03-parts.Rmd

# Examples {#exemple}

## Inverse closed Banach \*-algebras from topological partial actions {#restrix}

We introduce now certain partial crossed product $C^*$-algebras
associated to partial topological actions of the discrete group. We do
not consider the most general case; in particular, we restrict
generality in such a way to have certain natural faithful Hilbert space
representations. In this way, transparent results on inverse closedness
will be available.

So let $(X,\Theta,\textsf{G})$ be a *partial action* of the group $\textsf{G}$ (assumed
here to be countable) on the compact Hausdorff space $X$. In more
detail, the partial action is denoted by
$\big\{\Theta_g:X_{g^{-1}}\!\to X_g\,\big\vert\, g\in\textsf{G}\big\}$ , where
each $X_g$ is an open subset of $X$ and $\Theta_g$ is a homeomorphism
with domain $X_{g^{-1}}$ and image $X_g$ . One requires

-   $X_\textsf{e}=X$ and $\Theta_\textsf{e}=\textrm{id}_X$ ,

-   for every $g,h\in\textsf{G}$ , the homeomorphism $\Theta_{gh}$ extends
    $\Theta_{g}\circ\Theta_{h}$.

Observe that the composition $\Theta_g\circ\Theta_h$ is meant to refer
to the map whose domain is the set of all elements $x \in X$ for which
$\Theta_g\circ\Theta_h(x)$ is well defined, therefore the domain of
$\Theta_g\circ\Theta_h$ is the set $\Theta_h^{-1}( X_h\cap X_{g^{-1}})$.

In fact each locally compact partial dynamical system may be deduced by
restricting a global one in an essentially canonical way: there exists a
global action $\tau$ of the group $G$ on a topological space $\tilde{X}$
such that
$$X_g=\tau_g(\tilde{X})\cap \tilde{X} \quad \quad \forall g\in G$$ and
$\tau_g|_{X_{g^{-1}}}=\Theta_g$ [@Ex2 Ch. 3]. But this restriction of
the global action might fail in the case the total space $\tilde X$ is
not Hausdorff ([@Ex2 Prop. 5.6] and [@Ab Th. 4.44]).

We obtain from $(X,\Theta,\textsf{G})$ a partial $C^*$-dynamical system
$\big(C(X),\theta,\textsf{G}\big)$ , where the ideals are $$\label{santodomingo}
\mathcal A_g\!:=C\big(X_{g}\big)\equiv\big\{a\in C(X)\,\big\vert\,a(x)=0\,,\,\forall\,x\notin X_{g}\big\}$$
and the isomorphisms are $$\label{panama}
\theta_g:C\big(X_{g^{-1}}\big)\to C\big(X_{g}\big)\,,\quad\theta_g(a):=a\circ\Theta_{g^{-1}}\,.$$
Therefore, one can construct the Banach $^*$-algebra
$\ell^1_\theta\big(\textsf{G};C(X)\big)$ and its enveloping partial crossed product
$C(X)\!\rightthreetimes_\theta\!\textsf{G}$ . We deduce from Theorem (\@ref(thm:teoremix))

::: {.proposition #corolix} 
If $\,{\textsf{G}}$ is rigidly symmetric, $\ell^1_\theta\big({\textsf{G}},C(X)\big)$ is reduced,
symmetric and inverse closed in the partial crossed product.
:::

Assume now that under the partial action there is an open dense orbit
$Y$ with trivial isotropy. Thus, for some (and then any) $y\in Y$ one
has $$\label{guadelupa}
\textsf{G}(y):=\big\{g\in\textsf{G}\,\big\vert\,y\in X_{g^{-1}}\,,\Theta_g(y)=y\big\}=\{\textsf{e}\}\,.$$
Then $X$ will be a compactification of the discrete orbit $Y$ and for
any $y\in Y$ the orbit map $g\to\Theta_g(y)$ will be a homeomorphism on
its image.

