c}(H_{X})$ for $a\in A$.

There are even, $G$-continuous elements $M,N=1-M$ in $M(RL^{*}_{c}(H_{X}))$
such that

  1. (I)

$[M,D_{1}]\subset RL^{*}_{c}(H_{X})$,

  2. (II)

$[M,D_{2}]\subset RL^{*}_{c}(H_{X})$,

  3. (III)

$M(D_{1}\cap A)\subset RL^{*}_{c}(H_{X})$,

  4. (IV)

$ND_{3}\subset RL^{*}_{c}(H_{X})$,

  5. (V)

$g(M)-M\in RL^{*}_{c}(H_{X})$.

###### Proof.

We construct $M,N=1-M\in M(RL^{*}_{c}(H_{X}))$ following the standard argument
(see [7, Theorem 12.4.2], [23], [30, Section 1.4],[28, Section 3.8]). Let
$U_{n}$ be an increasing sequence of relatively compact, open subsets of $G$
such that $\cup U_{n}=G$ and let $K_{n}$ be the closure of $U_{n}$. Let $Y$ be
a compact total subset of $C_{0}(X)$. For $i=1,2,3$, let $Y_{i}$ be a compact
total subset of $D_{i}$ such that $Y_{1}\cap A$ is total in $D_{1}\cap A$. Let
$Z$ be a compact total subset of $C_{0}([1,\infty),\mathfrak{K}(H_{X}))$.

First, let $(a_{n})_{n\geq 0}$ ($a_{0}=0$) be an even, quasi-central
approximate unit in $A\subset M(A)$ so that $d_{n}=(a_{n+1}-a_{n})^{1/2}$
satisfies for $n\geq 1$,

  1. (1.1)

$\lVert[d_{n},x]\rVert<2^{-n}$ for $x\in Y_{1}$,

  2. (1.2)

$\lVert d_{n}x\rVert<2^{-n}$ for $x\in Y_{1}\cap A$,

  3. (1.3)

$\lVert g(d_{n})-d_{n}\rVert<2^{-n}$ for $g\in K_{n}\subset G$,

  4. (1.4)

$\lVert d_{n}x\rVert<2^{-n}$ for $x\in Z$.

Secondly, let $J$ be a separable $G$-$C^{*}$-subalgebra of $RL^{*}_{c}(H_{X})$
which contains the ideal $RL^{*}_{0}(H_{X})$, $[A,D_{2}]$ and $AD_{3}$. We
arrange $J$ so that elements in $D_{1}$, $D_{2}$ and $C_{0}(X)$ multiply (and
hence, derive) $J$. This is possible because $D_{1}$, $D_{2}$ and $C_{0}(X)$
multiply $RL^{*}_{c}(H_{X})$.

Thirdly, let $u_{n}$ be an even, quasi-central approximate unit in $J$ so that
$u_{n}$ satisfies

  1. (2.1)

$\lVert[u_{n},x]\rVert<2^{-n}$ for $x\in Y\cup Y_{1}\cup Y_{2}$,

  2. (2.2)

$\lVert(1-u_{n})x\rVert<2^{-n}$ for $x\in
d_{n}Y_{3}\cup[d_{n},Y_{2}]\cap[d_{n},Y_{2}]^{\ast}$,

  3. (2.3)

$\lVert g(d_{n})-d_{n}\rVert<2^{-n}$ for $g\in K_{n}\subset G$.

Now, we consider

$M=\sum_{n\geq 0}d_{n}u_{n}d_{n},\,\,\,N=1-M=\sum_{n\geq
0}d_{n}(1-u_{n})d_{n}.$

By (1.4), these infinite sums converge in the strict topology in
$M(\mathfrak{K}(H_{X})\otimes C_{0}[1,\infty))$ to define an element in
$\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$ (see [7, Proposition 12.1.2]) but
they may not converge in the strict topology in $M(RL^{*}_{c}(H_{X}))$. We
claim that $M\in M(RL^{*}_{c}(H_{X}))$ and so is $N=1-M$. This is because for
any $\phi$ in $C_{0}(X)$,

$[M,\phi]=\sum_{n\geq 0}d_{n}[u_{n},\phi]d_{n}\in
C_{0}([1,\infty),\mathfrak{K}(H_{X}))$

which absolutely converges in $C_{0}([1,\infty),\mathfrak{K}(H_{X}))$ for
$\phi\in Y$ by (2.1). Note that $[u_{n},\phi]\in
C_{0}([1,\infty),\mathfrak{K}(H_{X}))$ since $u_{n}\in J\subset
RL^{*}_{c}(H_{X})$.

We check $M,N$ satisfy the conditions (I) - (V).

  1. (I)

$[M,D_{1}]\subset RL^{*}_{c}(H_{X})$ follows from (1.1) and (2.1).

  2. (II)

$[M,D_{2}]=[N,D_{2}]\subset RL^{*}_{c}(H_{X})$ follows from (2.1) and (2.2).

  3. (III)

$M(D_{1}\cap A)\subset RL^{*}_{c}(H_{X})$ follows from (1.2).

  4. (IV)

$ND_{3}\subset RL^{*}_{c}(H_{X})$ follows from (2.2).

  5. (V)

$g(M)-M\in RL^{*}_{c}(H_{X})$ follows from (1.3) and (2.3).

That $M$ is $G$-continuous follows from (1.3) and (2.3). ∎

###### Proof of Proposition 14.5.

Let $y\in KK^{G}(\mathbb{C},A)$ and $z\in KK^{G}(A,\mathbb{C})$ so that
$x=y\otimes_{A}z$. By stabilizing $A$ by $\mathfrak{K}(H_{0})$ for a graded
Hilbert space $H_{0}$ if necessary, we can assume that $y$ is represented by
an odd, $G$-continuous, self-adjoint element $b\in M(A)$ such that $1-b^{2}\in
A$ and $g(b)-b\in A$ for $g\in G$. We can also assume that $z$ is represented
by a cycle $(H,\pi,F_{0})$ with $\pi$ non-degenerate. As in Lemma 14.7, we set
$H_{X}=H$ and let $F\in M(RL^{*}_{c}(H_{X}))$ be as given by the lemma. Now,
let $M,N=1-M$ in $M(RL^{*}_{c}(H_{X}))$ as given by Lemma 14.8 for unital
separable $G$-$C^{*}$-algebras $D_{1}\subset M(A)$ containing $b$ and
$D_{2}\subset M(RL^{*}_{c}(H_{X}))$ containing $F$, $[F,b]$. Note we have

$1-b^{2},g(b)-b\in D_{1}\cap A,\,\,\,1-F^{2},g(F)-F,[F,b]\in D_{3}$

for $g\in G$. Let

$T=M^{1/4}bM^{1/4}+N^{1/4}FN^{1/4}\in M(RL^{*}_{c}(H_{X})).$

It is routine to check that this odd, $G$-continuous, self-adjoint element $T$
satisfies $1-T^{2}\in RL^{*}_{c}(H_{X})$ and $g(T)-T\in RL^{*}_{c}(H_{X})$ for
any $g$ in $G$. Thus, $(H_{X},T)$ is an $X$-$G$-localized Kasparov cycle for
$KK^{G}(\mathbb{C},\mathbb{C})$. By the construction, and since
$a(F_{1}-F_{0})\in\mathfrak{K}(H_{X})$ for any $a\in A$, the pair
$(H_{X},T_{1})$ for $KK^{G}(\mathbb{C},\mathbb{C})$ represents the Kasparov
product of $y$ and $z$, that is $[H_{X},T_{1}]=x$.

∎

##  15\. Controlled algebraic aspect of the gamma element method

We recall for any $x\in R(G)=KK^{G}(\mathbb{C},\mathbb{C})$ and for any
separable (graded) $G$-$C^{*}$-algebra $B$, we have the following canonical
ring homomorphism

$\textstyle{KK^{G}(\mathbb{C},\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{B}}$$\textstyle{KK^{G}(B,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{G}_{r}}$$\textstyle{KK(B\rtimes_{r}G,B\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{End}(K_{\ast}(B\rtimes_{r}G))}$

where the last map is defined by the Kasparov product. Let us write the image
of $x$ by

$x^{B\rtimes_{r}G}_{\ast}\in\mathrm{End}(K_{\ast}(B\rtimes_{r}G)).$

In this section, we prove the following:

######  Theorem 15.1.

Let $X$ be a proper $G$-space. Suppose that an element $x\in R(G)$ is
$X$-$G$-localized, that is $x=[H_{X},T_{1}]$ for an $X$-$G$-localized Kasparov
cycle $(H_{X},T)$ for $KK^{G}(\mathbb{C},\mathbb{C})$. Then, there is a
natural group homomorphism

$\nu^{B,T}\colon K_{\ast}(B\rtimes_{r}G)\to\mathbb{D}^{B,G}_{\ast}(X)$

for any separable $G$-$C^{*}$-algebra $B$ such that

$x^{B\rtimes_{r}G}_{\ast}=\mathcal{F}\circ\nu^{B,T}\colon
K_{\ast}(B\rtimes_{r}G)\to\mathbb{D}^{B,G}_{\ast}(X)\to
K_{\ast}(B\rtimes_{r}G).$

In particular, $x^{B\rtimes_{r}G}_{\ast}$ factors through the Baum–Connes
assembly map $\mu_{r}^{B,G}$. Here, $\nu^{B,T}$ is natural with respect to a
$G$-equivariant $\ast$-homomorphism $B_{1}\to B_{2}$ in a sense that the
following diagram commutes

$\textstyle{K_{\ast}(B_{1}\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{B_{1},T}}$$\scriptstyle{\pi\rtimes_{r}1_{\ast}}$$\textstyle{\mathbb{D}^{B_{1},G}_{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\textstyle{K_{\ast}(B_{2}\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{B_{2},T}}$$\textstyle{\mathbb{D}^{B_{2},G}_{\ast}(X).}$

######  Theorem 15.2.

Let $X$ be a proper $G$-space. If there is an $X$-$G$-localized element $x\in
R(G)$ such that $x=1_{K}$ in $R(K)$ for any compact subgroup $K$, the
Baum–Connes assembly map $\mu_{r}^{B,G}$ is split-injective for any $B$ and in
this case the image of $\mu_{r}^{B,G}$ coincides with the image of
$x^{B\rtimes_{r}G}_{\ast}$. In particular, if $x=1_{G}$, BCC holds for $G$.

