# A Comparison of Takai and Treumann Dualities

Vikram Nadig Universität Regensburg/Universitá degli studi di Milano
vikram.nadig<EMAIL_ADDRESS>
###### Abstract

We prove a comparison result between two duality statements - Takai duality,
which is implemented by the crossed product functor $-\rtimes G:KK^{G}\to
KK^{\hat{G}}$ on equivariant Kasparov categories; and Treumann duality, which
asserts the existence of an exotic equivalence of stable $\infty$-categories
$\text{Mod}(KU_{p}[G])^{ft}\simeq\text{Mod}(KU_{p}[\hat{G}])^{ft}$ given by
tensoring with a particular $(G,\hat{G})$-bimodule $M_{E}$ and $p$-completing.

###### Contents

  1. 1 Introduction
  2. 2 The Crossed Product Functor
  3. 3 Takai Duality
  4. 4 Treumann Duality
  5. 5 A Comparison of Takai Duality with Treumann Duality

##  1 Introduction

Let $G$ be a locally compact abelian group and $\hat{G}$ be its Pontryagin
dual. In this paper, we study and compare two different duality results on
algebraic structures with actions of $G$ and $\hat{G}$.

The first of these is in the realm of $C^{*}$-algebras. Given a
$G$-$C^{*}$-algebra (a $C^{*}$-algebra together with a continuous $G$-action)
$A$, we can form the crossed product $A\rtimes G$, which is a $C^{*}$-algebra
that acquires a natural action of $\hat{G}$. It is natural to ask, via the
Pontryagin duality isomorphism, what the $G$-$C^{*}$-algebra $(A\rtimes
G)\rtimes\hat{G}$ is. This question was answered by Takai in the 1970s:

######  Theorem 1.1 (Takai Duality).

There is a $G$-equivariant isomorphism of $C^{*}$-algebras

$(A\rtimes G)\rtimes\hat{G}\cong A\otimes K(L^{2}(G))$

Here, $K(L^{2}(G))$ is given the $G$-action of conjugation with the left-
regular representation of $G$ on $L^{2}(G)$.

The other duality result we take up is in the modern realm of higher algebra.
For this theorem, we assume that $G$ is a finite abelian $p$-group.

Let $KU_{p}$ denote the $p$-completed complex $K$-theory spectrum, and let
$KU_{p}[G]$ denote the commutative ring spectrum
$KU_{p}\wedge\Sigma_{+}^{\infty}G$. One can form the stable $\infty$-category
of modules of finite type $\text{Mod}(KU_{p}[G])^{ft}$ over $KU_{p}[G]$. In
[Tre15], a module $M_{E}$ over $KU_{p}[G\times\hat{G}]$ is constructed, whose
underlying $KU_{p}$-module is $KU_{p}$ itself. Tensoring by $M_{E}$ over
$KU_{p}[G]$ and $KU_{p}[\hat{G}]$, respectively, induces functors
$\text{Mod}(KU_{p}[G])^{ft}\to\text{Mod}(KU_{p}[\hat{G}])^{ft}$ and
$\text{Mod}(KU_{p}[\hat{G}])^{ft}\to\text{Mod}(KU_{p}[G])^{ft}$. Let $L_{p}$
denote the $p$-completion functor. Treumann proved the following duality
result (Theorem 1.5.1 of [Tre15]):

######  Theorem 1.2 (Treumann Duality).

The composites

${{\textup{Mod}(KU_{p}[G])^{ft}}}$${{\textup{Mod}(KU_{p}[\hat{G}])^{ft}}}$${{\textup{Mod}(KU_{p}[G])^{ft}}}$${{\textup{Mod}(KU_{p}[\hat{G}])^{ft}}}$${{\textup{Mod}(KU_{p}[G])^{ft}}}$${{\textup{Mod}(KU_{p}[\hat{G}])^{ft}}}$$\scriptstyle{L_{p}(-\otimes_{KU_{p}[G]}M_{E})}$$\scriptstyle{L_{p}(-\otimes_{KU_{p}[\hat{G}]}M_{E})}$$\scriptstyle{L_{p}(-\otimes_{KU_{p}[\hat{G}]}M_{E})}$$\scriptstyle{L_{p}(-\otimes_{KU_{p}[G]}M_{E})}$

are equivalent to the respective identities.

This theorem produces an equivalence of stable $\infty$-categories
$\text{Mod}(KU_{p}[G])^{ft}\simeq\text{Mod}(KU_{p}[\hat{G}])^{ft}$ that is not
induced by any choice of isomorphism $G\cong\hat{G}$ (for example, the
$KU_{p}[G]$-module $KU_{p}$ with trivial $G$-action is mapped to
$KU_{p}[\hat{G}]$ under this equivalence).

The article [BEL23] bridges these two different worlds. A stable
$\infty$-category $KK^{G}$ is constructed, which represents Kasparov’s
$G$-equivariant $KK$-theory for $C^{*}$-algebras, in the following sense.
Denote by $KK^{G}_{*}(-,-)$ Kasparov’s original equivariant $KK$-groups. There
is a functor $kk^{G}:GC^{*}\text{Alg}^{nu}\to KK^{G}$ (where
$GC^{*}\text{Alg}^{nu}$ is the category of $G$-$C^{*}$-algebras) such that

$\pi_{*}KK^{G}(kk^{G}(A),kk^{G}(B))\cong KK^{G}_{*}(A,B)$

for all separable $G$-$C^{*}$-algebras $A$ and $B$. That is, for separable
algebras, Kasparov’s classical $G$-equivariant $KK$-groups can be read off
from the stable homotopy groups of the mapping spectra in $KK^{G}$. In Section
2.1, we recall the explicit construction of $KK^{G}$ and $kk^{G}$ by a series
of Dwyer-Kan localizations and state their universal properties.

In the language of equivariant $KK$-theory, Theorems 1.1 and 1.2 are similar
statements. To motivate why, we consider a simple example. Consider the
$C^{*}$-algebra $\mathbb{C}$ with trivial $G$-action (here we assume that $G$
is a finite abelian group). Its $p$-completed $K$-theory spectrum is nothing
but $KU_{p}$ itself, and it inherits the trivial $G$-action. On the other
hand, the crossed product $\mathbb{C}\rtimes G$ can be identified, under the
Fourier transform, with $C_{0}(\hat{G})$. This can be identified with a
$\hat{G}$-indexed direct sum $\bigoplus_{\hat{G}}\mathbb{C}$, and $\hat{G}$
acts on it by permuting the summands. Consequently, the $p$-completed
$K$-theory spectrum of $\mathbb{C}\rtimes G$ is $KU_{p}[\hat{G}]$ with the
usual $\hat{G}$-action, and so one recovers the association $KU_{p}\mapsto
KU_{p}[\hat{G}]$ of Theorem 1.2 by taking the crossed product as illustrated.

In Section 2.2, we observe that the crossed product gives a functor $-\rtimes
G:GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$. We subsequently show
that it descends essentially uniquely to a functor $-\rtimes
G:KK^{G}\to\hat{K}K^{\hat{G}}$. Here, “descends” means that we have a
commutative square of functors

${{GC^{*}\text{Alg}^{nu}}}$${{\hat{G}C^{*}\text{Alg}^{nu}}}$${{KK^{G}}}$${{KK^{\hat{G}}}}$$\scriptstyle{\rtimes
G}$$\scriptstyle{kk^{G}}$$\scriptstyle{\rtimes G}$$\scriptstyle{kk^{\hat{G}}}$

If we aspire to formulate a reasonable comparison statement between the two
duality results stated above, we must begin by stating Theorem 1.1 in
categorical language, which we do in Section 3. From this and the
$K_{G}$-stability of $KK$-theory, we conclude that $-\rtimes G:KK^{G}\to
KK^{\hat{G}}$ is an equivalence (Corollary 3.3). The rest of the section is
dedicated to providing a different formula for the functor $-\rtimes
G:KK^{G}\to KK^{\hat{G}}$ which we use later to compare the two duality
results.

In Section 4, we precisely formulate Theorem 1.2 as Theorem 4.5. The only
novelty here is a description of the module ME using $C^{*}$-categories. This
proves to be very suitable for our intentions.

In Section 5, we make precise what we mean by comparing Theorems 1.1 and 1.2.
We briefly summarize this below:

Taking $K$-theory and $p$-completing yields a natural functor

$r_{G}:KK^{G}\to\text{Mod}(KU_{p}[G])$

We define $KK^{G,ft}$ to be the full subcategory of $KK^{G}$ that is mapped by
$r_{G}$ to $\text{Mod}(KU_{p}[G])^{ft}$. We show that $KK^{G,ft}$ is, in fact,
a thick subcategory of $KK^{G}$. Further, we show that the crossed product
functor $-\rtimes G$ restricts to an equivalence $KK^{G,ft}\simeq
KK^{\hat{G},ft}$ (Corollary 5.5).

We conclude the paper with our comparison result (Theorem 5.7):

######  Theorem 1.3.

We have a commutative square of functors

${{KK^{G,ft}}}$${{KK^{\hat{G},ft}}}$${{\textup{Mod}(KU_{p}[G])^{ft}}}$${{\textup{Mod}(KU_{p}[\hat{G}])^{ft}}}$$\scriptstyle{-\rtimes
G}$$\scriptstyle{\sim}$$\scriptstyle{r_{G}}$$\scriptstyle{r_{\hat{G}}}$$\scriptstyle{\sim}$$\scriptstyle{L_{p}(-\otimes_{KU_{p}[G]}M_{E})}$

Both the horizontal arrows are equivalences - the upper one is that of Takai
duality, and the lower one is that of Treumann duality.

Acknowledgements: The author is very grateful to Ulrich Bunke for proposing
this problem, his extensive help with ideas and suggestions, and reviewing
preliminary drafts in detail. The author also thanks Benjamin Dünzinger for
helpful discussions, Christoph Winges for providing a proof of Proposition
4.1., and Paul Arne Østvær for useful comments.

The author was supported by scholarships from ERASMUS+ and the ALGANT
consortium.

##  2 The Crossed Product Functor

Throughout this section, we denote by $G$ a locally compact, second countable,
abelian group with identity element $e$ and its Pontryagin dual by $\hat{G}$.
The dual group $\hat{G}$ is endowed with the compact-open topology. We fix
Haar measures $\mu$ and $\hat{\mu}$ on $G$ and $\hat{G}$, respectively. These
are unique up to multiplication by a positive scalar.

###  2.1 A stable $\infty$-category for equivariant $KK$-theory

In this subsection, we will introduce the definition and state the universal
property of the stable $\infty$-category $KK^{G}$ referenced in the
introduction. All the material here is contained in [BEL23]111This reference
deals with discrete groups. and [Bun23].

We will denote by $C^{*}\text{Alg}$ the category with objects the unital
$C^{*}$-algebras and morphisms the unital $*$-algebra homomorphisms, whereas
$C^{*}\text{Alg}^{nu}$ will refer to the category with objects the
$C^{*}$-algebras (not necessarily unital) and morphisms the $*$-algebra
homomorphisms222The notation $C^{*}\text{Alg}^{nu}$ is not to be confused with
that of nuclear $C^{*}$-algebras. We are following the convention in [BEL23]..
We also recall the notion of a pre-$C^{*}$-algebra - it is a $*$-algebra over
$\mathbb{C}$ each element of which has a finite maximal norm333Let $A$ be a
$*$-algebra over $\mathbb{C}$. For an element $a$ of $A$, we define its
maximal norm to be $\text{sup}||\rho(a)||$, where the supremum is taken over
all representations $\rho:A\to B$, where $B$ is a $C^{*}$-algebra.. One has a
category $C^{*}_{\text{pre}}\text{Alg}^{nu}$ of pre-$C^{*}$-algebras and
$*$-algebra homomorphisms. The fully faithful inclusion
$C^{*}\text{Alg}^{nu}\hookrightarrow C^{*}_{\text{pre}}\text{Alg}^{nu}$ admits
a left adjoint
$\hypertarget{shreehari}{\text{compl}}:C^{*}_{\text{pre}}\text{Alg}^{nu}\to
C^{*}\text{Alg}^{nu}$, which acts on objects by completing with respect to the
maximal norm (see Section 2 of [Bun24]).

We denote by $GC^{*}\text{Alg}^{nu}$ the following category. The objects are
pairs of the form $(A,\alpha)$, where $A$ is an object of
$C^{*}\text{Alg}^{nu}$ and $\alpha:G\to\text{Aut}_{C^{*}\text{Alg}^{nu}}(A,A)$
is a group homomorphism, such that for each element $a$ of $A$, the function
$G\to A$ given by $g\mapsto\alpha_{g}(a)$ is continuous. At times we will be
sloppy and simply write $A$ instead of $(A,\alpha)$. The morphisms of
$GC^{*}\text{Alg}^{nu}$ are the $G$-equivariant $C^{*}$-algebra homomorphisms.
When $G$ is discrete, $GC^{*}\text{Alg}^{nu}$ can be identified with the
functor category $\text{Fun}(BG,C^{*}\text{Alg}^{nu})$. The category
$GC^{*}\text{Alg}^{nu}$ has two different symmetric monoidal structures given
by the minimal and maximal tensor products. However, in this paper, for
simplicity, we will work exclusively with the maximal tensor product, which we
denote simply by $-\otimes-$.

Let $GC^{*}\text{Alg}^{nu}_{sep}$ denote the full subcategory of
$GC^{*}\text{Alg}^{nu}$ consisting of the separable algebras.

######  Definition 2.1.

Let $D$ be an $\infty$-category, and $F:GC^{*}\textup{Alg}^{nu}_{sep}\to D$ a
functor.

  1. 1.

The functor $F$ is said to be homotopy invariant if it sends homotopy
equivalences to equivalences. The notion of homotopy is defined with respect
to the point-norm topology on the mapping sets in
$GC^{*}\textup{Alg}^{nu}_{sep}$.

  2. 2.

Let $K_{G}:=K(L^{2}(G)\otimes l^{2})\in GC^{*}\textup{Alg}^{nu}$ be the
algebra of compact operators on the Hilbert space $L^{2}(G)\otimes l^{2}$,
with group action induced by conjugation of the natural action on $L^{2}(G)$.
A morphism $f:A\to B$ in $GC^{*}\textup{Alg}^{nu}_{sep}$ is a
$K_{G}$-equivalence if $f\otimes K_{G}$ is a homotopy equivalence.

  3. 3.