In the Hilbert space $\ell^2\big(Y\big)$ we introduce the representation
by multiplication operators $$\label{mexic}
\pi:C(X)\to\mathbb B\big[\ell^2(Y)\big]\,,\quad\big[\pi(a)\xi\big](y):=a(y)\xi(y)\,.$$
For any $h\in\textsf{G}\,,\,y\in Y$ and $\xi\in\ell^2(Y)$ one defines
$$\label{partilix}
\big[u_h(\xi)\big](y):=
\begin{cases}
\xi\big(\Theta_{h^{-1}}(y)\big) &\textrm{if}\ \;y\in X_h\,,\\
0\, &\textrm{if}\ \;y\notin X_h\,.
\end{cases}$$ Then $\big(\pi,u,\ell^2(Y)\big)$ *is a covariant
representation of* $\big(C(X),\theta,\textsf{G}\big)$ and *the integrated form
representation*
$$\Pi\equiv\pi\!\rightthreetimes u\!:C(X)\!\rightthreetimes_\theta\!\textsf{G}\to\mathbb B\big[\ell^2(Y)\big]\,,$$
which acts on $\ell^1_\theta\big(\textsf{G};C(X)\big)$ as
$\Pi(\Phi)=\sum_{h\in\textsf{G}}\pi\big[\Phi(h)\big]u_h$ and which satisfies
\begin{equation}
\Pi\big(a\otimes\delta_h\big)=\pi(a)u_h\,,\quad\forall\,h\in\textsf{G}\,,\,a\in C(X_h)\,.(\#eq:lemikainnen)
\end{equation}
To express it explicitly, we need some notations. For $y,z\in Y$ we set
$g_{yz}$ for the unique element of the group such that
$\Theta_{g_{yz}}(y)=z$ ; we use the fact that $Y$ is an orbit with
trivial isotropy. Note the relations 
\begin{equation}
g_{yz}^{-1}=g_{zy}\,,\quad g_{zy}=g_{xy}g_{zx}\,,\quad\forall\,x,y,z\in Y. (\#eq:dominicana)
\end{equation}
A direct computation, relying on (\@ref(eq:dominicana)), shows that on $\ell^1_\theta\big(\textsf{G};C(X)\big)$ the
representation $\Pi$ reads 
\begin{equation}
\big[\Pi(\Phi)\xi\big](y)=\sum_{z\in Y}\Phi\big(g_{zy},y\big)\xi(z)\,, (\#eq:flochax)
\end{equation}
where we set $\big[\Phi(k)\big](y)=:\Phi(k,y)$ .

::: {.theorem #pohjola} 
Suppose that ${\textsf{G}}$ is countable and rigidly symmetric and that $Y$ is a open dense orbit of the space
$X$, with trivial isotropy. Let
$\Phi\in\ell^1_\theta\big({\textsf{G}};C(X)\big)$ .

  (i).  If the operator $\,\Pi(\Phi)$ is invertible, there exists $\Psi\in\ell^1_\theta\big({\textsf{G}};C(X)\big)$ such that $\Pi(\Phi)^{-1}\!=\Pi(\Psi)$ .

  (ii).  If $\,\Pi(\Phi)$ is Fredholm, at least one of its inverses $T$
    modulo $\mathbb K\big[\ell^2(Y)\big]$ belongs to
    $\Pi[\ell^1_{\theta}\big({\textsf{G}};C(X)\big)]$ .
:::

::: proof
(i). Having in view Proposition (\@ref(prp:corolix)), we only need to check that the representation $\Pi$ is faithful; see also Theorem (\@ref(thm:teoremix))(iv). The universal and the reduced partial crossed products here are the same, cf. Remark (\@ref(cnj:lantors)). Note that the restriction of $\Pi$ to $C(X)$ coincides with $\pi$ (set $h=\textsf{e}$ in (\@ref(eq:lemikainnen)). By [@ELQ Th. 2.6], injectivity of $\Pi$ would
follow if we prove that $\pi$ is injective and the action is topologically free. The injectivity of $\pi$ is obvious: $\pi(a)=0$ means that the restriction of $a$ to $Y$ is null, which implies that $a$ is null, since the orbit $Y$ was supposed dense. Topological freeness (cf. [@ELQ Def. 2.1.]) is clear, since $Y$ is dense: having trivial isotropy, it is disjoint from any set of the form
$$F_g:=\big\{x\in X_{g^{-1}}\,\big\vert\,\Theta_g(x)=x\big\}\,,\quad g\ne\textsf{e}\,,$$
so this one must have empty interior.

(ii). Since $Y$ is discrete, the elements of $C_0(Y)$ are transformed by the representation $\pi$ into compact multiplication operators in $\ell^2(Y)$ . The $\theta$-invariant ideal $C_0(Y)$ gives rise to the ideal
$C_0(Y)\!\rightthreetimes_\theta\!\textsf{G}$ of the partial crossed product $C(X)\!\rightthreetimes_\theta\!\textsf{G}$ and one has $\Pi\big[C_0(Y)\!\rightthreetimes_\theta\!\textsf{G}\big]=\mathbb K\big[\ell^2(Y)\big]$ .
Therefore one can apply Corollary (\@ref(cor:honduras)) and conclude that
$\Pi[\ell^1_{\theta}\big(\textsf{G};C(X)\big)]\subset\mathbb B\big[\ell^2(Y)\big]$
is Fredholm inverse closed. Then the assertion follows easily from Atkinson's Theorem; see also Example (\@ref(exm:fredholmix)).
:::

## Toeplitz-Bunce-Deddens operators {#dedenix}

In the Hilbert space $\ell^2(\mathbb{N})$ , with canonical orthonormal base
$\{\delta_n\}_{n\in\mathbb{N}}$ , each function $a\in\ell^\infty(\mathbb{N})$ defines a
bounded multiplication operator $\textsf{M}_a$ . Let us denote by
$\textsf{S}$ the right shift uniquely determined by
$\textsf{S}\,\delta_n\!:=\delta_{n+1}$ for every $n\in\mathbb{N}$ . For
$q\in\mathbb{N}$ , we say that $a\in\ell^\infty(\mathbb{N})$ is $q$-periodic if
$a(n+q)=a(n)\,,\,\forall\,n\in\mathbb{N}$ . We denote by $\ell^\infty(\mathbb{N};q)$ the
$C^*$-subalgebra of all $q$-periodic functions. The starting point in
constructing a Toeplitz-Bunce-Deddens algebra is an infinite family
$\mathbf q:=\{q_i\!\mid\!i\in\mathbb{N}\}$ of strictly increasing positive
integers such that each $q_i$ divides $q_{i+1}$ , i.e. $q_{i+1}=r_i q_i$
with $r_i\ge 2$ .