The proof of these theorems will be quite simple and formal. We start with the
construction of $\nu^{B,T}$. We should think that the map $\nu^{B,T}$ is
obtained by sending an $X$-$G$-localized cycle $(H_{X},T)$ through

$KK^{G}(\mathbb{C},RL^{*}_{c}(H_{X}))\to KK^{G}(B,RL^{*}_{c}(H_{X}\otimes
B))\to KK(B\rtimes_{r}G,RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)$
$\to\mathrm{End}(K_{\ast}(B\rtimes_{r}G),K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G))$

where the first map is $\sigma_{B}$ followed by the inclusion
$RL^{*}_{c}(H_{X})\otimes B\to RL^{*}_{c}(H_{X}\otimes B)$, the second map is
Kasparov’s descent $j^{G}_{r}$ and the last map is given by the Kasparov
product. Although we can take it as the definition of $\nu^{B,T}$, we reduce
this to the separable $G$-$C^{*}$-algebras setting.

Recall, for any $C^{*}$-subalgebra $A$ of $RL^{*}_{c}(H_{X})$ containing the
ideal $RL^{*}_{0}(H_{X})$, the multiplier algebra $M(A)$ is naturally a
subalgebra of $M(RL^{*}_{0}(H_{X}))$. Thus, it makes sense to ask if an
operator $T$ in $M(RL^{*}_{c}(H_{X}))\subset M(RL^{*}_{0}(H_{X}))$ belongs to
$M(A)$.

######  Lemma 15.3.

Let $(H_{X},T)$ be an $X$-$G$-localized Kasparov cycle for
$KK^{G}(\mathbb{C},\mathbb{C})$. Then, there is a separable, graded
$G$-$C^{*}$-subalgebra $A_{X}$ of $RL^{*}_{c}(H_{X})$ containing the ideal
$RL^{*}_{0}(H_{X})$ such that

$T\in M(A_{X})\subset M(RL^{*}_{0}(H_{X}))$

and for any $g\in G$,

$1-T^{2},g(T)-T\in A_{X}$

Note, $T$ is automatically a $G$-continuous element in $M(A_{X})$.

###### Proof.

Just take any separable, graded $G$-$C^{*}$-subalgebra $A_{X}$ of
$RL^{*}_{c}(H_{X})$ containing the ideal $RL^{*}_{0}(H_{X})$, such that it
contains $1-T^{2}$ and $T-g(T)$ for all $g\in G$ and such that $T$ multiplies
$A_{X}$. ∎

Now, for any $X$-$G$-localized cycle $(H_{X},T)$ for
$KK^{G}(\mathbb{C},\mathbb{C})$, let $A_{X}$ be a separable, graded
$G$-$C^{*}$-subalgebra $A_{X}$ of $RL^{*}_{c}(H_{X})$ as in the previous
lemma. Then, the pair $(A_{X},T)$ is a cycle for $KK^{G}(\mathbb{C},A_{X})$
and hence by applying the usual homomorphism

$\textstyle{KK^{G}(\mathbb{C},A_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{G}_{r}\circ\sigma_{B}}$$\textstyle{\mathrm{End}(K_{\ast}(B\rtimes_{r}G),K_{\ast}((A_{X}\otimes
B)\rtimes_{r}G)),}$

we obtain a group homomorphism

$(j^{G}_{r}\circ\sigma_{B})[A_{X},T]_{\ast}\colon K_{\ast}(B\rtimes_{r}G)\to
K_{\ast}((A_{X}\otimes B)\rtimes_{r}G)$

which is natural with respect to a $G$-equivariant $\ast$-homomorphism
$B_{1}\to B_{2}$. The desired group homomorphism

$\nu^{B,T}\colon K_{\ast}(B\rtimes_{r}G)\to K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)$

is obtained as the composition of $(j^{G}_{r}\circ\sigma_{B})[A_{X},T]_{\ast}$
with the $\ast$-homomorphism

$(A_{X}\otimes B)\rtimes_{r}G\to(RL^{*}_{c}(H_{X})\otimes B)\rtimes_{r}G\to
RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G.$

It is clear, from the naturality of Kasparov product and the descent map, that
$\nu^{B,T}$ is natural with respect to a $G$-equivariant $\ast$-homomorphism
$B_{1}\to B_{2}$.

Note that for a graded $X$-$G$-module $H_{X}$, the K-theory group
$K_{\ast}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)$ of the graded
$C^{*}$-algebra $RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G$ is functorial with
respect to a graded, $G$-equivariant continuous cover $(V_{t}\colon H_{X}\to
H_{Y})_{t\in[1,\infty)}$ for a $G$-equivariant map $f\colon X\to Y$. Such a
cover exists for any $H_{X}$ whenever $H_{Y}$ is universal, that is if both
the even space $H_{Y}^{(0)}$ and the odd space $H_{Y}^{(1)}$ is universal.
Moreover, if $H^{0}_{X}$ is a (ungraded) $X$-$G$-module and $H_{X}$ is a
graded $X$-$G$-module given by $H_{X}^{(0)}=H_{X}^{(1)}=H^{0}_{X}$, the
inclusion

$RL^{*}_{c}(H^{0}_{X}\otimes B)\to RL^{*}_{c}(H_{X}\otimes B)$

induces an isomorphism

$K_{\ast}(RL^{*}_{c}(H^{0}_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G).$

Therefore, we may redefine the functor $\mathbb{D}^{B,G}_{\ast}(X)$ by using a
graded universal $X$-$G$-module $H_{X}$ instead, and this functor is naturally
equivalent to the original one. Moreover, in this case, for any graded
$X$-$G$-module $H_{X}$, we have a natural group homomorphism

$K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\to\mathbb{D}^{B,G}_{\ast}(X).$

The forget control map

$\mathcal{F}\colon\mathbb{D}^{B,G}_{\ast}(X)\to K_{\ast}(B\rtimes_{r}G)$

is defined as before. With these in mind, the natural group homomorphism
$\nu^{B,T}\colon K_{\ast}(B\rtimes_{r}G)\to K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)$ extends to

$\nu^{B,T}\colon K_{\ast}(B\rtimes_{r}G)\to\mathbb{D}^{B,G}_{\ast}(X).$

######  Lemma 15.4.

Let $(H_{X},T)$ be an $X$-$G$-localized cycle for
$KK^{G}(\mathbb{C},\mathbb{C})$ and $x=[H_{X},T_{1}]$ in $R(G)$. Then, we have

$\mathcal{F}\circ\nu^{B,T}=x^{B\rtimes_{r}G}_{\ast}$

on $K_{\ast}(B\rtimes_{r}G)$.

###### Proof.

The evaluation $\mathrm{ev}_{1}\colon RL^{*}_{c}(H_{X})\to\mathfrak{K}(H_{X})$
restricts to $A_{X}$ and it extends to the map

$\mathrm{ev}_{1}\colon M(A_{X})\to\mathfrak{L}(H_{X})$

which sends $T$ to $T_{1}$. The evaluation map $\mathrm{ev}_{1}\colon
A_{X}\to\mathfrak{K}(H_{X})$ defines an element $[H_{X},\mathrm{ev}_{1},0]\in
KK^{G}(A_{X},\mathbb{C})$ whose Kasparov product with $[A_{X},T]\in
KK^{G}(\mathbb{C},A_{X})$ is just $x=[H_{X},T_{1}]$ in
$KK^{G}(\mathbb{C},\mathbb{C})$. It is easy to see that the composition
$\mathcal{F}\circ\nu^{B,T}$ coincides with the composition of

$(j^{G}_{r}\circ\sigma_{B})[A_{X},T]_{\ast}\colon K_{\ast}(B\rtimes_{r}G)\to
K_{\ast}((A_{X}\otimes B)\rtimes_{r}G)$

and

$(j^{G}_{r}\circ\sigma_{B})[H_{X},\mathrm{ev}_{1},0]_{\ast}\colon
K_{\ast}((A_{X}\otimes B)\rtimes_{r}G)\to K_{\ast}(B\rtimes_{r}G)$

defined by the element $[H_{X},\mathrm{ev}_{1},0]\in KK^{G}(A_{X},\mathbb{C})$
through

$\textstyle{KK^{G}(A_{X},\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{G}_{r}\circ\sigma_{B}}$$\textstyle{\mathrm{End}(K_{\ast}(A_{X}\otimes
B\rtimes_{r}G),K_{\ast}(B\rtimes_{r}G)).}$

From this, we can see that
$\mathcal{F}\circ\nu^{B,T}=x^{B\rtimes_{r}G}_{\ast}$. ∎

###### Proof of Theorem 15.1.

The first part of the theorem follows from Lemma 15.4. The second part follows
from the first part and from Theorem 9.13. Note that we do not need the
isomorphism Theorem 13.2. ∎