The functor $F$ is said to be $G$-stable if for every equivariant isometric
embedding of separable unitary444Here, unitary means that $G$ acts by
unitaries on $V^{\prime}$ and $V$. $G$-Hilbert spaces
$V^{\prime}\hookrightarrow V$, with $V^{\prime}\neq 0$, and for all objects
$A$ of $GC^{*}\text{Alg}^{nu}$, the induced map $F(A\otimes K(V^{\prime}))\to
F(A\otimes K(V))$ is an equivalence.

######  Proposition 2.2.

A homotopy invariant functor $F$ inverts $K_{G}$-equivalences if and only if
it is $G$-stable.

###### Proof.

First, suppose that $F$ is $G$-stable. Let $f:A\to B$ be a
$K_{G}$-equivalence. Write $\hat{K}_{G}$ for
$K(\mathbb{C}\oplus(L^{2}(G)\otimes l^{2}))$ with the obvious $G$-action, so
that one has equivariant isometric inclusions
$\mathbb{C}\hookrightarrow\hat{K}_{G}$ and $K_{G}\hookrightarrow\hat{K}_{G}$.
This gives a commutative diagram

${A}$${{A\otimes\hat{K}_{G}}}$${{A\otimes
K_{G}}}$${B}$${{B\otimes\hat{K}_{G}}}$${{B\otimes
K_{G}}}$$\scriptstyle{f}$$\scriptstyle{f\otimes\hat{K}_{G}}$$\scriptstyle{f\otimes
K_{G}}$

By assumption, $F$ inverts all the horizontal arrows and the right vertical
arrow (the latter because $f\otimes K_{G}$ is a homotopy equivalence and $F$
is homotopy invariant). Therefore, it inverts $f$.

Conversely, suppose that $F$ inverts $K_{G}$-equivalences. It suffices to show
that $K(V^{\prime})\to K(V)$ is a $K_{G}$-equivalence, where $V^{\prime}$ and
$V$ are as in Definition 2.1.3.

Let $K$ denote $K(l^{2})$. Suppose the $G$-actions on $V$ and $V^{\prime}$ are
trivial. Then $K(V^{\prime})\otimes K\to K(V)\otimes K$ is isomorphic to a
left upper corner inclusion, and therefore becomes a homotopy equivalence on
tensoring again by $K$. As $K_{G}\cong K_{G}\otimes K\otimes K$, we conclude
the result in this case.

In the general case, we will produce a natural equivariant isomorphism
$\phi_{H}:L^{2}(G)\otimes H\cong L^{2}(G)\otimes\text{Res}^{G}(H)$ for any
Hilbert space $H$ with $G$-action $\rho$. Here $\text{Res}^{G}(H)$ is the
Hilbert space $H$ with trivial $G$-action. This induces an isomorphism
$K(H)\otimes K_{G}\cong\text{Res}^{G}(K(H))\otimes K_{G}$, and we conclude by
the earlier case.

We can identify $L^{2}(G)\otimes H$ with $L^{2}(G,H)$ (the $G$-action given on
the latter by $g(f(h)):=\rho_{g}f(g^{-1}h)$), and similarly,
$L^{2}(G)\otimes\text{Res}^{G}(H)$ with $L^{2}(G,\text{Res}^{G}(H))$. We now
define $\phi_{H}$ by the formula $f(h)\mapsto\rho(h)^{-1}f(h)$, and one easily
checks that it is a natural equivariant isomorphism. ∎

We write $\hypertarget{hom}{GC^{*}\text{Alg}^{nu}_{h,sep}}$ for the
localization555We always refer to the Dwyer-Kan localization when we write
“localization”. See [Lur24, Tag 01N0] and [Lur24, Tag 01N1] for an existence
and uniqueness statement. of $GC^{*}\text{Alg}^{nu}_{sep}$ by the homotopy
equivalences, and
$\hypertarget{stable}{L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}}$ for the
localization of $GC^{*}\text{Alg}^{nu}_{sep}$ by the homotopy equivalences and
the $K_{G}$-equivalences. We have a sequence of localizations

${{L_{h,K_{G}}:GC^{*}\text{Alg}^{nu}_{sep}}}$${{GC^{*}\text{Alg}^{nu}_{h,sep}}}$${{L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}}}$$\scriptstyle{L_{h}}$$\scriptstyle{L_{K_{G}}}$

######  Definition 2.3.

Let $D$ be a pointed $\infty$-category, and
$F:GC^{*}\textup{Alg}^{nu}_{sep}\to D$ a reduced functor (that is, $F(0)$ is a
zero object). An exact sequence $0\to A\to B\to C\to 0$ in
$GC^{*}\textup{Alg}^{nu}_{sep}$ is semi-split if there exists a completely
positive and contractive $G$-equivariant section $C\to B$. The functor $F$ is
said to be semi-exact if it takes semi-split exact sequences to fibre
sequences.

We remark that completely analogous definitions can be made for
$GC^{*}\text{Alg}^{nu}$ in place of $GC^{*}\text{Alg}^{nu}_{sep}$.

If

$\mathcal{E}:0\to A\to B\xrightarrow{f}C\to 0$

is an exact sequence in $GC^{*}\text{Alg}^{nu}$, there is a natural induced
map $i_{f}:A\to C(f)$, where $C(f)$ is the cone on $f$. Let $W^{\prime}$
denote the class of morphisms of the form $L_{h,K_{G}}(i_{f})$ taken over all
semi-split exact sequences $\mathcal{E}$. Write $W$ for the closure of
$W^{\prime}$ by the 2-out-of-3 property and pullbacks.

Let

$L_{W}:L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}\to
L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}[W^{-1}]$

be the localization by the class $W$. The $\infty$-category
$L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}[W^{-1}]$ is left-exact and semi-
additive (see [Bun24, Cor. 4.3.2] and [Bun24, Prop. 4.4] in the case that $G$
is trivial and item 2 at the beginning of [BD24, Sec. 3.4] for discrete $G$).
There is a Bousfield localization

$\text{incl}:(L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}[W^{-1}])^{\text{grp}}\leftrightarrows
L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}[W^{-1}]:\Omega^{2}$

where the superscript grp indicates the full subcategory of group objects (see
[Bun24, Prop. 7.2] in the case that $G$ is trivial, and item 4 at the
beginning of [BD24, Sec 3.4] for an analogous statement for $E$-theory (for
discrete $G$)).

We define the $\infty$-category
$KK^{G}_{sep}:=(L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}[W^{-1}])^{\text{grp}}$,
and the functor $kk^{G}_{sep}$ as the composite

${{GC^{*}\text{Alg}^{nu}_{sep}}}$${{L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}}}$${{L_{K_{G}}GC^{*}\text{Alg}^{nu}_{h,sep}[W^{-1}]}}$${{KK^{G}_{sep}}}$$\scriptstyle{L_{h,K_{G}}}$$\scriptstyle{L_{W}}$$\scriptstyle{\Omega^{2}}$

The $\infty$-category $KK^{G}_{sep}$ is characterized essentially uniquely by
the following universal property.

######  Theorem 2.4.

$KK^{G}_{sep}$ is a stable $\infty$-category, the functor $kk^{G}_{sep}$ is
homotopy invariant, $G$-stable and semi-exact; and for any stable
$\infty$-category $D$, pre-composition by $kk^{G}_{sep}$ induces an
equivalence of functor categories

$\textup{Fun}^{ex}(KK^{G}_{sep},D)\simeq\textup{Fun}^{h,Gs,se}(GC^{*}\text{Alg}^{nu}_{sep},D)$

The superscripts $\\{h,Gs,se\\}$ refer to the full subcategory of homotopy
invariant, $G$-stable, semi-exact functors; the superscript $ex$ denotes the
full subcategory of exact funtors.

We define $KK^{G}:=\text{Ind}(KK^{G}_{sep})$ as the Ind-completion, with
$y:KK^{G}_{sep}\to KK^{G}$ the Yoneda embedding. We define the functor
$kk^{G}:GC^{*}\text{Alg}^{nu}\to KK^{G}$ by left-Kan extension as indicated
below:

${{GC^{*}\text{Alg}^{nu}_{sep}}}$${{GC^{*}\text{Alg}^{nu}}}$${{KK^{G}}}$${{KK^{G}_{sep}}}$$\scriptstyle{kk^{G}}$$\scriptstyle{kk^{G}_{sep}}$$\scriptstyle{y}$

We need another preliminary notion to characterize $KK^{G}$.

######  Definition 2.5.

Let $D$ be a cocomplete $\infty$-category, and $F:GC^{*}\textup{Alg}^{nu}\to
D$ a functor. $F$ is said to be $s$-finitary if the natural map

$\textup{colim}_{A^{\prime}\subset_{sep}A}F(A^{\prime})\to F(A)$

is an equivalence, for all objects $A$ of $GC^{*}\textup{Alg}^{nu}$. The
colimit here is taken over all separable $G$-$C^{*}$-subalgebras of $A$.

The $\infty$-category $KK^{G}$ is determined essentially uniquely by the
following universal property.

######  Theorem 2.6.

$KK^{G}$ is a stable and cocomplete $\infty$-category, the functor $kk^{G}$ is
homotopy invariant, $G$-stable, semi-exact and $s$-finitary; and for any
stable, cocomplete $\infty$-category $D$, pre-composition by $kk^{G}$ induces
an equivalence

$\textup{Fun}^{colim}(KK^{G},D)\simeq\textup{Fun}^{h,Gs,se,sfin}(GC^{*}\text{Alg}^{nu},D)$

The superscripts $\\{h,Gs,se,sfin\\}$ refer to the full subcategory of
homotopy invariant, $G$-stable, semi-exact and $s$-finitary functors; the
superscript $colim$ denotes the full subcategory of colimit-preserving
funtors.

We remark that $KK^{G}$ also admits a symmetric monoidal structure and that
$kk^{G}$ can be refined to a symmetric monoidal functor, and an analogous
universal property can be stated in this situation.

We denote $KK^{\\{e\\}}$ simply by $KK$, where $\\{e\\}$ is the trivial group.
For an object $A$ of $KK$, we write $K(A)$ “the $K$-theory spectrum of $A$”
for the mapping spectrum $KK(\mathbb{C},A)$. We denote the spectrum
$K(\mathbb{C})$ by $KU$; this coincides with the $2$-periodic topological
complex $K$-theory spectrum (see [Bun24, Rem. 9.19]). The spectrum-valued
functor $K(-)$ naturally refines to a functor valued in the $\infty$-category
$\text{Mod}(KU)$ of modules over the commutative ring spectrum $KU$.

A detailed account of the properties of $K(-)$ is presented in [BLP23].

###  2.2 The Crossed Product Functor

For a $G$-$C^{*}$-algebra $A$, one can consider the set $C_{c}(G,A)$ of
compactly supported continuous $A$-valued functions on $G$. This set acquires
the structure of a ${*}$-algebra over $\mathbb{C}$ with the following
definitions of the algebraic operations. For $f,f^{\prime}\in C_{c}(G,A)$ and
$g\in G$, we set:

  * •

$(f+f^{\prime})(g):=f(g)+f^{\prime}(g)$

  * •

$(f*f^{\prime})(g):=\int_{G}f(h)\alpha_{h}(f^{\prime}(h^{-1}g))\mu(h)$

  * •

$f^{*}(g):=\alpha_{g}(f(g^{-1})^{*})$

With this algebraic structure, one can check that $C_{c}(G,A)$ is a
pre-$C^{*}$-algebra - in fact, for $f\in C_{c}(G,A)$, the maximal norm of $f$
is bounded by its $L_{1}$-norm. We define the (maximal)666There is also a
variant called the reduced crossed product $A\rtimes_{r}G$ obtained by
completing $C_{c}(G,A)$ with respect to a particular representation. However,
we shall work exclusively with the maximal crossed product for simplicity, and
because of its compatibility with the maximal tensor product (Lemma 2.11).
crossed product $A\rtimes G$ to be the completion
$\hyperlink{shreehari}{\text{compl}}(C_{c}(G,A))$777This construction can also
be generalized for non-abelian $G$.. This yields a functor $-\rtimes
G:GC^{*}\text{Alg}^{nu}\to C^{*}\text{Alg}^{nu}$.888Throughout this paper, we
will abuse notation and use $-\rtimes G$ for different functors. By default,
we will mean the functor $GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$
of Proposition 2.8 or the functor $KK^{G}\to KK^{\hat{G}}$ of Theorem 2.19, as
should be clear from the context. In other cases, we will explicitly indicate
the source and target.

We will refine the functor $-\rtimes G$ to take values in
$\hat{G}C^{*}\text{Alg}^{nu}$, and show that it descends essentially uniquely
to a functor $KK^{G}\to KK^{\hat{G}}$.

To begin with, consider a character $\chi$ in $\hat{G}$, and an element $f$ of
$C_{c}(G,A)$, where $A$ is a $G$-$C^{*}$-algebra. We define $\chi f\in
C_{c}(G,A)$ by pointwise multiplication with the character value:

$(\chi f)(g):=\chi(g)f(g)$

######  Proposition 2.7.

The rule defined above gives a strongly continuous action of $\hat{G}$ on
$A\rtimes G$.

###### Proof.

By substituting into the definitions, one easily checks that we have an action
of $\hat{G}$ on the pre-$C^{*}$-algebra $C_{c}(G,A)$; applying compl, we get
an action of $\hat{G}$ on $A\rtimes G$.

We check that the action is continuous. We must show that for each element $f$
of $A\rtimes G$, the map $\hat{G}\to A\rtimes G$ given by $\chi\to\chi f$ is
continuous. To this end, we may assume that $f\in C_{c}(G,A)$, as $C_{c}(G,A)$
is dense in $A\rtimes G$. Let $S$ be the support of $f$. By assumption, $S$ is
compact, so $\mu(S)$ is finite. Suppose $\\{\chi_{i}\\}$ is a net converging
to the trivial character $1$. As the maximal norm is bounded by the
$L_{1}$-norm, it suffices to show that $||\chi_{i}f-f||_{L_{1}}$ goes to $0$.
We have

$||\chi_{i}f-f||_{L_{1}}=\int_{G}||\chi_{i}f-f||\mu\leq\int_{G}||\chi_{i}-1||||f||\mu$

To conclude, note that $||f||$ is bounded, and that by definition of the
compact-open topology, we can make the difference $||\chi_{i}-1||$ arbitrarily
small on $S$. ∎

It now follows that:

######  Proposition 2.8.

$-\rtimes G$ determines a functor
$GC^{*}\textup{Alg}^{nu}\to\hat{G}C^{*}\textup{Alg}^{nu}$.

We record the following important categorical fact.