::: {.definition #buncix} 
*The Toeplitz-Bunce-Deddens algebra $\mathfrak D(\mathbf q)$ associated to the multi-integer $\mathbf q$* is
the $C^*$-algebras of operators in $\ell^2(\mathbb{N})$ generated by the family
\begin{equation}
\Big\{\textsf{ S}_a:=\textsf{ S}\textsf{ M}_a\,\Big\vert\,a\in\bigcup_{i\in\mathbb{N}}\ell^\infty(\mathbb{N};q_i)\Big\}\,. (\#eq:generatrix)
\end{equation}
:::

::: {.proposition #propotrix} 
The Toeplitz-Bunce-Deddens algebra is topologically graded over $\mathbb{Z}$ . Its $\ell^1$-Banach $^*$-algebra
$\ell^1\big[\mathfrak D(\mathbf q)\big]$ is reduced, symmetric and
inverse closed in $\mathfrak D(\mathbf q)$ and in
$\mathbb B\big[\ell^2(\mathbb{N})\big]$ (an explicit description is included in
the proof).
:::

::: proof
By [@Exs1], a canonical action $\alpha$ of the torus
$\mathbb{T}=\widehat{\mathbb{Z}}$ on $\mathfrak D(\mathbf q)$ is defined as
follows: For every $z\in\mathbb T$ the unitary multiplication operator
determined by $U_z\delta_n:=z^n\delta_n$ ($n\in\mathbb{N}$) defines on
$\mathbb B\big[\ell^2(\mathbb{N})\big]$ the inner automorphism
$T\to U_z TU_z^*$. Because of 
\begin{equation}
\alpha_z(S_a)=U_z S_aU_z^*=zS_a\,,\quad\forall\,a\in\ell^\infty(\mathbb{N})\,, (eq:costarica)
\end{equation}
$\big(\mathfrak D(\mathbf q),\alpha,\mathbb{T}\big)$ is indeed a (full)
continuous action. We also write $\ell^\infty(\mathbb{N};\mathbf q)$ for the
(unital, Abelian) $C^*$-subalgebra of $\ell^\infty(\mathbb{N})$ generated by
$\bigcup_{i\in\mathbb{N}}\ell^\infty(\mathbb{N};q_i)$ . The spectral subspace
$\mathfrak D(\mathbf q)_k$ is generated by operators of the ordered form
\begin{equation}
S_{a_1}\dots S_{a_n}S^*_{b_1}\dots S^*_{b_m}\,,\quad a_1,\dots,a_n,b_1,\dots,b_m\in\ell^\infty(\mathbb{N};\mathbf q)\,,\ n-m=k\,. (\#eq:paracetamol)
\end{equation}
This follows easily from the relations $S^*_a S_b=M_{\overline ab}$ and
$M_a S_b=S_{\tilde ab}$ , where $\tilde a_n:=a_{n-1}$ if $n\ge 1$ .
Since $\tilde a_0$ does not appear in the definition of the weighted
shift operator, it can be fixed such that $\tilde a$ is periodic. Other
forms of the elements in $\mathfrak D(\mathbf q)_k$ could involve the
final projection $Q:=SS^*$. These having been settled, the assertions
concerning the $\ell^1$-Banach algebra follow from Theorem (\@ref(thm:exelix)).
:::

::: {.conjecture #toeplix} 
In [@Exs1] one gets the isomorphism
\begin{equation}
\mathfrak D(\mathbf q)\overset{\sim}{\longrightarrow}\big[c_0(\mathbb{N})\oplus\ell^\infty(\mathbb{N};\mathbf q)\big]\!\rightthreetimes_\theta\mathbb{Z}\cong C\big[\mathbb{N}(\mathbf q)\big]\!\rightthreetimes_\theta\mathbb{Z}\,. (\#eq:triplix)
\end{equation}
We refer to [@Exs1] for the Gelfand spectrum
$\mathbb{N}(\mathbf q)=\mathbb{N}\sqcup\Sigma(\mathbf q)$ of the Abelian $\alpha$-fixed
point $C^*$-algebra
$$\mathfrak D(\mathbf q)_0\!:=c_0(\mathbb{N})\oplus\ell^\infty(\mathbb{N};\mathbf q)$$
(a compactification of the discrete set $\mathbb{N}$ by a Cantor set
$\Sigma(\mathbf q)$) and for the interesting explicit form of the partial
action $\theta$ in the Gelfand realization (induced by a partial
homeomorphism of $\mathbb{N}(\mathbf q)$).
:::