###### Proof of Theorem 15.2.

Thanks to Theorem 15.1, there are group homomorphisms

$\nu^{B,T}\colon
K_{\ast}(B\rtimes_{r}G)\to\varinjlim_{Y\subset\underline{E}G,\mathrm{Gcpt}}K_{\ast}^{G}(C_{0}(Y),B)$

which are natural with respect to $B$ and we have

$\mu_{r}^{B,G}\circ\nu^{B,T}=x^{B\rtimes_{r}G}_{\ast}\colon
K_{\ast}(B\rtimes_{r}G)\to K_{\ast}(B\rtimes_{r}G).$

Under the assumption $x=1_{K}$ in $R(K)$ for any compact subgroup $K$ of $G$,
we can show that the other composition $\nu^{B,T}\circ\mu_{r}^{B,G}$ is the
identity for all $B$ just as in the proof of [38, Proposition 5.3] using the
naturality of $\mu_{r}^{B,G}$ and of $\nu^{B,T}$ with respect to a
$G$-equivariant $\ast$-homomorphism $\pi\colon B_{1}\to B_{2}$. It follows
that $\mu_{r}^{B,G}$ is split-injective for all $B$ and the image of
$\mu_{r}^{B,G}$ coincides with the idempotent
$\mu_{r}^{B,G}\circ\nu^{B,G}=x^{B\rtimes_{r}G}_{\ast}$ on
$K_{\ast}(B\rtimes_{r}G)$. ∎

Description of the splitting of the forget-control map Let $(H_{X},T)$ be an
$X$-$G$-localized Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$. We give
an explicit description of the map

$\nu^{B,T}\colon K_{0}(B\rtimes_{r}G)\to\mathbb{D}^{B,G}_{0}(X)$

which is a splitting of the forget-control map $\mathcal{F}$ if
$[H_{X},T_{1}]=1_{K}$ in $R(K)$ for any compact subgroup $K$ of $G$.

Let $\mathcal{H}_{0}=l^{2}(\mathbb{N})^{(0)}\oplus l^{2}(\mathbb{N})^{(1)}$ be
the standard graded Hilbert space. For any (not necessarily separable) graded
$C^{*}$-algebra $A$, the K-theory group $K_{0}(A)=KK(\mathbb{C},A)$ can be
identified (see [42, Section 3]) as the set of homotopy equivalence classes of
odd, self-adjoint operators $F\in
M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}A)\cong\mathfrak{L}(\mathcal{H}_{0}\hat{\otimes}A)$
such that
$1-F^{2}\in\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}A\cong\mathfrak{K}(\mathcal{H}_{0}\hat{\otimes}A)$.

We describe the image $\nu^{B,T}([F])\in\mathbb{D}^{B,G}_{0}(X)$ of $[F]\in
K_{0}(B\rtimes_{r}G)$ for $F\in
M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}B\rtimes_{r}G)$.

We have canonical representations

$M(RL^{*}_{c}(H_{X}))\to
M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}(RL^{*}_{c}(H_{X})\otimes
B)\rtimes_{r}G)$
$M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}B\rtimes_{r}G)\to
M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}(RL^{*}_{c}(H_{X})\otimes
B)\rtimes_{r}G)$

We still denote by $F$, resp. by $T$, the image of $F\in
M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}B\rtimes_{r}G)$, resp. $T\in
M(RL^{*}_{c}(H_{X}))$ in
$M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}(RL^{*}_{c}(H_{X})\otimes
B)\rtimes_{r}G)$ by these representations.

Let

$T\sharp F=F+(1-F^{2})^{1/4}T(1-F^{2})^{1/4}\in
M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}(RL^{*}_{c}(H_{X})\otimes
B)\rtimes_{r}G).$

It satisfies

$1-(T\sharp
F)^{2}\in\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}(RL^{*}_{c}(H_{X})\otimes
B)\rtimes_{r}G.$

Hence, the odd, self-adjoint element $T\sharp F$ defines the element $[T\sharp
F]$ in

$K_{0}((RL^{*}_{c}(H_{X})\otimes B)\rtimes_{r}G)\to
K_{0}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\to\mathbb{D}_{0}^{B,G}(X).$

We have

$\nu^{B,T}[F]=[T\sharp F]\in\mathbb{D}^{B,G}_{0}(X).$

In this sense, $\nu^{B,T}$ is just given by the Kasparov product by $T$. Of
course, the forget control map $\mathcal{F}\colon\mathbb{D}^{B,G}_{0}(X)\to
K_{0}(B\rtimes_{r}G)$ sends this element $[T\sharp F]$ to

$[T_{1}\sharp F]\in K_{0}(B\rtimes_{r}G)$

represented by an odd, self-adjoint operator

$T_{1}\sharp F=F+(1-F^{2})^{1/4}T_{1}(1-F^{2})^{1/4}$

in $M(\mathfrak{K}(\mathcal{H}_{0})\hat{\otimes}(\mathfrak{K}(H_{X})\otimes
B)\rtimes_{r}G)\cong\mathfrak{L}(\mathcal{H}_{0}\hat{\otimes}H_{X}\otimes
B\rtimes_{r}G)$, which represents the Kasparov product
$[F]\otimes_{B\rtimes_{r}G}j^{G}_{r}(\sigma_{B}[H_{X},T_{1}])$ in
$KK(\mathbb{C},B\rtimes_{r}G)$.

##  16\. Examples

In this section, we give some examples of $X$-$G$-localized Kasparov cycles
for $KK^{G}(\mathbb{C},\mathbb{C})$.

Recall from [3] that an unbounded cycle for $KK^{G}(\mathbb{C},\mathbb{C})$ is
a pair $(H,D)$ of a separable, graded $G$-Hilbert space $H$ and an odd, self-
adjoint unbounded operator $D$ on $H$ such that

  * •

$(D\pm i)^{-1}\in\mathfrak{K}(H)$ ($D$ has compact resolvent) and

  * •

the $G$-action on $H$ preserves the domain of $D$, and $g(D)-D$ extends to a
strongly continuous, locally bounded, $\mathfrak{L}(H)$-valued function of
$g\in G$.

For any unbounded cycle $(H,D)$, the pair $(H,T)$ with $T=D(1+D^{2})^{-1}$ is
a (bounded) Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$ [3].

Now suppose we have a family $\\{D_{t}\\}_{t\geq 1}$ of odd, self-adjoint
unbounded operators on a graded $G$-Hilbert space $H$ which defines an odd,
regular, self-adjoint unbounded operator $D$ ([34, Chapter 10]) on a Hilbert
$C_{0}[1,\infty)$-module $H\otimes C_{0}[1,\infty)$. Then, the bounded
transform $T=D(1+D^{2})^{-1}$ of $D$ is an odd, self-adjoint operator in
$\mathfrak{L}(H\otimes C_{0}[1,\infty))$.

######  Proposition 16.1.

Let $H_{X}$ be a graded $X$-$G$-module and $D$ be an odd, regular self-adjoint
unbounded operator on a Hilbert $C_{0}[1,\infty)$-module $H_{X}\otimes
C_{0}[1,\infty)$ such that

  1. (1)

there is a dense subalgebra $B$ of $C_{0}(X)$ which preserves the domain of
$D$ such that for any $\phi\in B$, $[D,\phi]$ extends to a bounded operator
$S\in\mathfrak{L}(H\otimes C_{0}[1,\infty))$ with $\lVert S_{t}\rVert\to 0$ as
$t\to\infty$,

  2. (2)

the $G$-action on $H$ preserves the domain of $D$, and $g(D)-D$ extends to a
norm-continuous $\mathfrak{L}(H\otimes C_{0}[1,\infty))$-valued function of
$g\in G$, and

  3. (3)

$(D\pm i)^{-1}\in RL^{*}_{c}(H_{X})$.

Then, the pair $(H_{X},T)$ with $T=D(1+D^{2})^{-1}$ is an $X$-$G$-localized
Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$. If $D$ satisfies the
conditions (1), (2), then the condition (3) is equivalent to the following

  1. (3.1)

$(D\pm i)\in C_{b}([1,\infty),\mathfrak{K}(H_{X}))$ and

  2. (3.2)

for any $\epsilon>0$, there is $\phi\in C_{0}(X)$ such that
$\lVert(1-\phi)(D\pm i)\rVert<\epsilon$.

###### Proof.

The condition (3) implies $1-T^{2}=(1+D^{2})^{-1}\in RL^{*}_{c}(H_{X})$. We
have the following Baaj–Julg formula [3]

$T=\frac{\pi}{2}\int_{0}^{\infty}D(1+D^{2}+\lambda^{2})^{-1}d\lambda=\frac{\pi}{4}\int_{0}^{\infty}(D+\sqrt{1+\lambda^{2}}i)^{-1}+(D-\sqrt{1+\lambda^{2}}i)^{-1}d\lambda.$

Here, the integral converges in the strong topology on $\mathfrak{L}(H\otimes
C_{0}[1,\infty))$. For $\phi\in B$, we have

(16.1)
$[T,\phi]=\frac{\pi}{4}\int_{0}^{\infty}[(D+\sqrt{1+\lambda^{2}}i)^{-1},\phi]+[(D-\sqrt{1+\lambda^{2}}i)^{-1},\phi]d\lambda$

and

$[(D\pm\sqrt{1+\lambda^{2}}i)^{-1},\phi]=(D\pm\sqrt{1+\lambda^{2}}i)^{-1}[\phi,D](D\pm\sqrt{1+\lambda^{2}}i)^{-1}.$

From the condition (1), we see $[T_{t},\phi]\to 0$ as $t\to\infty$. For $g\in
G$, we have

(16.2) $g(T)-T=\frac{\pi}{4}\int_{0}^{\infty}A_{g}^{+}+A_{g}^{-}d\lambda$

where

(16.3)
$A_{g}^{\pm}=(g(D)\pm\sqrt{1+\lambda^{2}}i)^{-1}(D-g(D))(D\pm\sqrt{1+\lambda^{2}}i)^{-1}.$

