decomposes to $\psi^{\prime}(f)=\sum_{k=0}^{3}i^{k}\tau_{k}$ where
$\tau_{k}:A\to\mathbb{C}$, $k=0,...,3$, are positive functionals. Applying the
GNS-construction to each $\tau_{k}$, we get representations $\pi_{k}:A\to
B(H_{k})$ and cyclic vectors $\omega_{k}\in H_{k}$ such that
$\psi^{\prime}(f)(a)=\sum_{k=0}^{3}i^{k}\langle\pi_{k}(a)\omega_{k},\omega_{k}\rangle$.
Since $\\{\pi_{k}(\mu_{i})\\}_{i}$ is strongly convergent to the identity on
$H_{k}$ we conclude that $\\{f(\psi(\mu_{i}))\\}_{i}$ is convergent in
$\mathbb{C}$ to the number
$c_{f,\psi}:=\sum_{k=0}^{3}i^{k}\|\omega_{k}\|^{2}=\sum_{k=0}^{3}i^{k}\|\tau_{k}\|$
that depends only on $\psi$ and $f$ (it does not depend on
$\\{\mu_{i}\\}_{i}$). Now, since the net $\\{\psi(\mu_{i})\\}_{i}$ is bounded
in $E$, the Banach–Alaoglu Theorem says that there is a subnet
$\\{\psi(\mu_{i_{j}})\\}_{j}$ with an $E_{*}$-limit
$\psi(1_{A}):=E_{*}\mathchar 45\relax\lim_{j}\psi(\mu_{i_{j}})\in E$. So in
particular $f(\psi(1_{A}))=\lim_{j}f(\psi(\mu_{i_{j}}))=c_{f,\psi}$ for every
$f\in E_{*}$. This implies that every net $\\{\psi(\mu_{i})\\}_{i}$, where
$\\{\mu_{i}\\}_{i}$ is an approximate unit, is $E_{*}$-convergent to
$\psi(1_{A})$. ∎

######  Theorem 4.27 (Disintegration theorem).

Let $(\mathcal{G},\mathcal{L})$ be a twisted étale groupoid and let
$(\alpha,u)$ be a twisted inverse semigroup action of $S$ that models it. Then
$(\pi,v)\leftrightarrow\pi\rtimes v$ establishes a bijective correspondence
between representations $\psi:F^{S}(\mathcal{G},\mathcal{L})\to B$ in a Banach
algebra $B$ and

  1. (1)

$B^{\prime}$-normalized covariant representations $(\pi,v)$ of $(\alpha,u)$ in
$B$;

  2. (2)

$B_{*}$-normalized covariant representations $(\pi,v)$ of $(\alpha,u)$ in $B$,
if $(B,B_{*})$ is a dual Banach algebra;

  3. (3)

covariant representations $(\pi,v)$ of $(\alpha,u)$ on $E$, if $B=B(E)$ and
$E$ is a reflexive Banach space.

If each $X_{t}$, $t\in S$, is compact then the pairs $(\pi,v)$ in (1)–(3)
coincide with covariant representations $(\pi,v)$ of $(\alpha,u)$ in $B$ such
that $v_{t}=\pi(1_{X_{t}})v_{t}\in B$ for all $t\in S$.

###### Proof.

Let $\psi:F^{S}(\mathcal{G},\mathcal{L})\to B$ be a representation. Put
$\pi:=\psi|_{C_{0}(X)}$. For every $t\in S$, the map $C_{c}(X_{t})\ni
a_{t}\mapsto\psi(a_{t}\delta_{t})\in B$ extends to a linear contraction
$C_{0}(X_{t})\to B\subseteq B^{\prime\prime}$. Let $v_{t}\in B^{\prime\prime}$
be the element associated to this map in Lemma 4.26, where
$E=B^{\prime\prime}$ and $E_{*}=B^{\prime}$. Namely, for any approximate unit
$\\{\mu_{i}^{t}\\}_{i}\subseteq C_{0}(X_{t})$ we have
$v_{t}=B^{\prime}\mathchar 45\relax\lim_{i}\psi(\mu_{i}^{t}\delta_{t})$.
Recall that we consider multiplication in $B^{\prime\prime}$ which is
$B^{\prime}$-continuous in the second variable. For $a\in
C_{c}(X_{t})=C_{c}(X_{tt^{*}})$ we have
$(a\delta_{tt^{*}})*(\mu_{i}^{t}\delta_{t})\to a\delta_{t}$ in
$C_{0}(U_{t},\mathcal{L})\subseteq F^{S}(\mathcal{G},\mathcal{L})$. Thus