We may also state that
*$\ell^1\big[\mathfrak D(\mathbf q)\big]=\bigoplus_{k\in\mathbb{Z}}^{1,\alpha}\mathfrak D(\mathbf q)_k$
is Fredholm inverse closed in*
$\mathfrak D(\mathbf q)\subset\mathbb B\big[\ell^2(\mathbb{N})\big]$ . This
follows from Corollary (\@ref(cor:honduras)), where $\mathbb{K}=c_0(\mathbb{N})$ and $\Pi$ is the inverse of
the isomorphism appearing in (\@ref(eq:triplix)),
since
$\Pi\big[c_0(\mathbb{N})\!\rightthreetimes_\theta\!\mathbb{Z}\big]=\mathbb K\big[\ell^2(\mathbb{N})\big]$ . One
calls the quotient 
\begin{equation}
\mathfrak D(\mathbf q)/\mathbb K\big[\ell^2(\mathbb{N})\big]\cong\ell^\infty(\mathbb{N};\mathbf q)\!\rtimes_\theta\mathbb{Z}\cong C\big[\Sigma(\mathbf q)\big]\!\rtimes_\theta\mathbb{Z} (\#eq:triflix)
\end{equation}
*the Bunce-Deddens algebra*. The action \"at infinity\" of $\mathbb{Z}$ on the
Cantor set $\Sigma(\mathbf q)$ is induced by an odometer map. It is a
global action, so this quotient can be dealt with by usual crossed
products.

## UHF algebras {#afelliz}

*An UHF-algebra* is the inductive limit 
$$\label{limindiz}
\mathfrak C:=\underset{m\in\mathbb{N}}{\textrm{lim} \,\textrm{ind}\,\mathbf M^{p_m}}
$$ 
of
a sequence of full matricial $C^*$-algebras, with unital and injective
connecting morphisms, where each $p_m$ divides $p_{m+1}$. In [@Exs2]
Exel treated general approximately finite algebras. He defined on
$\mathfrak C$ a regular and semi-saturated $\mathbb T$-action and then
he applied his theory from [@Exs] to get an isomorphism between
$\mathfrak C$ and a partial crossed product $\mathcal{A}\rightthreetimes_\theta\mathbb{Z}$ , where $\mathcal{A}$
is an Abelian almost finite $AF$-algebra. To simplify, we decided to
treat only UHF algebras; the general case is similar, but it would
involve more complicated notations. To state a result on symmetric
subalgebras, we make use of Theorem (\@ref(thm:exelix)), only
invoking the circle action and its spectral subspaces; the partial
crossed product will not be mentioned.

On every full matrix algebra $\mathbf M^p$ one defines $$\label{inerix}
\alpha^p:\mathbb T\to\textrm{ Aut}\big(\mathbf M^p\big)\,,\quad\alpha^p_z\big[(c_{ij})_{i,j}\big]=\big(z^{i-j}c_{ij}\big)_{i,j}\,.$$
The connecting morphisms
$\big\{\mu^{m}\!:\!\mathbf M^{p_m}\to\mathbf M^{p_{m+1}}\big\}$ are
covariant with respect to the actions $\big\{\alpha^{p_m}\big\}$. The
inductive limit comes with the canonical monomorphisms
$\big\{\nu^{m}\!:\mathbf M^{p_m}\!\to\mathfrak C\big\}$ and
$\bigcup_m\nu^{(m)}\!\big(\mathbf M^{p_m}\big)$ is dense in
$\mathfrak C$ . For every $z\in\mathbb T\,,m\in\mathbb{N}$ and
$\Phi\in\mathbf M^{p_m}$ we set $$\label{tetaniz}
\alpha_z\big[\nu^{m}(\Phi)\big]:=\nu^{m}\big[\alpha^{p_m}_z(\Phi)\big]\,.$$
It is shown in [@Exs2 Sect. 2] that $\alpha$ extends to an action on
$\mathfrak C$ (which is semi-saturated and regular).

To make Theorem (\@ref(thm:exelix)) concrete, one needs the spectral subspaces. For each
$|k|\le p$ , let us denote by $\mathbf M^p_k$ the vector subspace of
$p\!\times\!p$-matrices with non-empty entries only on the $k'$th
diagonal ($c_{ij}=0$ if $i-j\ne k$). If $|k|>p$ , we set simply
$\mathbf M^p_k\!:=\{0\}$ . Then $\mathbf M^p_k$ is the $k'$th spectral
subspace of the action $\alpha^p$. It is follows from the definitions
and from the fact that each canonical morphism $\nu^{m}$ is injective
that the spectral subspaces of the action $\alpha$ are the closures in
$\mathfrak C$ of the unions over $m$ of the images through $\nu^{m}$ of
the subspaces $\mathbf M^{p_m}_k$.