Using the conditions (2), (3) and
$\lim_{t\to\infty}\lVert[T_{t},\phi]\rVert=0$ for $\phi\in C_{0}(X)$, we see
that $T$ is a $G$-continuous element in $M(RL^{*}_{c}(H_{X}))$ and that

$g(T)-T\in C_{b}([1,\infty),\mathfrak{K}(H_{X})).$

To see $g(T)-T\in RL^{*}_{c}(H_{X})$, since we have
$\lim_{t\to\infty}[T_{t},\phi]=0$, it is enough to show that for any
$\epsilon>0$, there is $\phi\in C_{0}(X)$ such that
$\lVert(g(T)-T)(1-\phi)\rVert<\epsilon$. This follows from (16.2), (16.3),
using $(D\pm\sqrt{1+\lambda^{2}}i)^{-1}\in RL^{*}_{c}(H_{X})$ for any
$\lambda\geq 0$ (which follows from the condition (1)). We may also use
$g(D)-D\in M(RL^{*}_{c}(H_{X}))$ (which follows from the conditions (1), (3))
to see $g(T)-T\in RL^{*}_{c}(H_{X})$. We explain that for $D$ satisfying (1)
and (2), the condition (3) is equivalent to (3.1) and (3.2). That (3) implies
(3.1) and (3.2) is obvious. On the other hand, the conditions (1) and (3.1)
imply $[(D\pm i)^{-1},\phi]\in C_{0}([1,\infty),\mathfrak{K}(H_{X}))$ and the
condition (2) implies that $(D\pm i)^{-1}$ is $G$-continuous. Together with
(3.2), these imply (3).

∎

######  Example 16.2.

(c.f. [30, Section 5] [33, Section 3] [47, Example 9.6]) Let $X$ be a
complete, simply connected Riemannian manifold of non-positive sectional
curvature, with curvature bounded below, on which a locally compact group $G$
acts properly and isometrically. Let $x_{0}\in X$, $d_{X}$ be the Riemannian
distance function on $X$ and $\rho\in C^{\infty}(X)$ be defined by

$\rho(x)=\sqrt{d_{X}(x_{0},x)^{2}+1}.$

Let $d\rho=\xi$ be the exterior derivative of $\rho$. The norm
$\lVert\xi\rVert_{x}$ of the co-vector $\xi$ at $x$ satisfies
$\lVert\xi\rVert_{x}=\frac{d_{X}(x_{0},x)}{\sqrt{d_{X}(x_{0},x)^{2}+1}}\to 1$
as $x$ goes to infinity. Let $H_{X}=\Omega^{*}_{L^{2}}(X)$ be the
$L^{2}$-space of the de-Rham complex on $X$ and for $t>0$, let

$D_{t}=t^{-1}(d+d^{\ast})+\mathrm{ext}(\rho d\rho)+\mathrm{int}(\rho d\rho)$

where $d$ is the de-Rham differential operator and $\mathrm{ext}(v)$ and resp.
$\mathrm{int}(v)$ are the exterior, resp. interior, multiplication by the co-
vector $v$. The odd, unbounded operators $D_{t}$ defined on the space
$\mathcal{E}_{c}^{\infty}$ of compactly supported differential forms are
essentially self-adjoint and have compact resolvent. We have

$D_{t}^{2}=t^{-2}\Delta+\rho^{2}\lVert\xi\rVert^{2}+t^{-1}A$

where $\Delta=dd^{\ast}+d^{\ast}d$ and $A=[(d+d^{\ast}),\mathrm{ext}(\rho
d\rho)+\mathrm{int}(\rho d\rho)]$ is an order-zero, smooth bundle-
endomorphism. There are constants $C_{0},C_{1}>0$ so that we have for $x\in
X$,

$\lVert A_{x}\rVert\leq C_{0}+C_{1}\rho(x).$

Using this, we see that there is $N>0$ so that we have for $t\geq 1$,

(16.4) $D_{t}^{2}+N^{2}\geq t^{-2}\Delta+\rho^{2}/2.$

The family $\\{D_{t}\\}_{t\geq 1}$ defines an odd, essentially self-adjoint,
regular operator $D$ on $H_{X}\otimes C_{0}[1,\infty)$ with domain
$C_{c}([1,\infty),\mathcal{E}_{c}^{\infty})$. We have

$(D\pm i)^{-1}\in C_{b}([1,\infty),\mathfrak{K}(H_{X})).$

Let $T=D(1+D^{2})^{-1}$ in $\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$.

######  Proposition 16.3.

The pair $(H_{X},T)$ is an $X$-$G$-localized Kasparov cycle for
$KK^{G}(\mathbb{C},\mathbb{C})$.

###### Proof.

We check that $D$ satisfies the conditions (1) -(3) of Proposition 16.1. (1):
Let $B=C_{c}^{\infty}(X)\subset C_{0}(X)$. We have for any $\phi\in B$,

$[D_{t},\phi]=t^{-1}(\mathrm{ext}(d\phi)-\mathrm{int}(d\phi))$

from which the condition (1) holds for $D$. (2): we have

$g(D_{t})-D_{t}=\mathrm{ext}(\rho^{\prime}d\rho^{\prime}-\rho
d\rho)+\mathrm{int}(\rho^{\prime}d\rho^{\prime}-\rho d\rho)$

where $\rho^{\prime}(x)=\sqrt{d(g(x_{0}),x)^{2}+1}$. We have $\lVert
d\rho^{\prime}-d\rho\rVert_{x}\leq
6d(g(x_{0}),x_{0})(\rho(x)+\rho^{\prime}(x))^{-1}$ [30, Lemma 5.3]. Using
this, we see that the condition (2) holds for $D$. (3): Since $(D\pm
i)^{-1}\in C_{b}([1,\infty),\mathfrak{K}(H_{X}))$, it is enough to show that
for any $\epsilon>0$, there is $\phi\in C_{0}(X)$, such that
$\lVert(1-\phi)(D_{t}\pm i)^{-1}\rVert<\epsilon$ for all $t\geq 1$. For this,
we may instead prove the same claim not for $(D_{t}\pm i)^{-1}$ but for
$(D_{t}\pm Ni)^{-1}$ for the constant $N>0$ as in (16.4). Take any
$v=(D_{t}\pm Ni)v_{0}$ for $v_{0}\in\mathcal{E}_{c}^{\infty}$. Then, we have

$(1-\phi)(D_{t}\pm i)^{-1}v=(1-\phi)v_{0}$

and

$\lVert v\rVert^{2}=\lVert(D_{t}\pm
Ni)v_{0}\rVert^{2}=\langle(D_{t}^{2}+N^{2})v_{0},v_{0}\rangle\geq\frac{1}{2}\langle\rho^{2}v_{0},v_{0}\rangle.$

Using these, it is not hard to see that for any $\epsilon>0$, there is
$\phi\in C_{0}(X)$ with $\lVert(1-\phi)(D_{t}\pm Ni)^{-1}\rVert<\epsilon$ for
all $t\geq 1$. Hence, the condition (3) holds for $D$. ∎

One can show that $[H_{X},T_{1}]=1_{K}$ in $R(K)$ for any compact subgroup $K$
of $G$. By Theorem 15.2, we see that the Baum–Connes assembly map
$\mu^{B,G}_{r}$ is split-injective for any $G$ which acts properly and
isometrically on a complete, simply connected Riemannian manifold of non-
positive sectional curvature, with curvature bounded below. Of course,
$[H_{X},T_{1}]$ in $KK^{G}(\mathbb{C},\mathbb{C})$ is nothing but the gamma
element for $G$ constructed by Kasparov [30].

######  Example 16.4.

(c.f. [29]) Let $X$ be a locally finite tree on which a locally compact group
$G$ acts properly (and continuously) by automorphisms. Let

$H^{\mathrm{JV}}_{X}=l^{2}(X^{0})\oplus l^{2}(X^{1})$

be the direct sum of $l^{2}$-spaces of the set $X^{0}$ of vertices of $X$ and
of the set $X^{1}$ of the edges. More precisely, we define $l^{2}(X^{1})$ as
the quotient of $l^{2}$-space of oriented edges by the closed subspaces
spanned by $[x,y]+[y,x]$ for oriented edges $[x,y]$ of $X$. The Hilbert space
$H^{\mathrm{JV}}_{X}$ is naturally a graded $X$-$G$-module. Fix $x_{0}\in X$
and $T_{\mathrm{JV}}$ be the associated Julg–Valette operator on
$H^{\mathrm{JV}}_{X}$. The operator $T_{\mathrm{JV}}$ can be defined as
follows. For any $x\in X^{0}\backslash\\{x_{0}\\}$, let $e_{x}=[x^{\prime},x]$
be the unique (oriented) edge which is closest to $x_{0}$ among the all edges
containing $x$. Then, the Hilbert space $H^{\mathrm{JV}}_{X}$ has the
following direct sum decomposition

$H^{\mathrm{JV}}_{X}=\mathbb{C}\delta_{x_{0}}\oplus\bigoplus_{x\in
X^{0}\backslash\\{x_{0}\\}}\mathbb{C}\delta_{x}\oplus\mathbb{C}\delta_{e_{x}}.$

Under this decomposition of $H^{\mathrm{JV}}_{X}$, the Julg–Valette operator
$T_{\mathrm{JV}}$ is expressed as the block-diagonal operator

$0+\sum_{x\in X^{0}\backslash\\{x_{0}\\}}\begin{bmatrix}0&1\\\
1&0\end{bmatrix}.$

The Julg–Valette operator $T_{\mathrm{JV}}$ is an, odd, self-adjoint, bounded,
$G$-continuous operator on $H^{\mathrm{JV}}_{X}$ such that for any $g\in G$,

$1-T_{\mathrm{JV}}^{2},\,\,\,g(T_{\mathrm{JV}})-T_{\mathrm{JV}}\in\mathfrak{K}(H^{\mathrm{JV}}_{X})$

(in fact they are of finite rank). Thus,
$(H^{\mathrm{JV}}_{X},T_{\mathrm{JV}})$ is a (bounded) Kasparov cycle for
$KK^{G}(\mathbb{C},\mathbb{C})$ and we have
$[H^{\mathrm{JV}}_{X},T_{\mathrm{JV}}]=1_{G}$ in
$KK^{G}(\mathbb{C},\mathbb{C})$ [29, Proposition 1.6] (for this, the
$G$-action does not have to be proper).