######  Proposition 2.9.

The functor $-\rtimes
G:GC^{*}\textup{Alg}^{nu}\to\hat{G}C^{*}\textup{Alg}^{nu}$ preserves filtered
colimits.

###### Proof.

As this result appears to be folklore, we only sketch the idea.

Let $I$ be a small filtered category, and $A:I\to GC^{*}\text{Alg}^{nu}$ a
functor. We first assume that $A$ is unital - i.e., it factors through the
inclusion $GC^{*}\textup{Alg}\hookrightarrow GC^{*}\textup{Alg}^{nu}$. One can
show by hand that the canonical map $\text{colim}_{I}(A\rtimes
G)\to(\text{colim}_{I}A)\rtimes G$ is surjective. To produce an inverse, we
use the observation that if $A_{i}\to A_{j}$ is a unital homomorphism, then
$A_{i}\rtimes G\to A_{j}\rtimes G$ is an essential homomorphism and thus
functorially induces a map on multiplier algebras $M(A_{i}\rtimes G)\to
M(A_{j}\rtimes G)$ together with [Wil07, Prop. 2.34]. The extension from the
unital to the non-unital case is carried out exactly as in the proof of
[BEL23, Thm 7.13.1]. ∎

We now proceed to show that $-\rtimes
G:GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$ descends to a functor
$-\rtimes G:KK^{G}\to KK^{\hat{G}}$, by showing that it descends through the
intermediate localizations introduced in Section 2.1. Recall the
$\infty$-categories GC∗Algnuh,sep and LKGGC∗Algnuh,sep introduced there.

Observe that the functor $-\rtimes
G:GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$ preserves separable
algebras. We recall the following criterion to check for topological
enrichment.

######  Lemma 2.10.

Let $F:GC^{*}\textup{Alg}^{nu}_{sep}\to
G^{\prime}C^{*}\textup{Alg}^{nu}_{sep}$ be a functor. Suppose that for all
commutative algebras $B$ and for all objects $A$ of
$GC^{*}\textup{Alg}^{nu}_{sep}$, there exists a natural isomorphism

$F(A\otimes B)\cong F(A)\otimes B$

(where $B$ is given trivial group actions). Further, assume that the composite
isomorphism

$F(A)\cong F(A\otimes\mathbb{C})\cong F(A)\otimes\mathbb{C}\cong F(A)$

is the identity for all objects $A$ of $GC^{*}\textup{Alg}^{nu}_{sep}$. Then
$F$ is a topologically enriched functor and hence descends essentially
uniquely to a functor $F:GC^{*}\textup{Alg}^{nu}_{h,sep}\to
G^{\prime}C^{*}\textup{Alg}^{nu}_{h,sep}$.

###### Proof.

Write $\text{Hom}_{G}(-,-)$ and $\text{Hom}_{G^{\prime}}(-,-)$ for the mapping
spaces in $GC^{*}\textup{Alg}^{nu}_{sep}$ and
$G^{\prime}C^{*}\textup{Alg}^{nu}_{sep}$, respectively.

For objects $A,B$ of $GC^{*}\textup{Alg}^{nu}_{sep}$, and a compact
topological space $X$, consider the composite map

$\displaystyle\text{Hom}_{\textbf{Top}}(X,\text{Hom}_{G}(A,B))$
$\displaystyle\cong\text{Hom}_{G}(A,B\otimes C(X))$
$\displaystyle\to\text{Hom}_{G}(F(A),F(B\otimes C(X)))$
$\displaystyle\cong\text{Hom}_{G^{\prime}}(F(A),F(B)\otimes C(X))$
$\displaystyle\cong\text{Hom}_{\textbf{Top}}(X,\text{Hom}_{G^{\prime}}(F(A),F(B)))$

The condition on the isomorphism in the statement of the lemma implies that
the underlying map of sets
$\text{Hom}_{\textbf{Top}}(X,\text{Hom}_{G}(A,B))\to\text{Hom}_{\textbf{Top}}(X,\text{Hom}_{G^{\prime}}(F(A),F(B)))$
constructed above is induced by $F$, from which the lemma follows because the
point-norm topology on mapping sets between separable algebras is compactly
generated. ∎

######  Lemma 2.11.

Let $A$ be an object of $GC^{*}\textup{Alg}^{nu}$ and $B$ an object of
$C^{*}\textup{Alg}^{nu}$. There is a natural isomorphism of algebras

$\phi_{A,B}:(A\rtimes G)\otimes B\simeq(A\otimes B)\rtimes G$

given by $f\otimes b\mapsto(g\mapsto f(g)\otimes b)$ on the dense subset
$C_{c}(G,A)\otimes B$.

###### Proof.

See [Wil07, Lem. 2.75]. ∎

######  Proposition 2.12.

The functor $-\rtimes
G:GC^{*}\textup{Alg}^{nu}_{sep}\to\hat{G}C^{*}\textup{Alg}^{nu}_{sep}$
descends essentially uniquely to a functor $-\rtimes
G:GC^{*}\textup{Alg}^{nu}_{h,sep}\to\hat{G}C^{*}\textup{Alg}^{nu}_{h,sep}$.

###### Proof.

We check the condition of Lemma 2.10. Indeed, the isomorphism $\phi_{A,B}$ of
Lemma 2.11 gives us the desired isomorphism: it is $\hat{G}$-equivariant, and
the composition

$A\rtimes G\cong(A\rtimes
G)\otimes\mathbb{C}\xrightarrow{\phi_{A,\mathbb{C}}}(A\otimes\mathbb{C})\rtimes
G\cong A\rtimes G$

is the identity. ∎

######  Lemma 2.13.

Let $B$ be a $C^{*}$-algebra, and let $\rho:G\to U(M(B))$ be a strongly
continuous group homomorphism. Then $B$ becomes a $G$-$C^{*}$-algebra with
action $\beta_{g}(b):=\rho_{g}b\rho_{g^{-1}}$. Denote by $\underline{B}$ the
algebra $B$ with trivial $G$-action. For any $G$-$C^{*}$-algebra $(A,\alpha)$,
there is a natural (in $A$) algebra isomorphism

$\psi_{A,B}:(A\otimes B)\rtimes G\to(A\otimes\underline{B})\rtimes G$

given by $f(g)\mapsto f(g)(\textup{id}\otimes\rho_{g})$ on the dense subset
$C_{c}(G,A\otimes B)$.

###### Proof.

Note that $A\otimes B$ is an ideal in $M(A)\otimes M(B)$, so by the universal
property of the multiplier algebra, we obtain a unital map $M(A)\otimes
M(B)\to M(A\otimes B)$. We can therefore define $\textup{id}\otimes\rho_{g}$
as the composite

${G}$${{U(M(A))\times U(M(B))}}$${{U(M(A)\otimes M(B))}}$${{U(M(A\otimes
B))}}$$\scriptstyle{(\text{id},\rho_{g})}$$\scriptstyle{(a,b)\mapsto a\otimes
b}$

We check that the formula $f(g)\mapsto f(g)(\textup{id}\otimes\rho_{g})$
defines a $*$-algebra isomomorphism

$\psi^{\prime}_{A,B}:C_{c}(G,A\otimes B)\to C_{c}(G,A\otimes\underline{B})$

One can easily check from the definitions that it defines a $*$-algebra
homomorphism. The inverse $\psi^{\prime-1}_{A,B}$ is given by $f(k)\mapsto
f(k)(\text{id}\otimes\rho_{k^{-1}})$.

We now set $\psi_{A,B}:=\text{compl}(\psi^{\prime}_{A,B})$. ∎

######  Proposition 2.14.

The functor $-\rtimes
G:GC^{*}\textup{Alg}^{nu}_{h,sep}\to\hat{G}C^{*}\textup{Alg}^{nu}_{h,sep}$
descends essentially uniquely to a functor $-\rtimes
G:L_{K_{G}}GC^{*}\textup{Alg}^{nu}_{h,sep}\to
L_{K_{\hat{G}}}\hat{G}C^{*}\textup{Alg}^{nu}_{h,sep}$.

###### Proof.

We will show that the composition

$F:GC^{*}\text{Alg}^{nu}_{sep}\xrightarrow{-\rtimes
G}\hat{G}C^{*}\text{Alg}^{nu}_{sep}\xrightarrow{L_{K_{\hat{G}},h}}L_{K_{\hat{G}}}\hat{G}C^{*}\text{Alg}^{nu}_{h,sep}$

is $G$-stable. Since $F$ is homotopy invariant by Lemma 2.12, this would imply
the assertion by the universal property of
$L_{K_{G}}GC^{*}\textup{Alg}^{nu}_{h,sep}$.

Let $V^{\prime}$ and $V$ be $G$-Hilbert spaces together with an equivariant
unitary embedding $V^{\prime}\hookrightarrow V$. We have to show that for any
$A\in GC^{*}\text{Alg}^{nu}_{sep}$, the induced map $A\otimes K(V^{\prime})\to
A\otimes K(V)$ is sent to an equivalence by $F$.

Using Lemma 2.11 and Lemma 2.13, we have the following commutative diagram of
algebras, with the horizontal arrows being natural isomorphisms:

${{(A\otimes K(V^{\prime}))\rtimes
G}}$${{(A\otimes\underline{K(V^{\prime})})\rtimes G}}$${{(A\rtimes
G)\otimes\underline{K(V^{\prime})}}}$${{(A\otimes K(V))\rtimes
G}}$${{(A\otimes\underline{K(V)})\rtimes G}}$${{(A\rtimes
G)\otimes\underline{K(V)}}}$$\scriptstyle{\psi_{A,K(V^{\prime})}}$$\scriptstyle{\psi_{A,K(V)}}$$\scriptstyle{\phi_{A,K(V^{\prime})}}$$\scriptstyle{\phi_{A,K(V)}}$

We can upgrade this to a commutative diagram of $\hat{G}$-algebras by
introducing the appropriate $\hat{G}$ actions on the algebras in the middle
and right columns. It is easy to see that this will be the usual crossed
product action on the middle column, and $\hat{\alpha}\otimes\text{id}$ on the
right column, where $\hat{\alpha}$ is the usual action on $A\rtimes G$.

The right vertical map is induced by the identity on $A\rtimes G$ and the
embedding $V^{\prime}\hookrightarrow V$. Therefore, it becomes an equivalence
on passage to $L_{K_{\hat{G}}}\hat{G}C^{*}\text{Alg}^{nu}_{h,sep}$
(Proposition 2.2). Hence, so does the left vertical map, and this is what we
wanted to show. ∎

######  Proposition 2.15.

The functor $-\rtimes G:L_{K_{G}}GC^{*}\textup{Alg}^{nu}_{sep,h}\to
L_{K_{\hat{G}}}\hat{G}C^{*}\textup{Alg}^{nu}_{sep,h}$ descends essentially
uniquely to an exact functor $-\rtimes G:KK^{G}_{sep}\to KK^{\hat{G}}_{sep}$.

###### Proof.

First, we show that $-\rtimes
G:GC^{*}\text{Alg}^{nu}_{sep}\to\hat{G}C^{*}\text{Alg}^{nu}_{sep}$ preserves
semi-exact sequences.

So, suppose that we are given a semi-exact sequence of $G$-$C^{*}$-algebras

${0}$${I}$${A}$${Q}$${0}$

with a $G$-equivariant completely positive and contractive split $r:Q\to A$.
It is known (Proposition 9 of [Mey07]) that this yields a semi-exact sequence
of algebras

${0}$${{I\rtimes G}}$${{A\rtimes G}}$${{Q\rtimes G}}$${0}$

with completely positive and contractive split $\hat{r}:Q\rtimes G\to A\rtimes
G$ induced by post-composition with $r$ on $C_{c}(G,Q)\to C_{c}(G,A)$.

It only remains to check that $\hat{r}$ is $\hat{G}$-equivariant - but this is
clear from the definition of the action and the $\mathbb{C}$-linearity of $r$.

Thus, the composition $kk^{\hat{G}}\circ\rtimes G:GC^{*}\text{Alg}^{nu}\to
KK^{\hat{G}}_{sep}$ is semi-exact, and we now conclude using the universal
property of $KK^{G}_{sep}$. ∎

To proceed, we need the notion of an Ind-$s$-finitary functor:

######  Definition 2.16.

([BEL23, Def. 4.1])999This definition and the following two lemmas are stated
in [BEL23] for discrete groups. However, the arguments provided there work
almost verbatim in our generality.
A functor $F:GC^{*}\textup{Alg}^{nu}\to HC^{*}\textup{Alg}^{nu}$ that
preserves separable algebras is said to be Ind-$s$-finitary if:

  1. 1.

For all $A\in GC^{*}\textup{Alg}^{nu}$, the inductive system
$F(A^{\prime})^{F(A)}_{A^{\prime}\subset_{sep}A}$ is cofinal in the system of
all separable algebras of $F(A)$.

  2. 2.

The canonical map
$F(A^{\prime})_{A^{\prime}\subset_{sep}A}\to(F(A^{\prime})^{F(A)})_{A^{\prime}\subset_{sep}A}$
is an isomorphism of inductive systems in
$\textup{Ind}(HC^{*}\textup{Alg}^{nu})$.

The notation $F(A^{\prime})^{F(A)}$ denotes the image of $F(A^{\prime})$ in
$F(A)$.

######  Lemma 2.17.

Let $F:GC^{*}\textup{Alg}^{nu}\to HC^{*}\textup{Alg}^{nu}$ be a functor that
preserves separable algebras and satisfies item 1 of Definition 2.16. If $F$
preserves countably filtered colimits, then it is Ind-$s$-finitary.

###### Proof.

This is [BEL23, Lem. 4.3.1]. ∎

The notion of Ind-$s$-finitariness is related to that of $s$-finitariness by
the following result:

######  Lemma 2.18.

Let $F^{\prime}:HC^{*}\textup{Alg}^{nu}\to D$ be an $s$-finitary functor, and
$F:GC^{*}\textup{Alg}^{nu}\to HC^{*}\textup{Alg}^{nu}$ an Ind-$s$-finitary
functor. Then, the composite $F^{\prime}\circ F:GC^{*}\textup{Alg}^{nu}\to D$
is an $s$-finitary functor.

###### Proof.

This is [BEL23, Lem. 4.2]. ∎

We can now conclude this section with its main result:

######  Theorem 2.19.

The functor $-\rtimes G:KK^{G}_{sep}\to KK^{\hat{G}}_{sep}$ extends
essentially uniquely to a colimit-preserving functor $-\rtimes G:KK^{G}\to
KK^{\hat{G}}$.