::: {.theorem #cioflix} 
The Banach $^*$-algebra $$\label{perru}
\ell^1(\mathfrak C)=\bigoplus^{1,\alpha}_{k\in\mathbb{N}}\overline{\bigcup_{m\in\mathbb{Z}}\nu^{m}\big(\mathbf M^{p_m}_k\big)}$$
is reduced, symmetric and inverse closed in $\mathfrak C$ .
:::

::: {.conjecture #takesix} 
In [@Ta] Takesaki (see also [@FD2 VIII.17])
provided an approach to UHF algebras which would lead to a different
grading, and consequently also to a symmetric $\ell^1$-type Banach
$^*$-algebra, this time over the infinite restricted product
$\textsf{G}:=\prod^\prime_{m\in\mathbb{N}}\textsf{G}_m$ with the discrete topology, where each
$\textsf{G}_m$ is a cyclic group of a certain order $q_m$ . Actually, the UHF
algebra $\mathfrak C$ is isomorphic to an infinite tensor product
$\otimes_m\mathbf M^{q_m}$, where $q_m:=p_{m+1}/p_m$ . In its turn, this
one is shown to be isomorphic to the full crossed product
$C(\widehat{\textsf{G}}\,)\!\rtimes\textsf{G}$ . The dual $\widehat{\textsf{G}}$ can be identified with the product group $\prod_{m\in\mathbb{N}}\textsf{G}_m$ .
:::

## CAR algebras {#afellix}

Let $\big(\mathcal R,(\cdot|\cdot)\big)$ be an infinitely dimensional
separable Hilbert space and $a:\mathcal R\to\mathbb B(\mathcal H)$ *a
representation of the canonical anticommutation relations* in the
Hilbert space $\mathcal{H}$ , generating the unital $C^*$-algebra
$\textrm{ CCR}(\mathcal R)\subset\mathbb B(\mathcal H)$ . Thus $a$ is
lineal and for every $r,s\in\mathcal R$ one has $$\label{anticom}
a(r)a(s)+a(s)a(r)=0\,,\quad a^*(r)a(s)+a^*(s)a(r)=(r|s)\,.$$ It is known
that $\textrm{ CCR}(\mathcal R)$ is isomorphic to the UHF algebra
$\mathbf M(2^\infty):=\underset{m\in\mathbb{N}}{\textrm{ lim\,ind}\,\mathbf M^{2^m}}$
as well as with the infinite tensor product
$\bigotimes_{m\in\mathbb{N}}\mathbf M^2$.

Let also $V:\widehat{\textsf{G}}\to\mathbb B(\mathcal R)$ be a strongly continuous
unitary representation of the compact Abelian group $\widehat{\textsf{G}}$ . Associated
to $V$ there is a continuous action
$\upsilon:\widehat{\textsf{G}}\to\textrm{ Aut}\big[\textrm{ CCR}(\mathcal R)\big]$ ,
uniquely defined on generators by $$\label{creatrix}
\upsilon_\chi\big[a(r)\big]=a\big[V_\chi(r)\big]\,,\quad\forall\,r\in\mathcal R\,,\,\chi\in\widehat{\textsf{G}}\,.$$

We denote by $\textsf{G}$ the dual of $\widehat{\textsf{G}}$ ; it is a discrete Abelian group. We
already have the grading $$\label{carnatix}
\textrm{ CCR}(\mathcal R)=\widetilde\bigoplus_{g\in\textsf{G}}\textrm{ CCR}(\mathcal R)^{\upsilon}_g$$
in terms of spectral subspaces of the action $\upsilon$ . The next
corollary follows directly from Theorem (\@ref(thm:teoremix)).
Note the generality: any representation $\big(\widehat{\textsf{G}},V\big)$ provides a
result.

::: {.corollary #spermioni} 
The corresponding $\ell^1$-algebra
\begin{equation}
\ell^1\big(\textrm{CCR}(\mathcal R)\big)=\bigoplus_{g\in{\textsf{G}}}^{1,\upsilon}\textrm{ CCR}(\mathcal R)^{\upsilon}_g (\#eq:sperioni)
\end{equation}
is symmetric and inverse closed in $\textrm{ CCR}(\mathcal R)$ and in
$\mathbb B(\mathcal H)$.
:::

It is clear that $a(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_g$ if and
only if $r\in\mathcal R^V_g$, meaning by definition that
$V_\chi(r)=\chi(g)r$ for all $\chi\in\widehat{\textsf{G}}$ . Similarly,
$a^*(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_g$ if and only if
$r\in\mathcal R^V_{g^{-1}}$ .

::: {.example #sfarnak} 
If $V:\mathbb{T}\to\mathbb B(\mathcal R)$ is given
by $V_\tau(r):=e^{2\pi i\tau}r$ and the group duality is implemented by
$\tau(k):=e^{2\pi ik\tau}$ ($k\in\mathbb{Z}$) , then
$\mathcal R^V_1=\mathcal R$ , while $\mathcal R^V_k=\{0\}$ for any
$k\ne 1$ . Thus $a(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_1$ and
$a^*(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_{-1}$ , from which it
follows that $$\label{followix}
a^*(r_1)\dots a^*(r_n)a(s_1)\dots a(s_m)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_{m-n}\,,\quad\forall\,r_1,\dots,r_n,s_1,\dots s_m\in\mathcal R\,.$$
:::

## Wiener-Hopf algebras associated to quasi-lattice ordered groups {#quasilatix}

One can treat this subject by using partial crossed products, as in
Subsection \@ref(cerculix), due to results from [@QR; @ELQ; @Ex2]. We found
it easier for our purposes to use the initial article [@Ni], combined
with Theorem (\@ref(thm:teoremix)).