Now, we recall some idea from [9]. Instead of the $X$-$G$-module
$H^{\mathrm{JV}}_{X}$ which is rather “discrete”, we consider the Hilbert
space $H^{\mathrm{dR}}_{X}$ of the de-Rham complex on $X$ which is more
“continuous”. The Hilbert space $H^{\mathrm{dR}}_{X}$ is defined as

$H^{\mathrm{dR}}_{X}=\Omega^{\ast}_{L^{2}}(X)=\bigoplus_{i=0,1}\bigoplus_{\sigma\in
X^{i}}\Omega^{\ast}_{L^{2}}(\sigma)$

where for a vertex $\sigma\in X^{0}$, $\Omega^{\ast}_{L^{2}}(\sigma)$ is just
the one-dimensional space $\mathbb{C}\delta_{\sigma}$ and for a (non-oriented)
edge $\sigma\in X^{1}$, $\Omega^{\ast}_{L^{2}}(\sigma)$ is the space of
$L^{2}$-sections of the de-Rham complex on $\sigma$ equipped with the standard
metric so that $\sigma\cong[0,1]$ isometrically. The space
$H^{\mathrm{dR}}_{X}$ is naturally a graded $X$-$G$-module and we have an
embedding

$H^{\mathrm{JV}}_{X}\to H^{\mathrm{dR}}_{X}$

of $X$-$G$-modules which sends $\delta_{x}$ for $x\in X^{0}$ to
$\delta_{x}\in\Omega^{0}_{L^{2}}(x)$ and $\delta_{e}$ for an oriented edge
$e=[x,y]$ to the constant 1-form in $\Omega^{1}_{L^{2}}(e)$ which corresponds
to $ds$ under the isomorphism

$\Omega^{1}_{L^{2}}(e)\cong\Omega^{1}_{L^{2}}[0,1]=L^{2}[0,1]ds$

under which the vertex $x$ is identified as $0$ and $y$ is identified as $1$
in $[0,1]$. Fix $x_{0}\in X$, we have the following direct sum decomposition

(16.5) $H^{\mathrm{dR}}_{X}=\mathbb{C}\delta_{x_{0}}\oplus\bigoplus_{x\in
X^{0}\backslash\\{x_{0}\\}}\mathbb{C}\delta_{x}\oplus\Omega^{\ast}_{L^{2}}(e_{x})$

where $e_{x}=[x^{\prime},x]$ as before. Furthermore, we have an isomorphism

(16.6)
$\Omega^{\ast}_{L^{2}}(e_{x})\cong\Omega^{\ast}_{L^{2}}[0,1]=L^{2}[0,1]\oplus
L^{2}[0,1]ds$

under which $x^{\prime}$ is identified as $0$ and $x$ is identified as $1$ in
$[0,1]$. For any $t>0$, let

$d_{t}=t^{-1}d+\mathrm{ext}(ds)\colon\Omega^{0}_{0}[0,1]\to\Omega^{1}[0,1]$

be the Witten-type perturbation $t^{-1}(e^{-ts}de^{ts})$ of the de-Rham
differential operator $d$ on $\Omega^{\ast}_{L^{2}}[0,1]$ and consider it as
an unbounded operator on $\Omega^{\ast}_{L^{2}}(e_{x})$. Here,
$\Omega^{\ast}[0,1]$ is the space of (smooth) forms and

$\Omega_{0}^{0}[0,1]=\\{s\in C^{\infty}[0,1]\mid s(0)=s(1)=0\\}.$

Let

$D_{t}=d_{t}+d_{t}^{\ast}\colon\Omega^{\ast}_{L^{2}}[0,1]\to\Omega^{\ast}_{L^{2}}[0,1]$

with domain $\Omega_{0}^{0}[0,1]\oplus\Omega^{1}[0,1]$. Then, $D_{t}$ is an
odd, essentially self-adjoint, unbounded operator on
$\Omega^{\ast}_{L^{2}}[0,1]$ with compact resolvent. For each integer $k>0$,
the span of the forms

$\sin(\pi ks),\,\,\,d_{t}\sin(\pi ks)=t^{-1}\pi k\cos(\pi ks)ds+\sin(\pi
ks)ds$

is invariant under $D_{t}$ and we have

$D_{t}^{2}=t^{-2}\pi^{2}k^{2}+1$

on the span. These spans are mutually orthogonal in
$\Omega^{\ast}_{L^{2}}[0,1]$ and their mutual orthogonal complement
$\Omega^{\ast}_{L^{2}}[0,1]$ is spanned by

$\tilde{e}_{t}=e^{ts}ds\in\Omega^{1}[0,1]$

which lies in the kernel of $D_{t}$. Now, let

$S_{t}=\chi(D_{t})$

where $\chi\in C_{b}(\mathbb{R})$ is any odd, continuous, increasing function
on $\mathbb{R}$ such that $\chi(x)=1$ for $x\geq 1$. Then, we have

$1-S_{t}^{2}=P_{t}$

where $P_{t}$ is the rank one projection onto the span of $e_{t}$ in
$\Omega^{\ast}_{L^{2}}[0,1]$. Under the identification (16.6), $S_{t}$ defines
an, odd, self-adjoint operator $S_{t,x}$ on each
$\Omega^{\ast}_{L^{2}}(e_{x})$ for $x\in X^{0}\backslash\\{x_{0}\\}$. Now, let

$e_{t}=\frac{\tilde{e}_{t}}{\lVert\tilde{e}_{t}\rVert}=\sqrt{\frac{2t}{e^{2t}-1}}e^{ts}ds=\sqrt{\frac{2t}{1-e^{-2t}}}e^{t(s-1)}ds\in\Omega^{1}[0,1]$

and $e_{t,x}$ be the corresponding unit vector in
$\Omega^{\ast}_{L^{2}}(e_{x})$ under the identification (16.6). Then,
$1-S^{2}_{t,x}$ on $\Omega^{\ast}_{L^{2}}(e_{x})$ is the rank one projection
onto the span of $e_{t,x}$. We set

$R_{t,x}=\theta_{e_{t,x},\delta_{x}}+\theta_{\delta_{x},e_{t,x}}$

be the partial isometry on
$\mathbb{C}\delta_{x}\oplus\Omega^{\ast}_{L^{2}}(e_{x})$ which sends $e_{t,x}$
to $\delta_{x}$ and $\delta_{x}$ to $e_{t,x}$ and is zero on their complement.
We define the block-diagonal operator

$T_{t}=0+\sum_{x\in X^{0}\backslash\\{x_{0}\\}}S_{t,x}+R_{t,x}$

on $H^{\mathrm{dR}}_{X}$ with respect to the decomposition (16.5). Then, the
family $\\{T_{t}\\}_{t\geq 1}$ defines an odd, self-adjoint operator $T$ in
$\mathfrak{L}(H^{\mathrm{dR}}_{X}\otimes C_{0}[1,\infty))$.

######  Proposition 16.5.

The pair $(H^{\mathrm{dR}}_{X},T)$ is an $X$-$G$-localized Kasparov cycle for
$KK^{G}(\mathbb{C},\mathbb{C})$.

###### Proof.

It is clear that $1-T_{t}^{2}$ is the rank-one projection onto the span of
$\delta_{x_{0}}$. Moreover, as in the case of the Julg–Valette operator, for
any $g\in G$, the difference $g(T_{t})-T_{t}$ is contributed from only
finitely many blocks corresponding to the vertices $x$ on the geodesics from
$x_{0}$ to $g(x_{0})$. After noticing this, it is not hard to see that $T$ is
$G$-continuous and that for any $g\in G$,

$g(T)-T\in C_{b}([1,\infty),\mathfrak{K}(H^{\mathrm{dR}}_{X}))$

and $g(T)-T$ has uniform compact support with respect to $X$. Finally, we have
$\lim_{t\to\infty}\lVert[T_{t},\phi]\rVert=0$ for any $\phi$ in $C_{0}(X)$. To
see this, we take $\phi$ to be a compactly supported function on $X$ which is
smooth on each edge. Then, we have

$[D_{t},\phi]=t^{-1}(\mathrm{ext}(d\phi)-\mathrm{int}(d\phi))$

on each block $\Omega^{\ast}_{L^{2}}(e_{x})\cong\Omega^{\ast}_{L^{2}}[0,1]$
and this implies $\lim_{t\to\infty}\lVert[S_{t,x},\phi]\rVert=0$ for each $x$.
To see $\lim_{t\to\infty}\lVert[R_{t,x},\phi]\rVert=0$, we compute

$[\theta_{e_{t,x},\delta_{x}},\phi]=(\phi-\phi(x))\theta_{e_{t,x},\delta_{x}}.$

We have $\lVert(\phi-\phi(x))e_{t,x}\rVert\to 0$ as $t\to\infty$, uniformly in
$x\in X\backslash\\{x_{0}\\}$. This follows from the following elementary
estimate:

$\sqrt{\frac{2t}{1-e^{-2t}}}\lVert(1-s)e^{t(s-1)}\rVert_{L^{2}[0,1]}=\sqrt{\frac{2t}{1-e^{-2t}}}\lVert
se^{-ts}\rVert_{L^{2}[0,1]}$
$=\frac{1}{\sqrt{t}}\sqrt{\frac{2}{1-e^{-2t}}}\lVert
tse^{-ts}\rVert_{L^{2}[0,1]}\leq\frac{1}{\sqrt{t}}\sqrt{\frac{2}{1-e^{-2t}}}\sup\\{se^{-s}\mid
s\geq 0\\}$ $\leq\frac{1}{\sqrt{t}}\sqrt{\frac{2}{1-e^{-2t}}}\to 0$

as $t\to\infty$. We now see that $\lim_{t\to\infty}\lVert[T_{t},\phi]\rVert=0$
holds for any $\phi\in C_{0}(X)$. Hence, $T\in
M(RL^{*}_{c}(H^{\mathrm{dR}}_{X}))$ is an odd, self-adjoint, $G$-continuous
operator and we have for any $g\in G$,

$1-T^{2},\,\,\ g(T)-T\in RL^{*}_{c}(H^{\mathrm{dR}}_{X}).$

∎

One can show (see [9, Section 5]) that
$[H^{\mathrm{dR}}_{X},T_{1}]=[H^{\mathrm{JV}}_{X},T_{\mathrm{JV}}]=1_{G}$ in
$KK^{G}(\mathbb{C},\mathbb{C})$. In fact, $(H^{\mathrm{dR}}_{X},T_{t})$ for
$t\in[0,1]$ is a homotopy from $[H^{\mathrm{dR}}_{X},T_{1}]$ to
$[H^{\mathrm{JV}}_{X},T_{\mathrm{JV}}]$. By Theorem 15.2, we see that the
Baum–Connes assembly map $\mu^{B,G}_{r}$ is an isomorphism for any $G$ which
acts properly (and continuously) on a locally finite tree by automorphisms.
See [31] and [45] for the proof by the gamma element method.

##  17\. BCC for groups acting on finite-dimensional CAT(0)-cubical spaces

In this final section, we extend the recently obtained new proof of the
Baum–Connes conjecture with coefficients for groups $G$ which act properly
(and continuously) and co-compactly on a finite-dimensional CAT(0)-cubical
space with bounded geometry [9] to the not-necessarily co-compact setting. The
groundwork for this is already done in [9] (almost all results are proven in
the general, not-necessarily co-compact setting).

We first recall some materials and established results from [9] (see also
[8]). For the rest of this section, $X$ will be a bounded geometry
CAT(0)-cubical space, as in [37, Section 2.2]. In particular, $X$ is obtained
by identifying cubes, each of which is isometric to the standard Euclidean
cube $[0,1]^{q}$ of the appropriate dimension, by isometries of their faces.
We shall refer to $0$-cubes as vertices, $1$-cubes as edges, etc. The bounded
geometry condition asserts that there is a uniform bound on the number of
edges that contain a given vertex, and this implies that $X$ is finite
dimensional. Further, we shall assume all group actions on $X$ are by
automorphism of its structure as a CAT(0)-cubical space. In particular, a
group acting on $X$ permutes the vertices, the edges, and the $q$-cubes for
each fixed $q$.

The midplanes of a $q$-cube $C$ in $X$ correspond to the intersections of the
standard cube with the coordinate hyperplanes $x_{j}=1/2$ ($j=1,\cdots,q$). We
generate an equivalence relation on the collection of all midplanes of all
cubes in $X$ by declaring as equivalent any two midplanes whose intersection
is itself a midplane. The union of all midplanes in a given class is called a
hyperplane, and each hyperplane $H$ satisfies the following important
properties: (i) $H$ is connected, and (ii) the complement $X\backslash H$ has
precisely two path components. We shall say that a hyperplane $H$ separates
two subsets of $X$ if these subsets lie in distinct components of $X\backslash
H$. A group acting on $X$ necessarily permutes the hyperplanes. A hyperplane
$H$ is adjacent to a cube $C$ if it is disjoint from $C$, but intersects a
cube that includes $C$ as a codimension-one face.

On the cube $[0,1]^{q}$, the smooth $p$-forms can be written as
$\alpha=\sum_{I}f_{I}dx_{I}$ where $I=\\{i_{1}<\cdots<i_{p}\\}$ is a multi-
index, and $f_{I}$ is a smooth function on the cube $[0,1]^{q}$. For each
$q$-cube $C$, the space $\Omega^{p}(C)$ of $p$-forms carries a natural inner
product and we let $\Omega_{L^{2}}^{p}(C)$ be the Hilbert space completion.

We fix a base vertex ($0$-cube) $P$ of $X$. Let $C$ be a cube in $X$ and $H$
be a hyperplane that intersects $C$. A coordinate function $x_{H}\colon
D\to\mathbb{R}$ is defined as the affine function on $C$ which takes $1$ on
the codimension-one face of $C$, which is disjoint from $H$ and is separated
by $H$ from the base vertex, while it takes $0$ on the opposite codimension-
one face disjoint from $H$ but not separated by $H$ from the base vertex. We
define $\alpha_{H}=dx_{H}$. This is a constant-coefficient one-form on $C$.

A weight function $w$ for $X$ is a positive function on the set of
hyperplanes. A weight function $w$ is called proper if the set $\\{H\mid
w(H)\leq M\\}$ is finite for any $M>0$. If a locally compact group $G$ acts on
$X$ by automorphisms, it is $G$-adapted if $\sup_{H}|w(H)-w(gH)|<\infty$ for
any $g\in G$ and for some open subgroup $G$, $w(gH)=w(H)$ for all $H$ and for
all $g$ in the subgroup. We take $w(H)=\mathrm{distace}(P,H)$ as our preferred
weight function which is both proper and $G$-adapted but is is helpful for us
not to fix a weight function in what follows.

For each cube $C$, and for each $p\geq 0$, we denote by
$\Omega_{0}^{p}(C)\subset\Omega^{p}(C)$ the space of those smooth $p$-forms on
$C$ that pull back to zero on each open face of $C$. We consider the de-Rham
differential $d$ as an unbounded operator from $\Omega_{L^{2}}^{p}(C)$ to
$\Omega_{L^{2}}^{p+1}(C)$ with domain $\Omega_{0}^{p}(C)$. We consider its
formal adjoint $d^{\diamond}$ as an unbounded operator with domain
$\Omega^{p+1}(C)$. We form the direct sum

$\Omega_{0}^{\ast}(C)=\bigoplus_{p}\Omega_{0}^{p}(C),\,\,\,\Omega_{L^{2}}^{\ast}(C)=\bigoplus_{p}\Omega_{L^{2}}^{p}(C),$

and then form the unbounded operator

$D=d+d^{\diamond}\colon\Omega_{L^{2}}^{\ast}(C)\to\Omega_{L^{2}}^{\ast}(C).$

The operator $D$ on $\Omega_{L^{2}}^{\ast}(C)$ is essentially self-adjoint on
the domain $\Omega_{0}^{\ast}(C)$, and the self-adjoint closure of $D$ has
compact resolvent. The kernel of the closure is precisely the one-dimensional
space of top-degree constant coefficient forms (see [9, Lemma 4.2]).

Let $w$ be a weight function for $X$. For each cube $C$ in $X$, an affine
function $w_{C}\colon C\to\mathbb{R}$ is defined by the formula

$w_{C}=\sum_{H\in\mathrm{Mid}(C)}w(H)x_{H}$

where $\mathrm{Mid}(C)$ is the set of hyperplanes that contain a midplane of
$C$. We form the Witten-type perturbation

(17.1)
$d_{w}=e^{-w_{c}}de^{w_{c}}\colon\Omega_{0}^{\ast}(C)\to\Omega^{\ast}(C).$

We have

$d_{w}\colon\beta\mapsto
d\beta+\sum_{H\in\mathrm{Mid}(C)}w(H)x_{H}\wedge\beta,$

so that $d_{w}$ is a bounded perturbation of $d$. Let $d_{w}^{\diamond}$ be
its formal adjoint considered as an unbounded operator with domain
$\Omega^{\ast}(C)$, then the unbounded operator

$D_{w}=d_{w}+d_{w}^{\diamond}\colon\Omega_{L^{2}}^{\ast}(C)\to\Omega_{L^{2}}^{\ast}(C)$

with domain $\Omega_{0}^{\ast}(C)$ is essentially self-adjoint and has compact
resolvent.

Let $C$ be a cube in $X$ and $H$ be a hyperplane that intersects $C$. We let

$y_{H}=x_{H}-1\colon C\to\mathbb{R}.$

Note that $dy_{H}=\alpha_{H}=dx_{H}$. We define a linear operator (see [9,
Definition 4.4])

$e_{w}\colon\Omega^{p}(X)\to\Omega^{p+1}(X)$

by the formula

$e_{w}\colon\beta\mapsto\sum_{H\in\mathrm{SAH}(C)}w(H)e^{w(H)y_{H}}\alpha_{H}\wedge\beta$

for $\beta\in\Omega^{\ast}(C)$ where:

  * •

$\mathrm{SAH}(C)$ is the set of all hyperplanes that are adjacent to $C$ and
separate $C$ from the base vertex $P$.

  * •

For a hyperplane $H\in\mathrm{SAH}(C)$, if $D_{H}$ is the cube that is
intersected by $H$ and that includes $C$ as a codimension-one face, then we
view $\alpha_{H}\wedge\beta$ as a differential form on $D_{H}$ by pulling back
$\beta$ along the orthogonal projection from $D_{H}$ to $C$.