###### Proof.

By Lemma 2.18, all we have to check is that $-\rtimes
G:GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$ is Ind-$s$-finitary.

We first verify the cofinality condition of Definition 2.16. Let $A$ be a
$G$-$C^{*}$-algebra, and $B$ a separable $\hat{G}$-subalgebra of $A\rtimes G$.
Let $\\{f_{i}\\}$ be a countable dense subset of $B$. It is a standard fact
(see [Wil07, Lem. 1.87]) that elements of the form $f\otimes a$, with $f\in
C_{c}(G)$ and $a\in A$, generate $A\rtimes G$. Thus, for each $i$, one can
find a sequence $\\{f_{ik}=\Sigma_{l=1}^{n_{k,i}}g_{l,k,i}\otimes
a_{l,k,i}\\}$ of linear combinations of elementary tensors that converges to
$f_{i}$. Consider the $G$-$C^{*}$-subalgebra $A^{\prime}$ of $A$ generated by
the collection $\\{a_{l,k,i}\\}$. Then $A^{\prime}$ is separable and $B$ is
contained in the image of $A^{\prime}\rtimes G$ in $A\rtimes G$.

By Proposition 2.9, $-\rtimes
G:GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$ preserves filtered
colimits.

The theorem now follows from Lemma 2.17. ∎

##  3 Takai Duality

We retain the notation and conventions from the previous section - we will
denote by $G$ a locally compact, second countable, abelian group with identity
element $e$ and its Pontryagin dual by $\hat{G}$. The dual group $\hat{G}$ is
endowed with the compact-open topology. We fix Haar measures $\mu$ and
$\hat{\mu}$ on $G$ and $\hat{G}$, respectively. These are unique up to
multiplication by a positive scalar.

Classically, the Takai duality statement is simply stated as the existence of
a certain equivariant isomorphism of algebras (Theorem 1.1); however, this is
not sufficient for our purposes. We shall begin by restating this in
categorical language.

######  Theorem 3.1.

Let $H:GC^{*}\textup{Alg}^{nu}\to GC^{*}\textup{Alg}^{nu}$ be the functor
$A\mapsto(A\rtimes G)\rtimes\hat{G}$, with the double-dual action, and let
$H^{\prime}:GC^{*}\textup{Alg}^{nu}\to GC^{*}\textup{Alg}^{nu}$ be the functor
$A\mapsto A\otimes K(L^{2}(G))$ with the tensor action (here $K(L^{2}(G))$ is
given the usual conjugation action). Then there is an isomorphism of functors
$F\simeq F^{\prime}$.

###### Proof.

The classical statement is that, for any $G$-$C^{*}$-algebra $A$, there exists
an equivariant isomorphism $(A\rtimes G)\rtimes\hat{G}\cong A\otimes
K(L^{2}(G))$; and this isomorphism is constructed as a sequence of
isomorphisms (see [Wil07, Sec. 7.1] for the proof). We will upgrade this to a
sequence of isomorphisms between functors and focus on the naturality,
omitting the well-established details of why each map that appears is an
equivariant isomorphism.

Let $A$ be a $G$-$C^{*}$-algebra with action $\alpha$. Classically, the major
part of the argument is the construction of an algebra isomorphism

$\phi_{1}^{A}:(A\rtimes
G)\rtimes\hat{G}\to({C_{0}(G)}\otimes\underline{A})\rtimes G$

where $\underline{A}$ is the algebra $A$ with trivial $G$-action. Let $F$ be
an element of $C_{c}(\hat{G}\times G,A)$, interpreted as an element of
$(A\rtimes G)\rtimes\hat{G}$ under the identification $C_{c}(\hat{G}\times
G,A)\cong C_{c}(\hat{G},C_{c}(G,A))\subset C_{c}(G,A\rtimes G)\subset(A\rtimes
G)\rtimes\hat{G}$. For such an element, $\phi_{1}^{A}$ is described by the
formula

$\phi_{1}^{A}(F)(s,r)=\int_{\hat{G}}\alpha_{r}^{-1}(F(\gamma,s))\overline{\gamma(s^{-1}r)}\hat{\mu}(\gamma)$

where $\phi_{1}^{F}(-,r)$ is interpreted as an element of
$C_{c}(G,C_{0}(G)\otimes\underline{A})\subset({C_{0}(G)}\otimes\underline{A})\rtimes
G$.

Consider the algebra $C_{0}(G)\rtimes G$, where $G$ acts on $C_{0}(G)$ by
left-multiplication. Functoriality induces an action $\kappa$ of $G$ on
$C_{0}(G)\rtimes G$. Let $\kappa^{\prime}$ be the unique action on
$({C_{0}(G)}\otimes\underline{A})\rtimes G$ such that the isomorphism
$\phi_{C_{0}(G),A}$ of Lemma 2.11 becomes equivariant with respect to the
action $\kappa\otimes\alpha$ on $(C_{0}(G)\rtimes G)\otimes A$. We obtain a
functor $H_{1}:GC^{*}\text{Alg}^{nu}\to GC^{*}\text{Alg}^{nu}$ given by

$(A,\alpha)\mapsto(({C_{0}(G)}\otimes\underline{A})\rtimes
G),\kappa^{\prime})$

One can check that $\phi_{1}^{A}$ is $G$-equivariant with the double-dual
action on $(A\rtimes G)\rtimes\hat{G}$ and the action $\kappa^{\prime}$ on
$({C_{0}(G)}\otimes\underline{A})\rtimes G$.

As integration commutes with bounded linear operators (in particular,
$C^{*}$-algebra morphisms), it follows that the family of isomorphisms
$\\{\phi_{1}^{A}\\}_{A\in GC^{*}\text{Alg}^{nu}}$ assembles to a natural
isomorphism of functors $H\simeq H_{1}$.

We now construct a natural isomorphism $H_{1}\simeq H^{\prime}$ as follows:

In the classical proof, an equivariant isomorphism

$\phi_{2}:({C_{0}(G)}\rtimes G,\kappa)\to(K(L^{2}(G)),\text{conjugation})$

is first constructed. On tensoring by $(A,\alpha)$ and using the isomorphism
$\phi_{C_{0}(G),A}$ of Lemma 2.11, we obtain an equivariant isomorphism

$\phi_{2}^{A}:({C_{0}(G)}\otimes\underline{A})\rtimes G\simeq A\otimes
K(L^{2}(G))$

The family of isomorphisms $\\{\phi_{2}^{A}\\}_{A\in GC^{*}\text{Alg}^{nu}}$
assembles to the desired natural isomorphism.

∎

######  Corollary 3.2.

The functor $(-\rtimes G)\rtimes\hat{G}:KK^{G}\to KK^{G}$ is equivalent to the
identity functor on $KK^{G}$.

###### Proof.

We have a natural (in $A$) zig-zag of morphisms

${A}$${{A\otimes\hat{K}_{G}}}$${{A\otimes K_{G}}}$${{A\otimes
K(L^{2}(G))\cong(A\rtimes G)\rtimes\hat{G}}}$

Each map above is inverted by $kk^{G}$ (see Proposition 2.2). ∎

######  Corollary 3.3.

The functor $-\rtimes G:KK^{G}\to KK^{\hat{G}}$ is an equivalence of
$\infty$-categories.

###### Proof.

Apply the previous corollary to $G$ and $\hat{G}$. ∎

Our aim in the remainder of this section is to provide a different formula for
the crossed product functor $-\rtimes G:KK^{G}\to KK^{\hat{G}}$ that is closer
in form to the duality result of Treumann.

The Hilbert space $L^{2}(G)$ admits a (unitary) $\hat{G}$-action defined as
follows. Let $f$ be an element of $L^{2}(G)$, interpreted as a function
$f:G\to\mathbb{C}$; then, for a character $\chi$ in $\hat{G}$, we set $(\chi
f)(g):=\chi(g)f(g)$. Conjugation now yields a $\hat{G}$-action on
$K(L^{2}(G))$. Also, recall the canonical $G$-action on $K(L^{2}(G))$ by
conjugation.

######  Lemma 3.4.

The actions of $G$ and $\hat{G}$ on $K(L^{2}(G))$ commute and thus yield a
$G\times\hat{G}$-action on $K(L^{2}(G))$.

###### Proof.

For $\chi\in\hat{G}$ and $g\in G$, let us denote by $S_{\chi}$ and $T_{g}$ the
linear operators representing the respective actions of $\chi$ and $g$ on
$L^{2}(G)$. Explicitly, for an element $f$ of $L^{2}(G)$, we have
$S_{\chi}(f)(h)=\chi(h)f(h)$ and $T_{g}(f)(h)=f(gh)$. We will show that for
any compact operator $A$ in $K(L^{2}(G))$, there is an equality
$S_{\chi}T_{g}^{-1}AT_{g}S_{\chi}^{-1}=T_{g}^{-1}S_{\chi}AS_{\chi}^{-1}T_{g}$.
It suffices to show that $S_{\chi}^{-1}T_{g}$ and $T_{g}S_{\chi}^{-1}$ are
scalar multiples of each other.

Indeed, for a function $f\in L^{2}(G)$, one has
$S_{\chi}^{-1}T_{g}(f)(h)=\chi(g^{-1}h^{-1})f(gh)$ and
$T_{g}S_{\chi}^{-1}(f)(h)=\chi(h^{-1})f(gh)$.

Thus, we can define an action of $G\times\hat{G}$ on $K(L^{2}(G))$ by
$(g,\chi)(A):=T_{g}S_{\chi}^{-1}AS_{\chi}T_{g}^{-1}$. ∎

We denote by B the algebra $K(L^{2}(G))$ together with this
$(G\times\hat{G})$-action. We can now define a functor
$F:GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$ by
$A\mapsto(A\otimes\textbf{B})\rtimes G$, where the $\hat{G}$-action is the one
induced by the action of $\hat{G}$ on B.

######  Proposition 3.5.

The functor $F:GC^{*}\textup{Alg}^{nu}\to\hat{G}C^{*}\textup{Alg}^{nu}$
descends essentially uniquely to a functor $F:KK^{G}\to KK^{\hat{G}}$.

###### Proof.

We realize $F$ as a composite of three functors -

$i:GC^{*}\text{Alg}^{nu}\hookrightarrow(G\times\hat{G})C^{*}\text{Alg}^{nu}$
$-\otimes\textbf{B}:(G\times\hat{G})C^{*}\text{Alg}^{nu}\to(G\times\hat{G})C^{*}\text{Alg}^{nu}$
$-\rtimes
G:(G\times\hat{G})C^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$

where, in the third functor, the $G$-action is used to form the crossed
product and the $\hat{G}$-action on the target is induced by functoriality.

One easily checks (using Proposition 2.2) that the first functor descends. The
second functor descends by the symmetric monoidality of $kk^{G\times\hat{G}}$.
That the third functor descends is seen through an identical101010To see why
the steps go through, the main point to note is that the isomorphisms
appearing in the proof of Proposition 2.14 are $\hat{G}$-equivariant if $A$ is
a $(G\times\hat{G})$-$C^{*}$-algebra, and if $V^{\prime}$ and $V$ are
$(G\times\hat{G})$-Hilbert spaces. The rest of the argument is trivial.
argument as in the descent of $-\rtimes
G:GC^{*}\text{Alg}^{nu}\to\hat{G}C^{*}\text{Alg}^{nu}$ explained in the
previous section. ∎

######  Theorem 3.6.

The descended functor $F:KK^{G}\to KK^{\hat{G}}$ implements Takai duality.

###### Proof.

Write $V$ for $L^{2}(G)$. Let $\underline{\textbf{B}}$ denote the
$G$-$C^{*}$-algebra $K(V)$ with trivial $G$-action. For a $G$-$C^{*}$-algebra
$A$, we have the isomorphism of Lemma 2.13 (since translation operators are
unitary)

$\psi_{A,\textbf{B}}:(A\otimes\textbf{B})\rtimes
G\to(A\otimes\underline{\textbf{B}})\rtimes G$

Identifying $A\otimes K(V)$ with $A$-valued convolution kernels of the form
$a(g^{\prime},g)$, the formula of Lemma 2.13 says that $\psi_{A,\textbf{B}}$
is given by $a_{k}(g^{\prime},g)\mapsto a_{k}(k^{-1}g^{\prime},g)$.

We give $(A\otimes\textbf{B})\rtimes G$ the $\hat{G}$-action of $F(A)$, and
define an action on $(A\otimes\underline{\textbf{B}})\rtimes G$ by
$\chi(a_{k}(g^{\prime},g)):=\chi(k(g^{\prime})^{-1}g)a_{k}(g^{\prime},g)$. One
can check that $\psi_{A,\textbf{B}}$ is $\hat{G}$-equivariant with respect to
these actions.

We have the natural algebra isomorphism of Lemma 2.11

$\phi_{A,\underline{\textbf{B}}}:(A\rtimes
G)\otimes\underline{\textbf{B}}\to(A\otimes\underline{\textbf{B}})\rtimes G$

which is given by $f(k)\otimes c(g^{\prime},g)\mapsto(k\mapsto f(k)\otimes
c(g^{\prime},g))$. We introduce a $\hat{G}$-action on $(A\rtimes
G)\otimes\underline{\textbf{B}}$ by setting $\chi(f(k)\otimes
c(g^{\prime},g)):=\chi(k)f(k)\otimes\chi((g^{\prime})^{-1}g)c(g^{\prime},g)$.
One can check that $\phi_{A,\underline{\textbf{B}}}$ is $\hat{G}$-equivariant
with respect to these actions.

We can now make the final comparison with the classical Takai duality functor
$-\rtimes G$. Consider the algebra $A\rtimes G$ with the “standard”
$\hat{G}$-action, i.e. given by $(\chi f)(k)=\chi(k)f(k)$. As in the proof of
Corollary 3.2, we define a zig-zag $\eta_{A}$ of $\hat{G}$-equivariant
morphisms

${{A\rtimes G}}$${{(A\rtimes G)\otimes K_{\hat{G}}}}$${{(A\rtimes G)\otimes
K_{G}}}$${{(A\rtimes G)\otimes K(L^{2}(G))=(A\rtimes
G)\otimes\underline{\textbf{B}}}}$

It is clear that $\eta_{A}$ is inverted by $kk^{\hat{G}}$ (Proposition 2.2).