We fix a sub-monoid $\textsf{P}$ of a discrete amenable group $\textsf{G}$ such that
$\textsf{P}\cap\textsf{P}^{-1}=\{\textsf{e}\}$ . One defines a left-invariant order relation
in $\textsf{G}$ by $g\leq h$ iff $g^{-1}h\in \textsf{P}$. The ordered group $(\textsf{G},\textsf{P})$
is *quasi-lattice ordered* if any $g\in\textsf{P}\textsf{P}^{-1}$ has a least upper
bound in $\textsf{P}$. Many examples may be found at [@Ni pag.23].

Consider the Hilbert space $\ell^2(\textsf{P})$ with its usual orthonormal
basis $\{e_q\}_{q\in\textsf{P}}$. For $p\in\textsf{P}$, consider the bounded linear
operator $W_p\in \mathbb{B}(\ell^2(\textsf{P}))$ defined by
$$W_p(e_q)=e_{pq}\,,\quad\forall\,q\in\textsf{P}.$$ We refer to $W$ as the
*regular semigroup of isometries* of $\textsf{P}$. The $C^*$-algebra of
operators on $\ell^2(\textsf{P})$ generated by the range of $W$ is called the
*Wiener-Hopf algebra of $\,\textsf{P}$* is denoted by $\mathfrak{W}(\textsf{P})$ .

For every $g\in\textsf{P}\textsf{P}^{-1}$ we define the closed subspace
$$\mathfrak{W}(\textsf{P})_g:=\overline{\textrm{span}}\,\big\{W_p W_q^*\,\big\vert\,pq^{-1}\!=g\big\}$$
and set $\mathfrak{W}(\textsf{P})_g=\{0\}$ if $g\notin \textsf{P}\textsf{P}^{-1}$. A
characterization of the Abelian $C^*$-subalgebra $\mathfrak{W}(\textsf{P})_\textsf{e}$
is
$$\mathfrak{W}(\textsf{P})_\textsf{e} =\left\{T\in\mathfrak{W}(\textsf{P})\!\mid\! T \textrm{ has a diagonal matrix relatively to the canonical basis of }\ell^2(\textsf{P})\right\}.$$
In Section 3 of Nica's paper [@Ni] it is shown that that *the collection
$\big\{\mathfrak{W}(\textsf{P})_g\,\big\vert\,g\in\textsf{G}\big\}$ provides a
topologically graded structure of the Wiener-Hopf algebra*. We can apply
now Theorem (\@ref(thm:teoremix)) and get

::: {.corollary #mimishor} 
The Banach $^*$-algebra
$\ell^1\Big( \widehat{\bigoplus}_{g\in{\textsf{G}}}\,\mathfrak{W}({\textsf{P}})_g \Big)$
is symmetric and inverse closed in
$\mathbb B\big[\ell^2({\textsf{P}})\big]$ .
:::

::: {.conjecture #spresfarsit} 
Suppose that ${\textsf{P}}$ is finitely generated (a slightly more general assumption is possible).
Then $\ell^1\Big( \widehat{\bigoplus}_{g\in{\textsf{G}}}\,\mathfrak{W}({\textsf{P}})_g \Big)$
*is also Fredholm inverse closed*. We only sketch a proof. As in
[@QR; @ELQ; @Ex2], one has an isomorphism
$\mathfrak W({\textsf{P}})\cong\mathfrak W({\textsf{P}})_\textsf{e}\rightthreetimes_\theta{\textsf{G}}$
for a certain partial action $\theta$ of ${\textsf{G}}$ on the diagonal
Abelian algebra. This one is induced from a (Bernoulli-type) partial
topological action $\Theta$ on its Gelfand spectrum $\Omega_\textsf{e}$ , which
is a compactification of the monoid ${\textsf{P}}$. This
compactification is regular (${\textsf{P}}$ is an *open* dense
orbit) under the stated hypothesis on ${\textsf{P}}$. It is shown in
[@Ni 6.3] that this regularity is equivalent with the fact that the
ideal of all compact operators, identified with
$C({\textsf{P}})\rightthreetimes_\theta{\textsf{G}}$ , is contained in
$\mathfrak W({\textsf{P}})\,.$ Then the result follows easily from
our Corollary (\@ref(cor:honduras)).
:::

## Higher rank graph algebras {#oroflix}

In this subsection we are going to use constructions from [@KP] and [@Ra].