$\pi(a)v_{t}=\psi(a\delta_{tt^{*}})B^{\prime}\mathchar
45\relax\lim_{i}\psi(\mu_{i}^{t}\delta_{t})=B^{\prime}\mathchar
45\relax\lim_{i}\psi(a\delta_{tt^{*}}*\mu_{i}^{t}\delta_{t})=\psi(a\delta_{t}).$

For $a\in C_{c}(X_{t^{*}})$ we have
$(\mu_{i}^{t}\delta_{t})*(a\delta_{t^{*}t})=(\alpha_{t}(a)\mu_{i}^{t})\delta_{t}\to\alpha_{t}(a)\delta_{t}$
in $C_{0}(U_{t^{*}},\mathcal{L})\subseteq F^{S}(\mathcal{G},\mathcal{L})$.
Multiplication in $B^{\prime\prime}$ is continuous in the first variable when
the second variable is in $B$, so

$v_{t}\pi(a)=B^{\prime}\mathchar
45\relax\lim_{i}\psi(\mu_{i}^{t}\delta_{t}*a\delta_{t^{*}t})=\psi\big{(}\alpha_{t}(a)\delta_{t}\big{)}=\pi(\alpha_{t}(a))v_{t}.$

If in addition $t=e\in E(S)$ then
$v_{e}\pi(a)=\psi(\alpha_{e}(a)\delta_{e})=\psi(a\delta_{e})=\pi(a)$. If $a\in
C_{c}(X_{st})$ for $s,t\in S$ then $\alpha_{s}^{-1}(a)=a\circ h_{s}\in
C_{c}(X_{s^{*}}\cap X_{t})$ and therefore
$(a\delta_{s})*(\mu_{i}^{t}\delta_{t})=a(\mu_{i}^{t}\circ
h_{s^{*}})u(s,t)\delta_{st}=\alpha_{s}(\alpha_{s}^{-1}(a)\mu_{i}^{t})u(s,t)\delta_{st}\to
au(s,t)\delta_{t}$ in $C_{0}(U_{st},\mathcal{L})\subseteq
F^{S}(\mathcal{G},\mathcal{L})$. Thus

$\pi(a)v_{s}v_{t}=B^{\prime}\mathchar
45\relax\lim_{i}\psi(a\delta)\psi(\mu_{i}^{t}\delta_{t})=\psi\big{(}au(s,t)\delta_{t}\big{)}=\pi\big{(}au(s,t)\big{)}v_{t}.$

By construction $v_{t}=B^{\prime}\mathchar
45\relax\lim_{i}\varphi(\mu_{i}^{t})v_{t})$. Hence we see that $(\pi,v)$ is a
$B^{\prime}$-normalized covariant representation of $(\alpha,u)$ with
$\psi=\pi\rtimes v$.

This immediately gives (1) as a $B^{\prime}$-normalized a covariant
representation $(\pi,v)$ of $(\alpha,u)$ with $\psi=\pi\rtimes v$ has to be
the one we constructed above. Similarly, we get (2) and (3) by passing to an
appropriate normalization as described in Proposition 3.15 and Corollary 3.23.
The last part of the assertion follows from Remark 3.9. ∎

######  Corollary 4.28.

We have a natural isometric isomorphism $F^{S}(\mathcal{G},\mathcal{L})\cong
C_{0}(X)\rtimes_{(\alpha,u)}S$. Moreover, if $\mathcal{E}$ is a class of
Banach spaces then denoting by $\mathcal{E}_{\operatorname{alg}}$ the class of
all covariant representations of $(\alpha,u)$ in Banach algebras $B(E)$, where
$E\in\mathcal{E}$, and by $\mathcal{E}_{\operatorname{spa}}$ the class of all
covariant representations of $(\alpha,u)$ on Banach spaces $E\in\mathcal{E}$,
then