To summarize, one has the following zig-zag of natural equivariant algebra
morphisms that become equivalences on descent to $KK^{\hat{G}}$, and thus
prove the theorem:

${{F(A)=(A\otimes\textbf{B})\rtimes
G}}$${{(A\otimes\underline{\textbf{B}})\rtimes G}}$${{(A\rtimes
G)\otimes\underline{\textbf{B}}}}$${{A\rtimes
G}}$$\scriptstyle{\psi_{A,\textbf{B}}}$$\scriptstyle{\phi_{A,\underline{\textbf{B}}}}$$\scriptstyle{\eta_{A}}$

∎

##  4 Treumann Duality

We begin with some preliminaries.

For a commutative ring spectrum $S$, we write $\text{Mod}(S)$ for the
presentable stable $\infty$-category of modules over $S$. Given a commutative
ring spectrum $R$ and a finite abelian group $G$, we define a commutative ring
spectrum

$R[G]:=R\wedge\Sigma_{+}^{\infty}G$

This is the $\infty$-categorical analogue of the group-ring construction, with
the ring structure on $R[G]$ coming from the ring structure of $R$ and the
group operation on $G$.

For finite abelian groups $G_{1}$ and $G_{2}$, we define a
$(G_{1},G_{2})$-bimodule over $R$ to be a module over $R[G_{1}\times G_{2}]$.
Each such bimodule $M$ determines a colimit-preserving functor
$-\otimes_{R[G_{1}]}M:\text{Mod}(R[G_{1}])\to\text{Mod}(R[G_{2}])$.
Conversely, every colimit preserving functor arises from some bimodule (see
[Tre15, Sec. 2.4.7]).

######  Proposition 4.1.

Let $G$ be a finite abelian111111The proof below works for any group, in fact.
group. We have an equivalence of $\infty$-categories

$\textup{Fun}(BG,\textup{Mod}(R))\simeq\textup{Mod}(R[G])$

###### Proof.

One argument is sketched at the beginning of [Tre15, Sec 2.4].

We sketch a more concrete argument121212This was explained to the author by
Christoph Winges.. Consider the colimit-preserving functor
$j^{\prime}:\text{Mod}(R)\to\text{Mod}(R[G])$ determined by the
$(\\{e\\},G)$-bimodule $R[G]$. Endowing $\text{Mod}(R)$ and $\text{Mod}(R[G])$
(regarded as objects in the $\infty$-category $\text{Pr}^{L}$ of presentable
$\infty$-categories and colimit-preserving functors) with the trivial
$G$-action, we see that $j^{\prime}$ induces a morphism in
$\text{Fun}(BG,\text{Pr}^{L})$. Its left adjoint is a morphism in
$\text{Pr}^{L}$

$j:\text{colim}_{BG}\text{Mod}(R)\to\text{Mod}(R[G])$

Since the colimit of a diagram in $\text{Pr}^{L}$ is computed by passing to
the right adjoint diagram and computing the limit in $\infty$-categories, we
have

$\displaystyle\text{colim}_{BG}\text{Mod}(R)$
$\displaystyle\simeq\text{lim}_{BG^{\text{op}}}\text{Mod}(R)$
$\displaystyle\simeq\text{lim}_{BG^{\text{op}}}\text{Fun}(*,\text{Mod}(R))$
$\displaystyle\simeq\text{Fun}(\text{colim}_{BG^{\text{op}}}*,\text{Mod}(R))$
$\displaystyle\simeq\text{Fun}(BG^{\text{op}},\text{Mod}(R))$
$\displaystyle\simeq\text{Fun}(BG,\text{Mod}(R))$

Moreover, the underlying functor
$c:\text{Mod}(R)\to\text{Fun}(BG,\text{Mod}(R))$ is given by taking the left
adjoint to the forgetful functor
$\text{Fun}(BG,\text{Mod}(R))\to\text{Mod}(R)$, i.e. by left-Kan extension
along the inclusion $*\hookrightarrow BG$. The object
$c(R)\in\text{Fun}(BG,\text{Mod}(R))$ generates
$\text{Fun}(BG,\text{Mod}(R))$, and one can compute that its underlying
$R$-module is $\bigoplus_{G}R$, with $G$-action given by permuting the
summands.

We need to show that $j$ is an equivalence. We will first show that $j$
induces an equivalence of mapping spectra

$\text{map}_{\text{Fun}(BG,\text{Mod}(R))}(c(R),c(R))\simeq\text{map}_{\text{Mod}(R[G])}(R[G],R[G])$

We have a commutative triangle of left adjoint functors:

${{\text{Mod}(R)}}$${{\text{Mod}(R[G])}}$${{\text{Fun}(BG,\text{Mod}(R))}}$$\scriptstyle{c}$$\scriptstyle{j}$$\scriptstyle{j^{\prime}}$

We denote by $(-)^{*}$ the right adjoint to $(-)$, where $(-)$ is one of
$j^{\prime},j$, or $c$. We observe that $(j^{\prime})^{*}$ and $c^{*}$ are
forgetful functors. The functor $j^{*}$ endows the underlying $R$-module of an
object of $\text{Mod}(R[G])$ the $G$-action granted by the symmetric
monoidality of the functor $R[-]$.

We reduce to showing that the unit $c(R)\to j^{*}j^{\prime}(R)$ induces an
equivalence $c^{*}c(R)\to(j^{\prime})^{*}j^{\prime}(R)$. We know that $c(R)$
and $j^{*}j^{\prime}(R)$ are abstractly isomorphic - the underlying $R$-module
is $\bigoplus_{G}R$ with $G$ acting by permuting the summands, in both cases.
Therefore it suffices to show that the map
$c^{*}c(R)\to(j^{\prime})^{*}j^{\prime}(R)$ takes the summand indexed by $e$
to the summand indexed by $e$ via the identity map. This follows from the fact
that the units $R\to(j^{\prime})^{*}j^{\prime}(R)$ and $R\to c^{*}cR$ can both
be described by the map $R\to\bigoplus_{G}R$ that takes $R$ via the identity
to the $e$-indexed summand of $\bigoplus_{G}R$, and one has a factorization of
the unit maps as

$R\to c^{*}cR\to(j^{\prime})^{*}j^{\prime}(R)$

We claim that this completes the proof - indeed, consider the full subcategory
$\mathcal{C}$ of $\text{Fun}(BG,\text{Mod}(R))$ spanned by objects $X$ for
which $j$ induces an equivalence

$\text{map}_{\text{Fun}(BG,\text{Mod}(R))}(c(R),X)\simeq\text{map}_{\text{Mod}(R[G])}(R[G],j(X))$

Then $\mathcal{C}$ contains $c(R)$, is closed under limits and filtered
colimits (because $c(R)$ is compact) and thus is closed under all colimits, so
$\mathcal{C}=\text{Fun}(BG,\text{Mod}(R))$. Now consider the full subcategory
$\mathcal{D}$ of $\text{Fun}(BG,\text{Mod}(R))$ spanned by objects $Y$ for
which $j$ induces an equivalence

$\text{map}_{\text{Fun}(BG,\text{Mod}(R))}(Y,X)\simeq\text{map}_{\text{Mod}(R[G])}(j(Y),j(X))$

for all objects $X$ of $\text{Fun}(BG,\text{Mod}(R))$. Then $\mathcal{D}$
contains $c(R)$ (as just argued) and is closed under colimits, so
$\mathcal{D}=\text{Fun}(BG,\text{Mod}(R))$ as desired.

Note that $j$ has the property of preserving the underlying $R$-module
structure. ∎

We write $\text{Mod}(R[G])^{ft}$ for the full subcategory of
$\text{Mod}(R[G])$ generated by those modules whose homotopy groups are
finitely generated over $\pi_{0}(R)$. We refer to the objects of
$\text{Mod}(R[G])^{ft}$ as modules of finite type over $R[G]$.

Fix a prime $p$ for the remainder of this section, and let $KU_{p}$ denote the
$p$-completion of $KU$.

######  Proposition 4.2.

  1. 1.

For a finite abelian group $G$, the equivalence of Proposition 4.1 induces an
equivalence

$\textup{Mod}(KU_{p}[G])^{ft}\simeq\textup{Fun}(BG,\textup{Mod}(KU_{p})^{\textup{perf}})$

where $\textup{Mod}(KU_{p})^{\textup{perf}}$ is the full, stable subcategory
of $\textup{Mod}(KU_{p})$ consisting of the perfect modules.

  2. 2.

For finite abelian groups $G_{1}$ and $G_{2}$, and a $KU_{p}[G_{1}\times
G_{2}]$-bimodule $M$ of finite type, the composite functor

${{\textup{Mod}(KU_{p}[G_{1}])}}$${{\textup{Mod}(KU_{p}[G_{2}])}}$${{\textup{Mod}(KU_{p}[G_{2}])}}$$\scriptstyle{-\otimes_{KU_{p}[G_{1}]}M}$$\scriptstyle{L_{p}}$

sends modules of finite type to modules of finite type, where $L_{p}$ is the
$p$-completion functor.

###### Proof.

  1. 1.

We first remark that the equivalence of Proposition 4.1 preserves the
underlying $R$-module. The finite-type property also only depends on the
underlying $R$-module - that is,
$\textup{Mod}(R[G])^{ft}\simeq\text{Fun}(BG,\textup{Mod}(R)^{ft})$. Thus, it
suffices to prove that over $KU_{p}$, the finite type modules are precisely
the perfect modules.

It is trivially seen that $\text{Mod}(KU_{p})^{ft}$ is closed under retracts.
If

${{M^{\prime}}}$${M}$${0}$${{M^{\prime\prime}}}$

is a pullback square in $\text{Mod}(KU_{p})$ with two of
$M^{\prime},M,M^{\prime\prime}$ in $\text{Mod}(KU_{p})^{ft}$, the long exact
sequence

${{\pi_{*+1}(M^{\prime\prime})}}$${{\pi_{*}(M^{\prime})}}$${{\pi_{*}(M)}}$${{\pi_{*}(M^{\prime\prime})}}$${{\pi_{*-1}(M^{\prime})}}$

yields that the third also belongs to $\text{Mod}(KU_{p})^{ft}$. Thus we
conclude that $\text{Mod}(KU_{p})^{ft}$ is a thick subcategory of
$\text{Mod}(KU_{p})$. In particular, it is stable.

We now show that
$\text{Mod}(KU_{p})^{ft}\subset\textup{Mod}(KU_{p})^{\textup{perf}}$. To this
end, first suppose $M$ is a $KU_{p}$-module with finitely generated and free
homotopy groups over $\mathbb{Z}_{p}$, so that $\pi_{0}(M)=\mathbb{Z}_{p}^{n}$
and $\pi_{1}(M)=\mathbb{Z}_{p}^{m}$. We choose a $\pi_{*}(KU_{p})$-basis of
$\pi_{*}M$. By the universal coefficient theorem and the $2$-periodicity of
homotopy groups, this determines an equivalence $KU_{p}^{n}\oplus\Sigma
KU_{p}^{m}\to M$, so $M$ is perfect.

Now, let $M$ be a $KU_{p}$-module of finite type. Choosing finite sets of
generators $\\{x_{1},...,x_{n}\\}$ and $\\{y_{1},...,y_{m}\\}$ for
$\pi_{0}(M)$ and $\pi_{1}(M)$, respectively, we obtain a map
$\alpha:KU_{p}^{n}\oplus\Sigma KU_{p}^{m}\to M$ that induces a surjection on
homotopy groups. In greater detail, we can write $M=M_{0}\oplus\Sigma M_{1}$
([Tre15, Prop. 2.1]), where $M_{0}$ and $M_{1}$ are even modules (that is,
they have non-trivial homotopy groups only in even degrees), and this gives us
maps (by the universal coefficient theorem) $\alpha_{0}:KU_{p}^{n}\to M_{0}$
and $\alpha_{1}:KU_{p}^{m}\to M_{1}$ that induce surjections on homotopy
groups. Let $P_{0}$ and $P_{1}$ be the fibres of $\alpha_{0}$ and
$\alpha_{1}$, respectively. They are both even modules, and since
$\mathbb{Z}_{p}$ is a noetherian PID, their homotopy groups are also free and
finitely generated. The square

${{P_{0}\oplus P_{1}}}$${{KU_{q}^{i}\oplus
KU_{q}^{j}}}$${0}$${M}$$\scriptstyle{\alpha=(\alpha_{0},\alpha_{1})}$

is cartesian, and by stability, co-cartesian as well. Thus $M$ is a cofibre of
a map of free modules and is hence perfect.

  2. 2.

We refer the reader to [Tre15, Sec. 3.5] for the proof. We only sketch the
argument provided there.

It is first argued that one can replace $L_{p}$ by $L_{K(1)}$ in the statement
of the proposition, as $KU_{p}$ is a $K(1)$-local spectrum.

Let $N\in\text{Mod}(KU_{p})[G_{1}]^{ft}$. The Künneth theorem yields that
$N\otimes_{KU_{p}}M\in\text{Mod}(KU_{p})[G_{2}]^{ft}$. This fact, in
conjunction with [Tre15, Lem. 3.2], implies that the homotopy $G_{1}$-fixed
points $(N\otimes_{KU_{p}}M)^{hG_{1}}\in\text{Mod}(KU_{p})[G_{2}]^{ft}$. We
can identify the tensor product over $KU_{p}[G_{1}]$ as the homotopy
$G_{1}$-orbit over the tensor product over $KU_{p}$; that is

$\text{colim}_{BG_{1}}(N\otimes_{KU_{p}}M)=:(N\otimes_{KU_{p}}M)_{hG_{1}}\simeq(N\otimes_{KU_{p}[G_{1}]}M)$

The proposition now follows from [HL13, Thm. 0.0.1], which implies that there
is an isomorphism

$L_{K(1)}(N\otimes_{KU_{p}}M)_{hG_{1}}\simeq
L_{K(1)}(N\otimes_{KU_{p}}M)^{hG_{1}}$

∎

We now briefly recall the setup for topological $K$-theory of
$C^{*}$-categories, as described in [BE21].