::: {.definition #hix} 
*A $\textrm{ k}$-graph* ( *higher rank graph*) is a countable category $\Lambda$ endowed with a functor
$\textrm{l}:\Lambda\to\mathbb{N}^\textrm{ k}$ ( *the degree functor*) satisfying
the *factorization property*: For every $\lambda\in\Lambda$ such that
$\textrm{ l}(\lambda)=\mathfrak m+\mathfrak n$ , there exist unique
elements $\mu,\nu\in\Lambda$ such that $\textrm{l}(\mu)=\mathfrak m$ ,
$\textrm{l}(\nu)=\mathfrak n$ and $\lambda=\mu\nu$ .
:::

If $\textrm{ k}=1$ , one identifies a $1$-graph with the path category
$E^*\!:=\bigsqcup_{n\in\mathbb{N}}E^n$ of a directed graph
$\big(s,r:E^1\to E^0\big)$ and $\textrm{ l}(\lambda)$ is the usual
length of a path $\lambda:=e_1\dots e_{\textrm{ l}(\lambda)}$ .
Accordingly, for the $\textrm{ k}$-graph
$\big(\textrm{ l}:\Lambda\to\mathbb{N}^\textrm{ k}\big)$ , one denotes by
$\textsf{s},\textsf{r}:\Lambda\to\Lambda^0\equiv\textrm{ Obj}(\Lambda)$ the source and
the range map in the category. One has the decomposition
$$\Lambda=\bigsqcup_{\mathfrak n\in\mathbb{N}^\textrm{ k}}\Lambda^{\mathfrak n}\equiv\bigsqcup_{\mathfrak n\in\mathbb{N}^\textrm{ k}}\textrm{ l}^{-1}(\mathfrak n)\,.$$
For elements from the object set $\Lambda^0$ (called *vertices*) one
prefers notations as $v,w$ . We also set set
$\Lambda^{\mathfrak n}(v):=\Lambda^{\mathfrak n}\cap\textrm{ r}^{-1}(v)$
(\"paths\" of \"length\" $\mathfrak n\in\mathbb{N}^\textrm{ k}$ ending in
$v\in\Lambda^0$ ).

::: {.definition #admisibilix} 
One says that the $\textrm{ k}$-graph is *admissible* if every $\Lambda^{\mathfrak n}(v)$
is finite (raw-finite $\textrm{ k}$-graph) and non-void (no sources) and
$\Lambda^0$ is finite (insuring that the $C^*$-algebra below is unital,
with unit $\sum_{v\in\Lambda^0}\textsf{ S}_v$).
:::

::: {.definition #cstarix} 
Let $\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}$ be an
admissible $\textrm{ k}$-graph. *A Cuntz-Krieger $\Lambda$-family* is a
set $\{\textsf{ T}_\lambda\!\mid\!\lambda\in\Lambda\}$ of partial
isometries in a $C^*$-algebra $\mathfrak D$ such that

1.  the elements $\big\{\textsf{ T}_v\!\mid\!v\in\Lambda^0\big\}$ are
    mutually orthogonal projections,

2.  if $\textrm{ s}(\lambda)=\textrm{ r}(\mu)$ , then
    $\textsf{ T}_\lambda\textsf{ T}_\mu=\textsf{ T}_{\lambda\mu}$ ,

3.  $\textsf{ T}^*_\lambda\textsf{ T}_\lambda=\textsf{ T}_{\textrm{ s}(\lambda)}$
    for every $\lambda\in\Lambda$ ,

4.  $\textsf{ T}_v=\sum_{\lambda\in\Lambda^{\mathfrak n}(v)}\textsf{ T}_\lambda\textsf{ T}^*_\lambda$
    for every $v\in\Lambda^0$ and $\mathfrak n\in\mathbb{N}^\textrm{ k}$.

We define $C^*(\Lambda,\textrm{ l})\equiv\mathfrak C(\Lambda)$ to be the
universal $C^*$-algebra generated by a Cuntz-Krieger $\Lambda$-family
$\{\textsf{ S}_\lambda\!\mid\!\lambda\in\Lambda\}$ . Universality means
that for every Cuntz-Krieger $\Lambda$-family
$\{\textsf{ T}_\lambda\!\mid\!\lambda\in\Lambda\}\subset\mathfrak D$ ,
there exists a unique $C^*$-morphism
$\pi_\textsf{ T}:\mathfrak C(\Lambda)\to\mathfrak D$ such that
$\pi_\textsf{ T}(\textsf{ S}_\lambda)=\textsf{ T}_\lambda$ for every
$\lambda$ .
:::

We identify now families of symmetric Banach $^*$-algebras, starting
with functors $\textsf{ F}:\Lambda\to\textsf{G}$ , where $\textsf{G}$ is supposed to be
a discrete Abelian group. One particular case is $\textsf{G}=\mathbb{N}^\textrm{ k}$ and
$\textsf{ F}=\textrm{ l}$ , but many others are possible. For every
$g\in\textsf{G}$ one sets $$\label{bebex}
\mathfrak{C}(\Lambda)_g:=\overline{\textrm{ span}}\big\{\textsf{ S}_\lambda\textsf{ S}^*_{\mu}\,{\big\vert\,\textsf{ F}}(\lambda)\textsf{ F}(\mu)^{-1}=g\big\}\,.$$
It turns out that these are spectral subspaces of a dual action
$\alpha^\textsf{ F}:\widehat{\textsf{G}}\to\textrm{ Aut}\big[\mathfrak C(\Lambda)\big]$ ,
uniquely determined (use the universal property of
$\mathfrak C(\Lambda)$) by
$$\alpha^\textsf{ F}_\chi(\textsf{ S}_\lambda)=\chi\big[\textsf{ F}(\lambda)\big]\textsf{ S}_\lambda\,,\quad\forall\,\lambda\in\Lambda\,,\,\chi\in\widehat{\textsf{G}}\,.$$