$F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L})\cong
C_{0}(X)\rtimes_{(\alpha,u),\mathcal{E}_{\operatorname{alg}}}S\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}C_{0}(X)\rtimes_{(\alpha,u),\mathcal{E}_{\operatorname{spa}}}S.$

The last homomorphism is isometric, for instance, if all spaces in
$\mathcal{E}$ are reflexive, or if each $X_{t}$, $t\in S$, is compact.

###### Proof.

Apply Theorem 4.27. ∎

######  Corollary 4.29.

For any $C^{*}$-algebra $B$, a homomorphism $F^{S}(\mathcal{G},\mathcal{L})\to
B$ is contractive (i.e. is a representation) if and only if it is
$*$-preserving, and then it factors through a $*$-homomorphism
$C^{*}(\mathcal{G},\mathcal{L})\to B$.

###### Proof.

Any representation $\psi:F^{S}(\mathcal{G},\mathcal{L})\to B$ is
$*$-preserving because any integrated representation is, by Proposition 4.25.
Conversely, if $\psi$ is a $*$-homomorphism it is
$\|\cdot\|_{C^{*}\max}$-contractive on
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ and thus it defines a
representation $\overline{\psi}:C^{*}(\mathcal{G},\mathcal{L})\to B$. Since
$\psi$ is the composition of the canonical representation
$F^{S}(\mathcal{G},\mathcal{L})\to C^{*}(\mathcal{G},\mathcal{L})$ and
$\overline{\psi}$, it is a representation itself. ∎

##  5\. Representations on $L^{p}$-spaces

Fix $p\in[1,\infty]$ and let $q\in[1,\infty]$ be the number satisfying
$1/p+1/q=1$, with the convention that $1/\infty=0$. Take a twisted étale
groupoid $(\mathcal{G},\mathcal{L})$ with a locally compact Hausdorff unit
space $X$. Regular representations of $\mathcal{G}$ on $\ell^{p}$-spaces (with
$p<\infty$) were considered in [19], and of $(\mathcal{G},\mathcal{L})$ (but
using a different picture of the twist) in [43]. We spell out some details
using our notation. We denote by $\ell^{p}(\mathcal{G},\mathcal{L})$ the
Banach space of all sections of $\mathcal{L}$ for which the norm
$\|f\|_{p}=(\sum_{\gamma\in\mathcal{G}}|f(\gamma)|^{p})^{1/p}$ when $p<\infty$
and $\|f\|_{\infty}=\sup_{\gamma\in\mathcal{G}}|f(\gamma)|$ when $p=\infty$,
is finite. We have an isometric isomorphism
$\ell^{p}(\mathcal{G},\mathcal{L})\cong\ell^{p}(\mathcal{G})$.

######  Proposition 5.1.

For each $p\in[1,\infty]$ the formula

$\Lambda_{p}(f)\xi(\gamma):=(f*\xi)(\gamma)=\sum_{r(\eta)=r(\gamma)}f(\eta)\cdot\xi(\eta^{-1}\gamma),\qquad
f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}),\
\xi\in\ell^{p}(\mathcal{G},\mathcal{L}),$

defines a representation $\Lambda_{p}:F(\mathcal{G},\mathcal{L})\to
B(\ell^{p}(\mathcal{G},\mathcal{L}))$, which is injective on
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, and

  1. (1)

for $p\in(1,\infty)$ we have
$\|\Lambda_{p}(f)\|\leq\|f\|_{*d}^{1/p}\cdot\|f\|_{*r}^{1/q}\leq\|f\|_{I}$,
while $\|\Lambda_{1}(f)\|=\|f\|_{*d}$ and
$\|\Lambda_{\infty}(f)\|=\|f\|_{*r}$, for all
$f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$;

  2. (2)

putting
$F^{p}_{\operatorname{r}}(\mathcal{G},\mathcal{L}):=\overline{\Lambda_{p}(\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}))}$,
we have
$F^{p}_{\operatorname{r}}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F^{q}_{\operatorname{r}}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$, with
the isometric isomorphism induced by the map $f\mapsto\widecheck{f}$. In
particular, $F^{p}_{\operatorname{r}}(\mathcal{G})^{\operatorname{op}}\cong
F^{q}_{\operatorname{r}}(\mathcal{G})$.