We begin with a recollection of some basic definitions as presented in Section
2 of loc.cit.. A possibly non-unital $\mathbb{C}$-linear $*$-category is a
possibly non-unital category enriched over the category of $\mathbb{C}$-vector
spaces together with an involution $*$ that fixes the objects, reverses the
direction of the morphisms, and acts complex anti-linearly on the morphism
spaces. We have a category ${}^{*}\text{Cat}^{nu}_{\mathbb{C}}$ with objects
the possibly non-unital $\mathbb{C}$-linear $*$-categories, and morphisms the
$\mathbb{C}$-linear functors of the underlying non-unital categories that
additionally preserve the involution $*$. For a Hilbert space $H$, the
$C^{*}$-algebra $B(H)$ can be regarded as a $\mathbb{C}$-linear $*$-category
with one object. We now define the maximal norm $||-||_{\text{max}}$ on the
morphism spaces of a possibly non-unital $\mathbb{C}$-linear $*$-category
$\mathcal{C}$. For a morphism $f$ of $\mathcal{C}$, we set

$||f||_{\text{max}}:=\text{sup}_{\rho:\mathcal{C}\to B(H)}||\rho(f)||$

where the supremum is taken over all morphisms $\rho$ in
${}^{*}\text{Cat}^{nu}_{\mathbb{C}}$, and $B(H)$ is given the operator norm.
An object of ${}^{*}\text{Cat}^{nu}_{\mathbb{C}}$ is called a $C^{*}$-category
if all its morphism spaces are complete with respect to $||-||_{\text{max}}$
131313This includes the assumption that $||-||_{\text{max}}$ is indeed a norm
on the morphism spaces.. We denote by $C^{*}\text{Cat}^{nu}$ the full
subcategory of ${}^{*}\text{Cat}^{nu}_{\mathbb{C}}$ spanned by the
$C^{*}$-categories 141414Henceforth, whenever we speak of functors between
$C^{*}$-categories, we always refer to morphisms in $C^{*}\text{Cat}^{nu}$.. A
$C^{*}$-algebra can be regarded as a $C^{*}$-category with a single object.
This provides an inclusion functor $i:C^{*}\text{Alg}^{nu}\hookrightarrow
C^{*}\text{Cat}^{nu}$, which admits a left adjoint
$A^{f}:C^{*}\text{Cat}^{nu}\to C^{*}\text{Alg}^{nu}$. We now define the
$K$-theory functor $K^{\textbf{C}}$ as the composite

${{K^{\textbf{C}}:C^{*}\text{Cat}^{nu}}}$${{C^{*}\text{Alg}^{nu}}}$${{\text{Mod}(KU)}}$$\scriptstyle{A^{f}}$$\scriptstyle{K}$

The composite $A^{f}\circ i$ is isomorphic to the identity functor, so for a
$C^{*}$-algebra $A$ one has $K^{\textbf{C}}(i(A))\simeq K(A)$.

Let $C^{*}\text{Cat}$ be the subcategory of $C^{*}\text{Cat}^{nu}$ comprising
of the unital $C^{*}$-categories and unital $C^{*}$-functors. We abuse
notation and write $K^{\textbf{C}}$ also for the composite

${{C^{*}\text{Cat}}}$${{C^{*}\text{Cat}^{nu}}}$${{\text{Mod}(KU)}}$$\scriptstyle{K^{\textbf{C}}}$

We next describe classes of morphisms in $C^{*}\text{Cat}$ that are inverted
by $K^{\textbf{C}}$. This helps felicitate computations as the functor $A^{f}$
has a complicated description.

We first describe the notion of a unitary equivalence between unital
$C^{*}$-categories ([BE20, Sec. 8.5.4]). Let C and D be unital
$C^{*}$-categories, and $f,g:\textbf{C}\to\textbf{D}$ be unital functors. We
say that $f$ and $g$ are unitarily isomorphic if there exists a natural
transformation $u:f\to g$ such that $u(c):f(c)\to g(c)$ is unitary for each
object $c$ of C. A unital functor $f:\textbf{C}\to\textbf{D}$ between unital
$C^{*}$-categories is a unitary equivalence if there exists a functor
$g:\textbf{D}\to\textbf{C}$ such that $f\circ g$ and $g\circ f$ are isomorphic
to the respective identity functors on D and C. If $f$ is a unitary
equivalence, then $K^{\textbf{C}}(f)$ is an equivalence ([BE20, Cor. 8.63]).

Thus, if $W$ denotes the class of unitary equivalences in $C^{*}\text{Cat}$,
one has the following factorization:

${{C^{*}\text{Cat}}}$${{\text{Mod}(KU)}}$${{C^{*}\text{Cat}[W^{-1}]}}$$\scriptstyle{K^{\textbf{C}}}$$\scriptstyle{L_{W}}$$\scriptstyle{K^{\mathcal{C}}}$

where $L_{W}$ is the localization functor.

In [Del12], the construction of a maximal tensor product $-\otimes-$, of two
unital $C^{*}$-categories is described, and it is shown that this endows
$C^{*}\text{Cat}$ with a symmetric monoidal structure such that the inclusion
$C^{*}\text{Alg}\hookrightarrow C^{*}\text{Cat}$ is symmetric monoidal. In
[BEL23], this was upgraded - a maximal tensor product $-\otimes-$ is
constructed on $C^{*}\text{Cat}^{nu}$ which gives it a symmetric monoidal
structure such that the inclusions $C^{*}\text{Cat}\hookrightarrow
C^{*}\text{Cat}^{nu}$ and $i:C^{*}\text{Alg}^{nu}\hookrightarrow
C^{*}\text{Cat}^{nu}$ are both symmetric monoidal.

######  Proposition 4.3.

  1. 1.

The localization functor $L_{W}$ admits a symmetric monoidal refinement.

  2. 2.

The functor $K^{\mathcal{C}}$ admits a lax symmetric monoidal refinement for
this symmetric monoidal structure on $C^{*}\textup{Cat}[W^{-1}]$.

###### Proof.

  1. 1.

This follows from the fact that for any unitary equivalence
$f:\textbf{C}\to\textbf{C}^{\prime}$ and any object D of $C^{*}\text{Cat}$,
$f\otimes\text{id}_{\textbf{D}}$ and $\text{id}_{\textbf{D}}\otimes f$ are
unitary equivalences.

  2. 2.

Write $kk^{\textbf{C}}$ for the composition

${{C^{*}\text{Cat}^{nu}}}$${{C^{*}\text{Alg}^{nu}}}$${KK}$$\scriptstyle{A^{f}}$$\scriptstyle{kk}$

[BEL23, Prop 7.9] states that $kk^{\textbf{C}}$ is symmetric monoidal.

Further, [BEL23, Cor. 6.14] states that $kk^{\textbf{C}}$ inverts unitary
equivalences between unital $C^{*}$-categories, so item 1 gives us a symmetric
monoidal functor $kk^{\mathcal{C}}:C^{*}\textup{Cat}[W^{-1}]\to KK$. We have
the following commutative diagram in the $\infty$-category of
$\infty$-categories:

${{C^{*}\text{Cat}}}$${{C^{*}\text{Cat}^{nu}}}$${KK}$${{\text{Mod}(KU)}}$${{C^{*}\text{Cat}[W^{-1}]}}$$\scriptstyle{kk^{\textbf{C}}}$$\scriptstyle{KK(\mathbb{C},-)}$$\scriptstyle{L_{W}}$$\scriptstyle{kk^{\mathcal{C}}}$$\scriptstyle{K^{\mathcal{C}}}$$\scriptstyle{K^{\textbf{C}}}$

As $K^{\mathcal{C}}$ is equivalent to the composite of the symmetric monoidal
functor $kk^{\mathcal{C}}$ followed by the lax symmetric monoidal functor
$KK(\mathbb{C},-)$, the statement follows.

∎

We now turn to Morita equivalences. We refer the reader to [BE21, Def. 5.5]
and [BE21, Def. 2.19] for the definitions of additive and idempotent complete
unital $C^{*}$-categories, respectively. We say that a morphism
$f:\mathcal{C}\to\mathcal{D}$ in $C^{*}\text{Cat}$ presents $\mathcal{D}$ as
the additive completion of $\mathcal{C}$ if $\mathcal{D}$ is additive, $f$ is
fully faithful, and every object of $\mathcal{D}$ is unitarily isomorphic to a
finite orthogonal sum of objects in the image of $f$ (Definition 16.1 of
loc.cit.). Similarly, we say that a morphism $f:\mathcal{C}\to\mathcal{D}$ in
$C^{*}\text{Cat}$ presents $\mathcal{D}$ as the idempotent completion of
$\mathcal{C}$ if $\mathcal{D}$ is idempotent complete, $f$ is fully faithful,
and for each object $d$ of $\mathcal{D}$, there is an isometry $d\to
d^{\prime}$, where $d^{\prime}$ is an object in the image of $f$ (Definition
16.4 of loc.cit.).

Let $C^{*}\text{Cat}^{\text{Idem}}_{\oplus}$ denote the full subcategory of
$C^{*}\text{Cat}$ spanned by the additive and idempotent complete
$C^{*}$-categories. There is a functor (see 16.1 of loc.cit)
$(-)^{\\#}:C^{*}\text{Cat}\to C^{*}\text{Cat}^{\text{Idem}}_{\oplus}$ together
with a natural transformation $u:\text{id}\to(-)^{\\#}$ such that for each
object $\mathcal{C}$ of $C^{*}\text{Cat}$, the map
$\mathcal{C}\xrightarrow{u_{\mathcal{C}}}\mathcal{C}^{\\#}$ presents
$\mathcal{C}^{\\#}$ as the additive and idempotent completion of
$\mathcal{C}$. We define a Morita equivalence to be a morphism in
$C^{*}\text{Cat}$ that is sent to a unitary equivalence by $(-)^{\\#}$
(Definition 16.7 of loc.cit.). Morita equivalences are inverted by
$K^{\textbf{C}}$ (Theorem 16.18 of loc.cit.).

Given a unital $C^{*}$-algebra $A$, one can consider the category of finitely
generated projective Hilbert $A$-modules, which we denote by
$\text{Hilb}(A)^{\text{fg,proj}}$. This acquires the structure of a unital
$C^{*}$-category in the obvious way. As argued in Example 16.9 of loc.cit.,
the inclusion $i(A)\to\text{Hilb}(A)^{\text{fg,proj}}$ is a Morita
equivalence, so it induces an equivalence
$K^{\textbf{C}}(\text{Hilb}(A)^{\text{fg,proj}})\simeq K(A)$.

######  Remark 4.4.

Let $A:=\textup{Mod}(\mathbb{C})$. For all $B\in C^{*}\textup{Cat}[W^{-1}]$,
the lax monoidality constraint

$K^{\mathcal{C}}(L_{W}(A))\otimes_{KU}K^{\mathcal{C}}(B)\to
K^{\mathcal{C}}(L_{W}(A)\otimes B)$

is an equivalence. To see why, note that if $A$ was $i(\mathbb{C})$, the
statement clearly holds; now we conclude using the Morita equivalence between
$i(\mathbb{C})$ and $\textup{Mod}(\mathbb{C})$.

In [Bun19], the existence of a simplicial model structure on $C^{*}\text{Cat}$
is shown, in which the weak equivalences are precisely the unitary
equivalences and every object is both cofibrant and fibrant. Therefore, the
localization $C^{*}\text{Cat}[W^{-1}]$ can be identified with the homotopy
coherent nerve $N_{h}(C^{*}\text{Cat})$. From the explicit description of the
simplicial structure in Definition 6.19 of loc.cit., one sees that
$N_{h}(C^{*}\text{Cat})$ is the nerve of a $2$-category.

We now give an explicit description of $N_{h}(C^{*}\text{Cat})$. Let $A\in
C^{*}\text{Cat}$, and let $\Pi(\Delta^{1})$ denote the fundamental groupoid of
the simplicial set $\Delta^{1}$. We describe the unital $C^{*}$-category
$A\otimes\Pi(\Delta^{1})$ that serves as the cylinder object on $A$. Its
objects are $\text{Ob}(A)\times\\{0,1\\}$. Its morphisms are given by, for
$j,j^{\prime}\in\\{0,1\\}$ and $a,a^{\prime}\in\text{Ob}(A)$,

$\text{Hom}_{A\otimes\Pi(\Delta^{1})}((a,j),(a^{\prime},j^{\prime}))=\text{Hom}_{A}(a,a^{\prime})$

with the obvious definitions of composition and $*$. We have unital functors
$i_{0},i_{1}:A\to A\otimes\Pi(\Delta_{1})$ given by $a\mapsto(a,0)$ and
$a\mapsto(a,1)$, respectively.

The data of a $2$-simplex in $N_{h}(C^{*}\text{Cat})$ is that of unital
$C^{*}$-categories $A,B,C$, unital functors $f_{0}:A\to B$, $f_{1}:B\to C$,
$f_{2}:A\to C$, and a unital functor $h:A\otimes\Pi(\Delta^{1})\to C$ such
that $h\circ i_{0}=f_{1}\circ f_{0}$ and $h\circ i_{1}=f_{2}$ (see [Lur24, Tag
00M5]).

In what follows, we denote by $U(1)^{\delta}$ the unitary group of
$\mathbb{C}$ endowed with the discrete topology. We define a functor
$\Phi:BBU(1)^{\delta}\to N_{h}(C^{*}\text{Cat})$ by specifying the data of the
object $\text{Mod}(\mathbb{C})\in C^{*}\text{Cat}$, the identity functor
$\text{Mod}(\mathbb{C})\to\text{Mod}(\mathbb{C})$, and the action of elements
of $U(1)$ as automorphisms of this identity functor by scalar multiplication.

Let $G$ be a finite abelian $p$-group. Suppose one has a map
$B(G\times\hat{G})\to BBU(1)^{\delta}$; on post-composing with $\Phi$, we get
a functor $B(G\times\hat{G})\to N_{h}(C^{*}\text{Cat})$. This gives an action
of $G\times\hat{G}$ on the category $\text{Mod}(\mathbb{C})$ regarded as an
object of $N_{h}(C^{*}\text{Cat})$. Further post-composing by
$K^{\mathcal{C}}$ gives a $G\times\hat{G}$-action on $KU$ regarded as an
object of $\text{Mod}(KU)$, and therefore determines a
$KU[G\times\hat{G}]$-module.

The equivalence classes of maps of spaces from $B(G\times\hat{G})\to
BBU(1)^{\delta}$ can be identified with the cohomology group
$H^{2}(B(G\times\hat{G}),U(1)^{\delta})$.

Consider the $U(1)$-valued $2$-cocycle
$E:=((g,\chi),(g^{\prime},\chi^{\prime}))\mapsto\chi^{\prime}(g)$. In the
manner explained above, this determines a $KU[G\times\hat{G}]$-module
$\hypertarget{cat}{M^{\prime}_{E}}$. On $p$-completing $M^{\prime}_{E}$, we
get a $KU_{p}[G\times\hat{G}]$-module which we shall denote by
$\hypertarget{M_E}{M_{E}}$.

We can now formulate Theorem 1.2 more precisely.