Applying Theorem (\@ref(thm:exelix)) one gets

::: {.corollary #ix} 
Let $\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}$ be an
admissible $\textrm{ k}$-graph and $\textsf{ F}:\Lambda\to\textsf{G}$ a functor
with values in an Abelian discrete group. Then
$$\ell^1\big(\mathfrak C(\Lambda)\big):=\bigoplus_{g\in\textsf{G}}^{1,\alpha^{\textsf{F}}}\,\overline{\textrm{ span}}\big\{\textsf{ S}_\lambda\textsf{ S}^*_\mu\,{\big\vert\,\textsf{F}}(\lambda)\textsf{ F}(\mu)^{-1}=g\big\}$$
is a reduced and symmetric Banach $^*$-algebra, which is inverse closed
in $\mathfrak C(\Lambda)$ .
:::

::: {.definition #aperiodix} 
On $\mathbb{N}^\textsf{ k}$ one considers the order $\mathfrak m\le\mathfrak n\,\Leftrightarrow\,\mathfrak m_i\le\mathfrak n_i\,,\forall\,i$ .
Let $\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}$ be an
admissible $\textrm{ k}$-graph.

1.  We say that $\rho$ is *a minimal common extension* of
    $\mu,\nu\in\Lambda$ , and we write $\rho\in\textrm{ MCE}(\mu,\nu)$ ,
    if $\textrm{ l}(\rho)=\max\{\textrm{ l}(\mu),\textrm{ l}(\nu)\}$ and
    $\rho=\mu\mu'=\nu\nu'$ for some $\mu',\nu'\in\Lambda$ .

2.  The $\textsf{ k}$-graph is called *aperiodic* if for every
    $\mu,\nu\in\Lambda$ with $\textrm{ s}(\mu)=\textrm{ s}(\nu)$ there
    exists $\lambda\in\Lambda$ with
    $\textrm{ r}(\lambda)=\textrm{ s}(\mu)$ and
    $\textrm{ MCE}(\mu\lambda,\nu\lambda)=\emptyset$ .
:::

The two notions are easier to visualize for $\textsf{ k}=1$ . In this
case, for instance, $\textrm{ MCE}(\mu,\nu)$ is either void, or a
singleton (one of the two elements) and aperiodicity boils down to
condition (L).

::: {.proposition #perdrix} 
Let $\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}$ be an
admissible and aperiodic $\textrm{ k}$-graph and
$\textsf{ F}:\Lambda\to{\textsf{G}}$ a functor with values in an
Abelian discrete group. Let
$\{\textsf{ T}_\lambda\!\mid\!\lambda\in\Lambda\}$ be a Cuntz-Krieger
$\Lambda$-family of partial isometries in the Hilbert space $\mathcal{H}$ such
that $\textsf{ T}_v\ne 0$ for every $v\in\Lambda^0$. Then (the quite
obvious action $\beta^\textsf{ F}$ is explained in the proof)
\begin{equation}
\bigoplus_{g\in{\textsf{G}}}^{1,\beta^{\textsf{ F}}}\,\overline{\textrm{ span}}\big\{\textsf{ T}_\lambda\textsf{ T}^*_{\mu}\,{\big\vert\,\textsf{ F}}(\lambda)\textsf{ F}(\mu)^{-1}\!=g\big\} (\#eq:fujimori)
\end{equation}
is inverse closed in $\mathbb B(\mathcal{H})$ .
:::

::: proof
Under the stated assumptions, \"the Cuntz-Krieger unicity
theorem\" (see [@KP; @Si] for example) states that the unique
representation $\pi_\textsf{ T}:\mathfrak C(\Lambda)\to\mathbb B(\mathcal{H})$
satisfying
$$\pi_{\textsf{T}}(\textsf{ S}_{\lambda})=\textsf{ T}_{\lambda}\,,\quad\forall\lambda\in\Lambda$$
(provided by the universal property) is injective. We denote by
$\mathfrak C(\textsf{ T})$ its range, so that $\mathfrak C(\Lambda)$ and
$\mathfrak C(\textsf{ T})$ are isomorphic. Theorem (\@ref(thm:teoremix)) states then that
$\pi_\textsf{ T}\Big[\ell^1\big(\mathfrak C(\Lambda)\big)\Big]$ is
inverse closed in $\mathfrak C(\textsf{ T})$ and in $\mathbb B(\mathcal{H})$ .
There is an action
$\beta^\textsf{ F}\!:\widehat{\textsf{G}}\to\textrm{ Aut}\big[\mathfrak C(\textsf{ T})\big]$
such that
$$\beta^\textsf{ F}_\chi(\textsf{ T}_\lambda)=\chi[\textsf{ F}(\lambda)]\textsf{ T}_\lambda\,,\quad\forall\,\chi\in\widehat{\textsf{G}}\,,\,\lambda\in\Lambda$$
and $\pi_T$ intertwines the two actions. It follows that
$\pi_\textsf{ T}\Big[\ell^1\big(\mathfrak C(\Lambda)\big)\Big]$
coincides with (\@ref(eq:fujimori)).
:::