We denote by $D^{\mathrm{dR}}_{w}$ (see [9, Definition 4.5], it was denoted by
$D_{\mathrm{dR}}$) the following symmetric operator on
$\Omega^{\ast}_{L^{2}}(X)$ with domain $\Omega^{\ast}_{0}(X)$

$D^{\mathrm{dR}}_{w}=d_{w}+e_{w}+d_{w}^{\diamond}+e_{w}^{\diamond}.$

By $d_{w}$, we mean here the direct sum of all the Witten-type operators
(17.1) over all cubes of $X$ and the same for the formal adjoint
$d_{w}^{\diamond}$. The operator $D^{\mathrm{dR}}_{w}$ is essentially self-
adjoint and it has compact resolvent [9, Theorem 4.6] if the weight function
$w$ is proper.

For any vertex $Q$ of $X$, a cube $C$ leads to $Q$ to the base vertex $P$ if
(i) the cube $C$ includes the vertex $Q$ and (ii) all the hyperplanes that
intersect $C$ also separate $Q$ from $P$. That is
$\mathrm{Mid}(C)\subset\mathrm{SAH}(Q)$. We denote by
$\Omega^{\ast}_{L^{2}}(X)_{Q}\subset\Omega^{\ast}_{L^{2}}(X)$ the direct sum
of the spaces $\Omega^{\ast}_{L^{2}}(C)$ over those cubes $C$ that lead $Q$ to
$P$. We define $\Omega^{\ast}_{0}(X)_{Q}$ and $\Omega^{\ast}(X)_{Q}$ similarly
(see [9, Definition 4.7]).

We have the following Hilbert space direct sum decomposition:

(17.2) $\Omega^{\ast}_{L^{2}}(X)=\bigoplus_{Q}\Omega^{\ast}_{L^{2}}(X)_{Q}.$

The operator $D^{\mathrm{dR}}_{w}$ is block-diagonal with respect to this
decomposition [9, Lemma 4.9]. We have the following useful estimate.

######  Lemma 17.1.

(See the proof of [9, Theorem 4.6]) For any
$\beta\in\Omega^{\ast}_{0}(X)_{Q}$,

$\lVert
D^{\mathrm{dR}}_{w}\beta\rVert^{2}\geq\sum_{H\in\mathrm{SAH}(Q)}\min\\{\frac{1}{2}w(H)(1-e^{-2w(H)}),\pi^{2}+w(H)^{2}\\}\lVert\beta\rVert^{2}$
$=\sum_{H\in\mathrm{SAH}(Q)}\frac{1}{2}w(H)(1-e^{-2w(H)})\lVert\beta\rVert^{2}.$

######  Lemma 17.2.

(See [9, Lemma 4.11]) If the weight function $w$ is $G$-adapted, then for any
$g$ in $G$, the difference $g(D^{\mathrm{dR}}_{w})-D^{\mathrm{dR}}_{w}$ is a
bounded operator. Moreover $\lVert
g(D^{\mathrm{dR}}_{w})-D^{\mathrm{dR}}_{w}\rVert$ is bounded by a constant
times

$\sup\\{gw(H)-w(H)\mid\text{ $H$ is a hyperplane of
$X$}\\}+\max\\{w(H)\mid\text{$H$ separates $P$ and $gP$}\\}$

The constant depends only on the dimension of $X$.

If $G$ acts on $X$ by automorphisms, for any proper, $G$-adapted weight
function $w$, the pair $(\Omega^{\ast}_{L^{2}}(X),D^{\mathrm{dR}}_{w})$ is an
unbounded Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$ and we denote by
$[D^{\mathrm{dR}}_{w}]$ its associated class in
$KK^{G}(\mathbb{C},\mathbb{C})$ (see [9, Theorem 4.12, Definition 4.13]).

######  Theorem 17.3.

(See [9, Theorem 5.1] and [8, Theorem 9.14]) If $G$ acts on $X$ by
automorphisms, the element $[D^{\mathrm{dR}}_{w}]\in
KK^{G}(\mathbb{C},\mathbb{C})$ satisfies

$[D^{\mathrm{dR}}_{w}]=1_{G}\,\,\,\text{in $KK^{G}(\mathbb{C},\mathbb{C})$}.$

Now, suppose the $G$-action on $X$ is proper. Let
$H_{X}=\Omega^{\ast}_{L^{2}}(X)$ which is naturally a graded $X$-$G$-module.
We shall construct an $X$-$G$-localized Kasparov cycle $(H_{X},T)$ for
$KK^{G}(\mathbb{C},\mathbb{C})$ such that
$[H_{X},T_{1}]=[D^{\mathrm{dR}}_{w}]=1_{G}$ in
$KK^{G}(\mathbb{C},\mathbb{C})$.

In what follows, for simplicity, we assume $w\geq 1/2$. For example, we can
use $w(H)=\mathrm{distace}(P,H)$.

For $t\geq 1$, we define an odd, symmetric unbounded operator

(17.3)
$\mathcal{D}_{t}=t^{-1}(d_{tw}+d_{tw}^{\diamond})+(\sqrt{t})^{-1}(e_{tw}+e_{tw}^{\diamond})\colon\Omega^{\ast}_{L^{2}}(X)\to\Omega^{\ast}_{L^{2}}(X)$

with domain $\Omega^{\ast}_{0}(X)$. Note that
$\mathcal{D}_{1}=D^{\mathrm{dR}}_{w}$ but $\mathcal{D}_{t}$ is slightly
different from $t^{-1}D^{\mathrm{dR}}_{tw}$. The operators $\mathcal{D}_{t}$
are block-diagonal with respect to the decomposition (17.2) and it is
essentially self-adjoint with compact resolvent on each block
$\Omega^{\ast}_{L^{2}}(X)_{Q}$. From these and from the formula (17.3) of
$\mathcal{D}_{t}$, we see that the family $\\{\mathcal{D}_{t}\\}_{t\geq 1}$
defines an odd, regular, symmetric unbounded operator $\mathcal{D}$ on
$H_{X}\otimes C_{0}[1,\infty)$ with domain
$C_{c}([1,\infty),\Omega^{\ast}_{0}(X))$ and that its closure is self-adjoint.

We have the following estimate.

######  Lemma 17.4.

For $t\geq 1$ and for any $\beta\in\Omega^{\ast}_{0}(X)_{Q}$,

$\lVert\mathcal{D}_{t}\beta\rVert^{2}\geq\sum_{H\in\mathrm{SAH}(Q)}\min\\{\frac{1}{2}w(H)(1-e^{-2tw(H)}),t^{-2}\pi^{2}+w(H)^{2}\\}\lVert\beta\rVert^{2}.$

###### Proof.

This can be proven as in the proof of [9, Theorem 4.6]. ∎

######  Lemma 17.5.

If the weight function $w$ is proper, The odd, regular, self-adjoint unbounded
operator $\mathcal{D}$ on $H_{X}\otimes C_{0}[1,\infty)$ satisfies

$(\mathcal{D}\pm i)^{-1}\in C_{b}([1,\infty),\mathfrak{K}(H_{X})).$

Moreover, for any $\epsilon>0$, there is $\phi\in C_{0}(X)$, such that
$\lVert(1-\phi)(\mathcal{D}\pm i)^{-1}\rVert<\epsilon$.

###### Proof.

The unbounded operator $\mathcal{D}$ is block diagonal with respect to (17.2)
and it is not hard to see that its restriction $\mathcal{D}_{Q}$ on each block
$\Omega^{\ast}_{L^{2}}(X)_{Q}$ satisfies

$(\mathcal{D}_{Q}\pm i)^{-1}\in
C_{b}([1,\infty),\mathfrak{K}(\Omega^{\ast}_{L^{2}}(X)_{Q})).$

Lemma 17.4 implies the sum

$(\mathcal{D}\pm i)^{-1}=\sum_{Q}(\mathcal{D}_{Q}\pm i)^{-1}$

is absolutely norm-convergent in $C_{b}([1,\infty),\mathfrak{K}(H_{X}))$. The
second claim also follows from Lemma 17.4. ∎

Note that the $G$-action preserves the domain
$C_{c}([1,\infty),\Omega^{\ast}_{0}(X))$ of $\mathcal{D}$.

######  Lemma 17.6.

If the weight function $w$ is $G$-adapted, then for any $g$ in $G$, the
difference $g(\mathcal{D})-\mathcal{D}$ is a bounded operator. Moreover
$\lVert g(\mathcal{D})-\mathcal{D}\rVert$ is bounded by a constant times

$\sup\\{gw(H)-w(H)\mid\text{ $H$ is a hyperplane of
$X$}\\}+\max\\{w(H)\mid\text{$H$ separates $P$ and $gP$}\\}$

The constant depends only on the dimension of $X$.