###### Proof.

It is routine to check that the formula in the assertion gives a well defined
homomorphism $\Lambda_{p}:\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})\to
B(\ell^{p}(\mathcal{G},\mathcal{L}))$, and
$\|\Lambda_{p}(f)\|\leq\|f\|_{\infty}$ for any $f\in C_{c}(U,\mathcal{L})$,
$U\in\operatorname{Bis}(\mathcal{G})$. Hence $\Lambda_{p}$ is
$\|\cdot\|_{\max}$-contractive and thus it uniquely extends to a
representation of $F(\mathcal{G},\mathcal{L})$, see also Remark 5.2 below. For
each $\gamma\in\mathcal{G}$ choose a norm one element $1_{\gamma}\in
L_{\gamma}$ and treat it as a section of $\mathcal{L}$ which is zero at
$\eta\neq\gamma$. Then $\\{1_{\gamma}\\}_{\gamma\in\mathcal{G}}$ is a Schauder
basis for $\ell^{p}(\mathcal{G},\mathcal{L})$ and for any
$f\in\mathfrak{C}_{c}(\mathcal{G})$ we have

$\|\Lambda_{p}(f)1_{\gamma}\|_{p}=\begin{cases}\left(\sum_{d(\eta)=d(\gamma)}|f(\eta\gamma^{-1})|^{p}\right)^{1/p},&\text{
if $p<\infty$}\\\ \sup_{d(\eta)=d(\gamma)}|f(\eta\gamma^{-1})|,&\text{ if
$p=\infty$}.\end{cases}$

This implies that $\Lambda_{p}$ is injective on
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$. Also using this one sees that
$\|\Lambda_{1}(f)\|=\|f\|_{*d}$ and $\|\Lambda_{\infty}(f)\|=\|f\|_{*r}$. Then
the inequalities
$\|\Lambda_{p}(f)\|\leq\|f\|_{*d}^{1/p}\cdot\|f\|_{*r}^{1/q}\leq\|f\|_{I}$
follow from the Riesz–Thorin interpolation theorem. This proves (1). To see
(2) it suffices to note that under the standard isomorphism
$\ell^{p}(\mathcal{G},\mathcal{L})^{\prime}\cong\ell^{q}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$
given by the paring
$\langle\xi,\eta\rangle:=\sum_{\gamma\in\mathcal{G}}\xi(\gamma)\eta(\gamma)$,
for $\xi\in\ell^{p}(\mathcal{G},\mathcal{L})$,
$\eta\in\ell^{q}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$, we have
$\Lambda^{p}(f)^{\prime}=\Lambda^{q}(\widecheck{f})$ for all
$f\in\mathfrak{C}_{c}(\mathcal{G})$ (note that $\xi(\gamma)\eta(\gamma)\in
L_{r(\gamma)}=\mathbb{F}$ because the bundle $\mathcal{L}$ is trivial over
$X$). ∎

######  Remark 5.2.

Let
$\tilde{h}:\operatorname{Bis}(\mathcal{G})\to\operatorname{PHomeo}(\mathcal{G})$
be the extension of the canonical action described in Example 2.18. The
formulas