######  Theorem 4.5 (Treumann Duality).

The functor
$\textup{Mod}(KU_{p}[G])^{ft}\to\textup{Mod}(KU_{p}[\hat{G}])^{ft}$ given by
$N\mapsto L_{p}(N\otimes_{KU_{p}[G]}M_{E})$ is an equivalence.

###### Proof.

The proof in [Tre15, Sec. 3.6] works mutatis mutandis; the only difference
between our statement and the one there is the description of $M_{E}$. The
argument is along the following lines.

Regarding $M_{E}$ as a $KU_{p}[\hat{G}\times G]$-module yields a functor
$\textup{Mod}(KU_{p}[\hat{G}])^{ft}\to\textup{Mod}(KU_{p}[G])^{ft}$. The cited
argument shows that this functor is the inverse of the functor of the theorem,
by showing that $L_{p}(M_{E\hat{G}}\otimes_{KU_{p}[\hat{G}]}M_{EG})$ and
$L_{p}(M_{EG}\otimes_{KU_{p}[G]}M_{E\hat{G}})$ are isomorphic to the diagonal
$(G,G)$ and $(\hat{G},\hat{G})$-bimodules over $KU_{p}$, respectively (here,
we write $M_{EG}$ for $M_{E}$ regarded as a $KU_{p}[\hat{G}\times G]$-module
and $M_{E\hat{G}}$ for $M_{E}$ regarded as a $KU_{p}[G\times\hat{G}]$-module).
This involves computations of group actions that are performed on the level of
the module category. These go through verbatim in our situation by Remark 4.4
and the fact that $\text{Mod}(\mathbb{C})$ is idempotent with respect to the
symmetric monoidal structure of the maximal tensor product. ∎

##  5 A Comparison of Takai Duality with Treumann Duality

In this section, we compare Takai duality with Treumann duality.

Fix a prime number $p$ and a finite abelian $p$-group $G$ for the rest of this
section.

We first define a functor
$r_{G}:GC^{*}\text{Alg}^{nu}\to\text{Mod}(KU_{p}[G])$ as the composite

${{GC^{*}\text{Alg}^{nu}}}$${{\text{Fun}(BG,C^{*}\text{Alg}^{nu})}}$${{\text{Fun}(BG,KK)}}$${{\text{Mod}(KU_{p}[G])}}$${{\text{Mod}(KU[G])}}$${{\text{Fun}(BG,\text{Mod}(KU))}}$$\scriptstyle{\sim}$$\scriptstyle{-\circ
kk}$$\scriptstyle{-\circ K(-)}$$\scriptstyle{\sim}$$\scriptstyle{L_{p}}$

One can check that the upper horizontal composite is homotopy invariant,
$G$-stable, semi-exact, and $s$-finitary, essentially because $kk$ has those
properties for $G=\\{e\\}$. Thus, one has an essentially unique descended
functor $\widehat{\text{Res}}_{G}:KK^{G}\to\text{Fun}(BG,KK)$ such that the
following diagram commutes:

${{GC^{*}\text{Alg}^{nu}}}$${{\text{Fun}(BG,C^{*}\text{Alg}^{nu})}}$${{KK^{G}}}$${{\text{Fun}(BG,KK)}}$$\scriptstyle{\sim}$$\scriptstyle{-\circ
kk}$$\scriptstyle{kk^{G}}$$\scriptstyle{r^{\prime}_{G}}$

It follows that $r_{G}$ also descends essentially uniquely to a functor
$KK^{G}\to\text{LMod}(KU_{p}[G])$, which we shall also denote by $r_{G}$.

We now restrict our attention to those objects of $KK^{G}$ that map to finite-
type modules. Let $KK^{G,ft}$ denote the full subcategory of $KK^{G}$ spanned
by objects $A$ such that $r_{G}(A)\in\text{Mod}(KU_{p}[G])^{ft}$.

######  Proposition 5.1.

$KK^{G,ft}$ is a thick subcategory of $KK^{G}$.

###### Proof.

Clearly, $0\in KK^{G,ft}$. It is also obvious that $KK^{G,ft}$ is closed under
retracts.

We observe that $r_{G}$ is exact. As proved in item 1 of Proposition 4.2,
$\text{LMod}(KU_{p}[G])^{ft}$ is stable. Therefore, $KK^{G,ft}$ is stable and
closed under fibers and cofibers. ∎

We now record the following crucial result, of which the case of cyclic groups
is shown in [Bun23]:

######  Theorem 5.2.

Let $G$ be a finite abelian $p$-group. For this theorem alone, regard
$-\rtimes G$ as a functor $KK^{G}\to KK$. There is an assembly map

$\textup{colim}_{BG}K(\widehat{\textup{Res}}_{G}(-))\to K(-\rtimes G)$

of functors $KK^{G}\to\textup{Mod}(KU)$ that becomes an equivalence after
$p$-completion.

###### Proof.

We begin with a lemma.

######  Lemma 5.3.

Let $H_{1}$ and $H_{2}$ be finite abelian groups, and set $G:=H_{1}\times
H_{2}$. There exists an essentially unique functor $F:KK^{G}\to KK^{H_{1}}$
such that the following diagram is commutative:

${{KK^{G}}}$${{\textup{Fun}(BH_{1},KK^{H_{2}})}}$${{\textup{Fun}(BH_{1},KK)}}$${KK}$${{KK^{H_{1}}}}$$\scriptstyle{\widehat{\textup{Res}}_{G,H_{2}}}$$\scriptstyle{-\circ(\rtimes
H_{2})}$$\scriptstyle{\widehat{\textup{Res}}_{H_{1}}}$$\scriptstyle{\rtimes
H_{1}}$$\scriptstyle{\rtimes G}$$\scriptstyle{F}$

###### Proof.

First, one checks that the following diagram is commutative:

${{GC^{*}\text{Alg}^{nu}}}$${{\text{Fun}(BH_{1},H_{2}C^{*}\text{Alg}^{nu})}}$${{\text{Fun}(BH_{1},C^{*}\text{Alg}^{nu})}}$${{C^{*}\text{Alg}^{nu}}}$${{H_{1}C^{*}\text{Alg}^{nu}}}$$\scriptstyle{\sim}$$\scriptstyle{-\circ(\rtimes
H_{2})}$$\scriptstyle{\sim}$$\scriptstyle{\rtimes H_{1}}$$\scriptstyle{\rtimes
G}$

It is easy to verify that each functor in this diagram can be descended to a
functor between the appropriate equivariant $KK$ categories. Using the
uniqueness properties of the descended functors several times, the lemma
follows. ∎

The assembly map alluded to in the theorem statement has been constructed in
the proof of [Bun23, Prop. 2.7]. We briefly recall this below.

First, a natural transformation of functors $KK^{G}\to\text{Fun}(BG,KK)$

$c^{\prime}_{G}:\widehat{\text{Res}}_{G}(-)\to\text{triv}^{G}(-\rtimes G)$

is constructed, where $\text{triv}^{G}:KK\to\text{Fun}(BG,KK)$ equips151515We
also use the notation $\text{triv}^{G}$ to indicate the functor
$KK^{G}\to\text{Fun}(BG,KK^{G})$ that equips an object with the trivial
$G$-action. It will be clear from the context as to which functor we have in
mind. an object with the trivial $G$-action. The map $c^{\prime}_{G}$ is
obtained by the following identifications (see loc.cit. for the details):

$\displaystyle\widehat{\text{Res}}_{G}(-)$
$\displaystyle\simeq(\text{triv}^{G}(-)\otimes C_{0}(G))\rtimes G$
$\displaystyle\simeq(\text{triv}^{G}(-)\otimes*_{G}\mathbb{C})\rtimes G$
$\displaystyle\xrightarrow{G\to*}\text{triv}^{G}(-\rtimes G)$

Its adjoint is a natural transformation

$c_{G}:\text{colim}_{BG}\widehat{\text{Res}}_{G}(-)\to-\rtimes G$

The desired assembly map is obtained by applying the colimit preserving
functor $K(-)$ to $c_{G}$.

Now, we write $G$ as a direct product of $n$ cyclic groups, each of prime
power order. We will proceed by induction on $n$.

If $n=1$, the result follows from [Bun23, Thm. 5.7].

So assume $G=H_{1}\times H_{2}$ and that the theorem holds for $H_{1}$ and
$H_{2}$. Let $F$ be the functor of Lemma 5.3.

To simplify the proof, we introduce the following notation and record some
obvious relations:

  * •

(Colimits) $L_{0}:=\text{colim}_{BG}:\text{Fun}(BG,KK)\to KK$,
$L_{1}:=\text{colim}_{BH_{1}}:\text{Fun}(BH_{1},KK)\to KK$,
$L_{2}:=\text{colim}_{BH_{2}}:\text{Fun}(BG,KK)\to\text{Fun}(BH_{1},KK)$;
$L_{0}\simeq L_{1}\circ L_{2}$.

  * •

(Restrictions) $M_{0}:=\widehat{\text{Res}}_{G}:KK^{G}\to\text{Fun}(BG,KK)$,
$M_{1}:=\widehat{\text{Res}}_{G,H_{2}}:KK^{G}\to\text{Fun}(BH_{1},KK^{H_{2}})$,
$M_{2}:=\widehat{\text{Res}}_{H_{2}}:\text{Fun}(BH_{1},KK^{H_{2}})\to\text{Fun}(BG,KK)$;
$M_{0}\simeq M_{2}\circ M_{1}$.

  * •

(Crossed Products) $N_{0}:=-\rtimes G:KK^{G}\to KK$, $N_{1}:=-\rtimes
H_{1}:KK^{H_{1}}\to KK$, $N_{2}:=-\rtimes H_{2}:KK^{G}\to KK^{H_{1}}$,
$N^{\prime}_{2}:=-\rtimes
H_{2}:\text{Fun}(BH_{1},KK^{H_{2}})\to\text{Fun}(BH_{1},KK)$; $N_{0}\simeq
N_{1}\circ N_{2}$.

Note that $c_{H_{2}}$ induces a map $L_{2}M_{2}(-)\to N^{\prime}_{2}(-)$. We
will abuse notation and also refer to this as $c_{H_{2}}$.

Abusing notation slightly, we observe that $c^{\prime}_{G}$ is equivalent to
the composite

$\displaystyle\widehat{\text{Res}}_{G}(-)$
$\displaystyle\simeq(\text{triv}^{G}(-)\otimes C_{0}(G))\rtimes G$
$\displaystyle\simeq(\text{triv}^{G}(-)\otimes C_{0}(H_{1})\otimes
C_{0}(H_{2}))\rtimes G$
$\displaystyle\simeq(\text{triv}^{H_{1}}((\text{triv}^{H_{2}}(-)\otimes
C_{0}(H_{2}))\rtimes H_{2})\otimes C_{0}(H_{1}))\rtimes H_{1}$
$\displaystyle\simeq(\text{triv}^{H_{1}}((\text{triv}^{H_{2}}(-)\otimes*_{H_{2}}\mathbb{C})\rtimes
H_{2})\otimes*_{H_{1}}\mathbb{C})\rtimes H_{1}$
$\displaystyle\xrightarrow{H_{2}\to*}\text{triv}^{H_{1}}((\text{triv}^{H_{2}}(-)\rtimes
H_{2})\otimes*_{H_{1}}\mathbb{C})\rtimes H_{1}$
$\displaystyle\xrightarrow{H_{1}\to*}\text{triv}^{G}(-\rtimes G)$

Therefore, the adjoint morphism $c_{G}:L_{0}M_{0}(-)\to N_{0}(-)$ is
equivalent to the composite

${{L_{1}L_{2}M_{2}M_{1}(-)}}$${{L_{1}N^{\prime}_{2}M_{1}(-)}}$${{L_{1}\widehat{\text{Res}}_{H_{1}}F(-)}}$${{N_{1}F(-)}}$${{N_{0}(-)}}$$\scriptstyle{L_{1}c_{H_{2}}M_{1}}$$\scriptstyle{\sim}$$\scriptstyle{c_{H_{1}}}$$\scriptstyle{\sim}$

where the identifications above are by Lemma 5.3.

The induction hypothesis implies that each individual map above induces an
equivalence on $p$-completed $K$-theory (we also use that $L_{p}$ commutes
with colimits, being a left adjoint functor), and therefore, so does $c_{G}$.
This completes the proof. ∎

######  Proposition 5.4.

$-\rtimes G:KK^{G}\to KK^{\hat{G}}$ restricts to a functor $-\rtimes
G:KK^{G,ft}\to KK^{\hat{G},ft}$.

###### Proof.

In the course of this proof, we will abuse notation by omitting restriction
functors. We recall an earlier remark that a module over $KU_{p}[G]$ or
$KU_{p}[\hat{G}]$ being of finite type depends only on the underlying
$KU_{p}$-module and not on the $G$ or $\hat{G}$-action on it.

Let $A$ be an object of $KK^{G}$, and suppose that $L_{p}(K(A))$ is of finite
type. We have to show that $L_{p}(K(A\rtimes G))$ is also of finite type.

Consider $KU_{p}$ with the trivial $(G\times\hat{G})$-action. Then, by item 2
of Proposition 4.2,

$L_{p}(\text{colim}_{BG}(L_{p}(K(A))))\simeq
L_{p}(L_{p}(K(A))\otimes_{KU_{p}[G]}KU_{p})$

is of finite type.

Using that $L_{p}$ is a left adjoint and idempotent functor, it follows that
$L_{p}(\text{colim}_{BG}(K(A)))$ is of finite type. We now conclude using
Theorem 5.2. ∎

######  Corollary 5.5.

$-\rtimes G:KK^{G,ft}\to KK^{\hat{G},ft}$ is an equivalence.

###### Proof.

This is immediate from the statement of Proposition 5.4 applied to both $G$
and $\hat{G}$, and Corollary 3.2. ∎

######  Lemma 5.6.

Let B be the $(G\times\hat{G})$-$C^{*}$-algebra defined just before
Proposition 3.5. We have an isomorphism
$L_{p}(K(\mathbf{B}))\simeq\hyperlink{M_E}{M_{E}}$ in
$\textup{Mod}(KU_{p}[G\times\hat{G}])$.

###### Proof.

We will show that $K(\mathbf{B})\simeq\hyperlink{cat}{M^{\prime}_{E}}$, so
that they are equivalent after $p$-completion.

Our first step is to establish a unitary equivalence of $C^{*}$-categories
$\text{Hilb}{(\mathbb{C})}^{\text{fg,proj}}\simeq\text{Hilb}{(K(L^{2}(G)))}^{\text{fg,proj}}$.