###### Proof.

This can be proven in the same way as the proof of [9, Lemma 4.11] with bit
more refined estimates. We write $\mathcal{D}_{t}=\mathcal{D}_{t,w,P}$ to make
it clear its dependence on the weight function $w$ and on the base vertex $P$
so that $g(\mathcal{D}_{t})=\mathcal{D}_{t,gw,gP}$. We bound the differences

$\mathcal{D}_{t,gw,gP}-\mathcal{D}_{t,w,gP}\,\,\,\text{and}\,\,\,\mathcal{D}_{t,w,gP}-\mathcal{D}_{t,w,P}$

separately, uniformly in $t\geq 1$ to obtain the estimate in the statement. We
have

$\mathcal{D}_{t,gw,gP}-\mathcal{D}_{t,w,gP}=t^{-1}(d_{tgw,gP}-d_{tw,gP})+t^{-1}(d_{tgw,gP}^{\diamond}-d_{tw,gP}^{\diamond})$
$+(\sqrt{t})^{-1}(e_{tgw,gP}-e_{tw,gP})+(\sqrt{t})^{-1}(e_{tgw,gP}^{\diamond}-e_{tw,gP}^{\diamond}).$

Since $w$ is $G$-adapted, there is $M\geq 0$ such that $|gw(H)-w(H)|\leq M$
for all $H$. Using the formula

$t^{-1}d_{tw,gP}\colon\beta\mapsto
t^{-1}d\beta+\sum_{H\in\mathrm{Mid}(C)}w(H)x_{H,gP}\wedge\beta,$

we see that

$\lVert t^{-1}(d_{tgw,gP}-d_{tw,gP})\rVert\leq\dim(X)\cdot M$

for all $t\geq 1$ and thus $\lVert
t^{-1}(d_{tgw,gP}^{\diamond}-d_{tw,gP}^{\diamond})\rVert\leq\dim(X)\cdot M$
for all $t\geq 1$.

For $w\geq 1/2$ and for $M\geq m\geq 0$, we have

(17.4) $\sqrt{t}\lVert we^{-twx}-(w+m)e^{-t(w+m)x}\rVert_{L^{2}[0,1]}\leq 2M$

for all $t\geq 1$ since for instance we can write

$|we^{-twx}-(w+m)e^{-t(w+m)x}|\leq
w(tmx)e^{-twx}\frac{(1-e^{-tmx})}{tmx}+me^{-t(w+m)x}$ $\leq
m(twx)e^{-twx}+me^{-t(w+m)x}$

and since (we use $w\geq 1/2$)

$\lVert
twxe^{-twx}\rVert^{2}_{L^{2}[0,1]}\leq(tw)^{-1}\int_{0}^{\infty}y^{2}e^{-2y}dy\leq\frac{1}{4wt}\leq\frac{1}{2t},$

and

$\lVert
e^{-t(w+m)x}\rVert^{2}_{L^{2}[0,1]}\leq\frac{1}{t(w+m)}\int_{0}^{\infty}e^{-2y}dy\leq\frac{1}{2tw}\leq\frac{1}{t}$

so that $\sqrt{t}\lVert we^{-twx}-(w+m)e^{-t(w+m)x}\rVert_{L^{2}[0,1]}$ is
bounded by

$\sqrt{t}m(\lVert twxe^{-twx}\rVert_{L^{2}[0,1]}+\lVert
e^{-t(w+m)x}\rVert_{L^{2}[0,1]})\leq\sqrt{t}m(\frac{1}{\sqrt{2t}}+\frac{1}{\sqrt{t}})\leq
2M.$

Using (17.4) and the formula

$(\sqrt{t})^{-1}e_{tw,gP}\colon\beta\mapsto\sum_{H\in\mathrm{SAH}(C)}\sqrt{t}w(H)e^{tw(H)y_{H,gP}}\alpha_{H,gP}\wedge\beta$

we obtain

$\lVert(\sqrt{t})^{-1}(e_{tgw,gP}-e_{tw,gP})\rVert\leq 2\dim(X)\cdot M$

for all $t\geq 1$ and thus
$\lVert(\sqrt{t})^{-1}(e_{tgw,gP}^{\diamond}-e_{tw,gP}^{\diamond})\rVert\leq
2\dim(X)\cdot M$ for all $t\geq 1$. Putting everything together, we obtain

$\lVert\mathcal{D}_{t,gw,gP}-\mathcal{D}_{t,w,gP}\rVert\leq 6\dim(X)\cdot M.$

Next, we consider the difference $\mathcal{D}_{t,w,gP}-\mathcal{D}_{t,w,P}$.
As in the proof [9, Lemma 4.11],

$(t^{-1}d_{tw,gP}-t^{-1}d_{tw,P})\beta=\sum_{H\in\mathrm{Mid}}w(H)(\alpha_{H,gP}-\alpha_{H,P})\wedge\beta$

where the difference $\alpha_{H,gP}-\alpha_{H,P}$ is zero unless the
hyperplane $H$ separates $P$ and $gP$ in which case the difference is
$2\alpha_{H,gP}$. Thus, we have

$\lVert t^{-1}d_{tw,gP}-t^{-1}d_{tw,P}\rVert\leq
2\dim(X)\cdot\max\\{w(H)\mid\text{$H$ separates $P$ and $gP$}\\}.$

The same is true for $\lVert
t^{-1}d_{tw,gP}^{\diamond}-t^{-1}d_{tw,P}^{\diamond}\rVert$. Lastly, we have

$(\sqrt{t})^{-1}(e_{tw,gP}-e_{tw,P})\beta=\sum_{H}\sqrt{t}w(H)(e^{tw(H)y_{H,gP}}\alpha_{H,gP}-e^{tw(H)y_{H,P}}\alpha_{H,P})\wedge\beta$

where again the summands are zero except when $H$ separates $P$ and $gP$ and
we can bound the sum by

$2\dim(X)\cdot\max\\{\sqrt{t}w(H)\lVert
e^{-tw(H)x}\rVert_{L^{2}[0,1]}\mid\text{$H$ separates $P$ and $gP$}\\}$ $\leq
2\dim(X)\cdot\max\\{w(H)\mid\text{$H$ separates $P$ and $gP$}\\}.$

The same is true for
$\lVert(\sqrt{t})^{-1}(e_{tw,gP}^{\diamond}-e_{t,w,P}^{\diamond})\rVert$.
Putting everything together, we obtain

$\lVert\mathcal{D}_{t,w,gP}-\mathcal{D}_{t,w,P}\rVert\leq
8\dim(X)\cdot\max\\{w(H)\mid\text{$H$ separates $P$ and $gP$}\\}$

so we are done. ∎

######  Theorem 17.7.

Let $w$ be a weight function for $X$ which is proper and $G$-adapted. Let
$H_{X}=\Omega^{\ast}_{L^{2}}(X)$ and $T=\mathcal{D}(1+\mathcal{D}^{2})^{-1}$.
Then, the pair $(H_{X},T)$ is an $X$-$G$-localized Kasparov cycle for
$KK^{G}(\mathbb{C},\mathbb{C})$ with
$[H_{X},T_{1}]=[D^{\mathrm{dR}}_{w}]=1_{G}$ in
$KK^{G}(\mathbb{C},\mathbb{C})$.

###### Proof.

We check that $\mathcal{D}$ satisfies the conditions (1) -(3) of Proposition
16.1. (1): Let $B\subset C_{0}(X)$ be the subalgebra of compactly supported
functions that are smooth on each cube. For any $\phi\in B$, we compute

$\displaystyle[\mathcal{D}_{t},\phi]$
$\displaystyle=[t^{-1}(d_{tw}+d_{tw}^{\diamond})+(\sqrt{t})^{-1}(e_{tw}+e_{tw}^{\diamond}),\phi]$
$\displaystyle=t^{-1}[d_{tw}+d_{tw}^{\diamond},\phi]+(\sqrt{t})^{-1}[e_{tw}+e_{tw}^{\diamond},\phi]$
$\displaystyle=t^{-1}c(\phi)+(\sqrt{t})^{-1}[e_{tw}+e_{tw}^{\diamond},\phi]$

where $c(\phi)$ denotes the Clifford multiplication by the gradient of $\phi$
in each cube which is a bounded operator. Let $C>0$ so that

$|\phi(x)-\phi(y)|\leq C\cdot\mathrm{distance}(x,y)\,\,\,\text{for any $x,y\in
X$}.$

We have (see the proof of [9, Theorem 5.2]) for all $\beta\in\Omega^{\ast}(C)$

$\displaystyle\lVert[e_{tw},\phi]\beta\rVert^{2}$
$\displaystyle\leq\sum_{H\in\mathrm{SAH}(C)}C^{2}\lVert
xtw(H)e^{-tw(H)x}\rVert^{2}_{L^{2}[0,1]}\cdot\lVert\beta\rVert^{2}$
$\displaystyle\leq\dim(X)\cdot C^{2}\cdot\max\\{(xe^{-x})^{2}\mid x\geq
0\\}\cdot\lVert\beta\rVert^{2}$ $\displaystyle\leq\dim(X)\cdot
C^{2}\cdot\lVert\beta\rVert^{2}$

for all $t\geq 1$. From these, we see that $[\mathcal{D},\phi]$ extends to a
bounded operator $S\in\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$ with $\lVert
S_{t}\rVert\to 0$ as $t\to\infty$. (2): This follows from Lemma 17.6. (3):
This follows from Lemma 17.5. Thus, by Proposition 16.1, $(H_{X},T)$ is an
$X$-$G$-localized Kasparov cycle for$KK^{G}(\mathbb{C},\mathbb{C})$. That
$[H_{X},T_{1}]=[D^{\mathrm{dR}}_{w}]$ follows from the definition of
$\mathcal{D}$. We have $[D^{\mathrm{dR}}_{w}]=1_{G}$ by Theorem 17.3. ∎

######  Theorem 17.8.

Let $G$ be a second countable, locally compact group $G$ which acts properly
and continuously on a finite-dimensional CAT(0)-cubical space with bounded
geometry by automorphisms. Then, the Baum–Connes assembly map $\mu^{B,G}_{r}$
is an isomorphism for any separable $G$-$C^{*}$-algebra $B$, i.e. BCC holds
for $G$.

###### Proof.

Combine Theorem 15.2 and Theorem 17.7. ∎

######  Remark 17.9.

Any group $G$ which acts properly on a CAT(0)-cubical space has the Haagerup
Approximation property [14, Section 1.2.7], so $G$ is a-T-menable. Thus, BCC
for these groups are already known by the Higson–Kasparov Theorem [25], [26].

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