Every linear operator on $L^{2}(G)$ is compact because $L^{2}(G)$ is finite-
dimensional. Let $S$ be the $K(L^{2}(G))$-module $L^{2}(G)$. It is a standard
fact that every finitely generated module over $K(L^{2}(G))$ is of the form
$\oplus_{i=1}^{n}S$, for some $n\geq 0$. Thus the underlying category of
$\text{Hilb}{(K(L^{2}(G)))}^{\text{fg,proj}}$ coincides with that of the
finitely generated modules over $K(L^{2}(G))$. One also sees that the
underlying category of $\text{Hilb}{(\mathbb{C})}^{\text{fg,proj}}$ is that of
the finite-dimensional $\mathbb{C}$-vector spaces. In what follows, we will
simply write $\text{Mod}(\mathbb{C})$ and $\text{Mod}(K(L^{2}(G)))$ for
$\text{Hilb}{(\mathbb{C})}^{\text{fg,proj}}$ and
$\text{Hilb}{(K(L^{2}(G)))}^{\text{fg,proj}}$, respectively.

Let $u:\mathbb{C}\hookrightarrow K(L^{2}(G))$ be the left upper corner
inclusion. This determines a (unital) functor
$u_{*}:\text{Mod}{(\mathbb{C})}\to\text{Mod}{(K(L^{2}(G)))}$, given by
$M\mapsto M\otimes_{\mathbb{C}}K(L^{2}(G))$. We claim that $u_{*}$ is a
unitary equivalence - let
$u^{*}:\text{Mod}(K(L^{2}(G)))\to\text{Mod}(\mathbb{C})$ be the functor given
by $M\mapsto u(\mathbb{C})M$; then one checks that $u_{*}\circ u^{*}$ and
$u^{*}\circ u_{*}$ are unitarily isomorphic to the respective identity
functors.

Using the functor $L_{W}(u)$, we can transport the $(G\times\hat{G})$-action
on $\text{Mod}(\mathbb{C})$ given by the cocycle $E$ to one on
$\text{Mod}(K(L^{2}(G)))$. We represent this action on
$\text{Mod}(K(L^{2}(G)))$ as a functor $\Psi^{\prime}:B(G\times\hat{G})\to
N_{h}(C^{*}\text{Cat})$. By construction,
$K^{\mathcal{C}}(\text{Mod}(K(L^{2}(G)))$ with this action is equivalent to
$M_{E}$ as a $KU[G\times\hat{G}]$-module.

Let $P$ be the $(G\times\hat{G})$-action on $K(L^{2}(G))$ that gives it the
structure of the $(G\times\hat{G})$-$C^{*}$-algebra B. With notation as in the
proof of Lemma 3.4, for $(g,\chi)\in G\times\hat{G}$, the $C^{*}$-algebra
homomorphism $P_{(g,\chi)}:K(L^{2}(G))\to K(L^{2}(G))$ is given by $A\mapsto
S_{\chi}^{-1}T_{g}AT_{g}^{-1}S_{\chi}$.

Consider the functor $j:C^{*}\text{Alg}\to C^{*}\text{Cat}$ given by
$A\mapsto\text{Hilb}(A)^{\text{fg,proj}}$. We obtain an action $j(P)$ on
$\text{Mod}(K(L^{2}(G))$. On further composing with $L_{W}$, we obtain a
functor $\Psi:B(G\times\hat{G})\to N_{h}(C^{*}\text{Cat})$.

The functor $\Psi$ is determined by the data of the object
$\text{Mod}(K(L^{2}(G)))$, and for each $(g,\chi)$, the endofunctor
$\Psi_{(g,\chi)}:\text{Mod}(K(L^{2}(G)))\to\text{Mod}(K(L^{2}(G)))$ given by
$M\mapsto M_{P_{(g,\chi)}}$, where $M_{P_{(g,\chi)}}$ is the module with the
same underlying abelian group as $M$ and $K(L^{2}(G))$-action twisted by the
isomorphism $P_{(g,\chi)}$. Because $K$-theory inverts Morita equivalences,
$K^{\mathcal{C}}(\text{Mod}(K(L^{2}(G)))$ with the resulting
$(G\times\hat{G})$-action is isomorphic to $K(\textbf{B})$ as a
$KU[G\times\hat{G}]$-module.

We will now construct an equivalence between the functors $\Psi^{\prime}$ and
$\Psi$. By the observations above, this would complete the proof.

We start by defining invertible natural transformations $\alpha_{(g,\chi)}$
between the identity functor and $\Psi_{(g,\chi)}$. For
$M\in\text{Mod}(K(L^{2}(G)))$, we define a module isomorphism
$\alpha_{(g,\chi)}:M\to\Psi_{(g,\chi)}(M)$ by the rule $m\mapsto
S_{\chi}^{-1}T_{g}m$. To make sense of this formula, note that the underlying
abelian groups of $\Psi_{(g,\chi)}(M)$ and $M$ are the same. One checks that
these maps indeed assemble to an invertible natural transformation
$\alpha_{(g,\chi)}$. Stated differently, we have the following invertible
$2$-cell in $N_{h}(C^{*}\text{Cat})$:

${{\text{Mod}(K(L^{2}(G)))}}$${{\text{Mod}(K(L^{2}(G)))}}$$\scriptstyle{\Psi_{(g,\chi)}}$id$\scriptstyle{\alpha_{(g,\chi)}}$

Now, we claim that
$\alpha_{(g,\chi)}\circ\alpha_{(g^{\prime},\chi^{\prime})}=E(g,\chi,g^{\prime},\chi^{\prime})\alpha_{(gg^{\prime},\chi\chi^{\prime})}=\chi^{\prime}(g)\alpha_{(gg^{\prime},\chi\chi^{\prime})}$.

Indeed, the composite
$\alpha_{(g,\chi)}\circ\alpha_{(g^{\prime},\chi^{\prime})}$ is given by the
matrix $S_{\chi}^{-1}T_{g}S_{\chi^{\prime}}^{-1}T_{g^{\prime}}$. From the
proof of Lemma 3.4, we see that
$T_{g}S_{\chi^{\prime}}^{-1}=\chi^{\prime}(g)S_{\chi^{\prime}}^{-1}T_{g}$. The
claim is immediate from this observation.

In what follows, we write $\mathcal{C}$ for $\text{Mod}(K(L^{2}(G)))$. We now
define a homotopy $\sigma:B(G\times\hat{G})\times\Delta^{1}\to
N_{h}(C^{*}\text{Cat})$. We start by doing this degree-wise:

  * •

Degree 0 Set $\sigma((*,0))=\mathcal{C}$ and $\sigma((*,1))=\mathcal{C}$.

  * •

Degree 1 Set $\sigma(((g,\chi),00))=\text{id}_{\mathcal{C}}$,
$\sigma(((g,\chi),01))=\Psi_{(g,\chi)}$ and
$\sigma(((g,\chi),11))=\Psi_{(g,\chi)}$.

  * •

Degree 2 Set $\sigma(((g,\chi),(g^{\prime},\chi^{\prime}),000))$ to be

${{\mathcal{C}}}$${{\mathcal{C}}}$${{\mathcal{C}}}$ididid$\scriptstyle{\chi^{\prime}(g)}$

Set $\sigma(((g,\chi),(g^{\prime},\chi^{\prime}),001))$ to be

${{\mathcal{C}}}$${{\mathcal{C}}}$${{\mathcal{C}}}$id$\scriptstyle{\Psi_{(g^{\prime}g,\chi^{\prime}\chi)}}$$\scriptstyle{\Psi_{(g,\chi)}}$$\scriptstyle{\alpha_{(g^{\prime},\chi^{\prime})}*\text{id}_{\Psi_{(g,\chi)}}}$

Set $\sigma(((g,\chi),(g^{\prime},\chi^{\prime}),011))$ to be

${{\mathcal{C}}}$${{\mathcal{C}}}$${{\mathcal{C}}}$$\scriptstyle{\Psi_{(g^{\prime},\chi^{\prime})}}$$\scriptstyle{\Psi_{(g,\chi)}}$$\scriptstyle{\Psi_{(g^{\prime}g,\chi^{\prime}\chi)}}$id

We set $\sigma(((g,\chi),(g^{\prime},\chi^{\prime}),111))$ similarly:

${{\mathcal{C}}}$${{\mathcal{C}}}$${{\mathcal{C}}}$$\scriptstyle{\Psi_{(g^{\prime},\chi^{\prime})}}$$\scriptstyle{\Psi_{(g,\chi)}}$$\scriptstyle{\Psi_{(g^{\prime}g,\chi^{\prime}\chi)}}$id

Since $B(G\times\hat{G})\times\Delta^{1}$ is the nerve of a $1$-category, it
is $2$-coskeletal. Similarly, $N_{h}(C^{*}\text{Cat})$, being the nerve of a
$2$-category, is $3$-coskeletal (see Theorem 5.2 of [Str87]161616The author
thanks Lyne Moser for pointing out this reference.). Therefore we only have to
check the following condition to see that the definition above extends
uniquely to the desired homotopy $\sigma$:

For two composable morphisms $f$ and $f^{\prime}$ in
$B(G\times\hat{G})\times\Delta^{1}$, denote by $\sigma_{f,f^{\prime}}$ the
image of the identity $2$-simplex on $f,f^{\prime}$ and $f^{\prime}\circ f$.
Let $f_{3},f_{2},f_{1}$ be a sequence of composable morphisms in
$B(G\times\hat{G})\times\Delta^{1}$. Then there is an equality of
$2$-simplices

$\sigma_{f_{1},f_{3}\circ
f_{2}}*(\sigma_{f_{2},f_{3}}\circ\text{id}_{f_{1}})=\sigma_{f_{2}\circ
f_{1},f_{3}}*(\text{id}_{f_{3}}\circ\sigma_{f_{1},f_{2}})$

We check this condition case-by-case on the sequence of composable morphisms,
according to the underlying sequence after projecting to $\Delta^{1}$:

  * •

The case $0000$ amounts to checking that $E$ is a cocycle.

  * •

The case $0001$ reduces to verifying the equality, for
$(g_{3},\chi_{3}),(g_{2},\chi_{2}),(g_{1},\chi_{1})\in G\times\hat{G}$,

$(\alpha_{(g_{2},\chi_{2})}\circ\alpha_{(g_{1},\chi_{1})})*\text{id}_{\Psi_{(g_{3},\chi_{3})}}=\chi_{1}(g_{2})\alpha_{(g_{1}g_{2},\chi_{1}\chi_{2})}*\text{id}_{\Psi_{(g_{3},\chi_{3})}}$

This follows from the claim proven just before the definition of $\sigma$.

  * •

The case $0011$ is just the trivial equality

$\alpha_{(g_{1},\chi_{1})}*\text{id}_{\Psi_{(g_{2}g_{3},\chi_{2}\chi_{3})}}=\alpha_{(g_{1},\chi_{1})}*\text{id}_{\Psi_{(g_{2}g_{3},\chi_{2}\chi_{3})}}$

  * •

The cases $0111$ and $1111$ are trivial because all the $2$-simplices involved
are identities.

Therefore, $\sigma$ defines a homotopy between $\Psi^{\prime}=\sigma|_{0}$ and
$\Psi=\sigma|_{1}$. It is, in fact, a homotopy equivalence, because it factors
through the maximal groupoid of $N_{h}(C^{*}\text{Cat})$.

This shows that the two actions are equivalent. Hence, $K(\textbf{B})$ with
its natural $(G\times\hat{G})$-action is equivalent to $KU$ with
$(G\times\hat{G})$-action given by the cocycle $E$, whence it follows that
$K(\textbf{B})\simeq M^{\prime}_{E}$, as desired. ∎

We are now ready to state and prove our main comparison result (Theorem 1.3) .

######  Theorem 5.7.

We have a commutative square of functors

${{KK^{G,ft}}}$${{KK^{\hat{G},ft}}}$${{\textup{Mod}(KU_{p}[G])^{ft}}}$${{\textup{Mod}(KU_{p}[\hat{G}])^{ft}}}$$\scriptstyle{-\rtimes
G}$$\scriptstyle{\sim}$$\scriptstyle{r_{G}}$$\scriptstyle{r_{\hat{G}}}$$\scriptstyle{\sim}$$\scriptstyle{L_{p}(-\otimes_{KU_{p}[G]}M_{E})}$

###### Proof.

By Theorem 3.6, we can replace the upper horizontal arrow by the descended
functor $F$ of Proposition 3.5. We recall that $F$ is given by
$A\mapsto(A\otimes\mathbf{B})\rtimes G$, with $\hat{G}$-action on the target
induced by that on B.

The down-right composite of the diagram is the functor $A\mapsto
L_{p}(L_{p}(K(A))\otimes_{KU_{p}[G]}M_{E})$, with $\hat{G}$-action induced by
that on ME. By the previous lemma, we have a natural isomorphism of
$KU_{p}[\hat{G}]$-modules

$L_{p}(L_{p}(K(A))\otimes_{KU_{p}[G]}M_{E})\simeq
L_{p}(\text{colim}_{BG}(L_{p}(K(A))\otimes_{KU_{p}}L_{p}(K(\textbf{B}))))$

The symmetric monoidality of $L_{p}$ gives a natural isomorphism of
$KU_{p}[\hat{G}]$-modules

$L_{p}(\text{colim}_{BG}(L_{p}(K(A))\otimes_{KU_{p}}L_{p}(K(\textbf{B}))))\simeq
L_{p}(\text{colim}_{BG}L_{p}(K(A)\otimes_{KU}K(\textbf{B})))$

By the Künneth theorem ($K(\textbf{B})\simeq KU$), the left-adjointness and
idempotence of $L_{p}$, we obtain yet another natural isomorphism171717The
colimit on the right-hand side is to be interpreted in $p$-complete
$KU_{p}$-modules. of $KU_{p}[\hat{G}]$-modules

$L_{p}(\text{colim}_{BG}L_{p}(K(A)\otimes_{KU}K(\textbf{B})))\simeq\text{colim}_{BG}(L_{p}(K(A\otimes\textbf{B})))$

Lastly, Theorem 5.2 with another application of the left-adjointness of
$L_{p}$ gives a natural $KU_{p}[\hat{G}]$-module isomorphism

$\text{colim}_{BG}(L_{p}(K(A\otimes\mathbf{B})))\simeq
L_{p}(K((A\otimes\mathbf{B})\rtimes G))$

It is easy to see that the composite $r_{\hat{G}}\circ F$ is $A\mapsto
L_{p}(K((A\otimes\mathbf{B})\rtimes G))$, so this completes the proof. ∎

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