$\pi(a)\xi(\gamma):=a(r(\gamma))\xi(\gamma),\qquad
v_{U}\xi(\gamma):=\begin{cases}\xi(\tilde{h}_{U^{*}}(\gamma)),&\gamma\in
r^{-1}(r(U)),\\\ 0,&\text{ otherwise},\end{cases}$

where $a\in C_{0}(X)$, $\xi\in\ell^{p}(\mathcal{G})$, $\gamma\in\mathcal{G}$,
$U\in\operatorname{Bis}(\mathcal{G})$, define a covariant representation
$(\pi,v)$ of the corresponding inverse semigroup action on $C_{0}(X)$ on
$\ell^{p}(\mathcal{G})$, and $\Lambda_{p}=\pi\rtimes v:F(\mathcal{G})\to
B(\ell^{p}(\mathcal{G}))$. More generally, if $(\alpha,u)$ is a twisted
inverse semigroup action of $S$ on $C_{0}(X)$ and $(\mathcal{G},\mathcal{L})$
is the associated twisted groupoid, then the formulas

$\pi(a)\xi[s,x]=a\big{(}h_{s}(x)\big{)}\xi([s,x]),\qquad
v_{t}\xi[s,x]=\begin{cases}u(t,t^{*}s)\big{(}h_{s}(x)\big{)}\xi\big{(}[t^{*}s,x]\big{)},&x\in
h_{s}^{-1}(X_{t}),\\\ 0,&\text{otherwise},\end{cases}$

where $a\in C_{0}(X)$, $\xi\in\ell^{p}(\mathcal{G})$, $x\in X_{s^{*}}$,
$t,s\in S$, define a covariant representation of $(\alpha,u)$ such that
$\pi\rtimes v:F(\mathcal{G},\mathcal{L})\to B(\ell^{p}(\mathcal{G}))$ is
equivalent to $\Lambda_{p}:F(\mathcal{G},\mathcal{L})\to
B(\ell^{p}(\mathcal{G},\mathcal{L}))$.

######  Definition 5.3.

We call $F^{p}_{\operatorname{r}}(\mathcal{G},\mathcal{L})$ the _reduced
$L^{p}$-operator algebra of $(\mathcal{G},\mathcal{L})$_.

######  Remark 5.4.

This definition is consistent with those in [19], [36] and [43] (where it is
assumed that $p<\infty$). Moreover,
$F^{2}_{\operatorname{r}}(\mathcal{G},\mathcal{L})=C_{\operatorname{r}}^{*}(\mathcal{G},\mathcal{L})$
is the reduced $C^{*}$-algebra of $(\mathcal{G},\mathcal{L})$, and we have
$F^{1}_{\operatorname{r}}(\mathcal{G},\mathcal{L})\cong
F_{d_{*}}(\mathcal{G},\mathcal{L})$ and
$F^{\infty}_{\operatorname{r}}(\mathcal{G},\mathcal{L})\cong
F_{r_{*}}(\mathcal{G},\mathcal{L})$ by Proposition 5.1(1).

We now use our main results to prove a general theorem for representations of
groupoid Banach algebras in $L^{p}$-operator algebras.

######  Theorem 5.5.

Let $p,q\in[1,\infty]$ with $1/p+1/q=1$ and let
$\psi:F^{S}(\mathcal{G},\mathcal{L})\to B(L^{p}(\mu))$ be a representation for
some measure $\mu$ and some inverse semigroup
$S\subseteq\operatorname{Bis}(\mathcal{G})$ covering $\mathcal{G}$. If
$\mathbb{F}=\mathbb{C}$, or $p=2$, or if $\psi(C_{0}(X))$ consists of
multiplication operators, then

$\|\psi(f)\|\leq\|f\|_{*d}^{1/p}\,\|f\|_{*r}^{1/q}\leq\|f\|_{I},\qquad
f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}).$

If $p=\infty$, the estimate $\|\psi(f)\|\leq\|f\|_{*r}$ holds also when
$L^{\infty}(\mu)$ is replaced by $C_{0}(\Omega)$ for a locally compact
Hausdorff space $\Omega$.

######  Remark 5.6.

We do not know if the above assertion holds for $\mathbb{F}=\mathbb{R}$
without any further restrictions. The reason is that representations as in
Example 2.13 may occur.

We first note that when $p<\infty$ we may assume that $\psi$ is non-
degenerate.

######  Proposition 5.7.

Let $p\in[1,\infty)$. For any representation
$\psi:F^{S}(\mathcal{G},\mathcal{L})\to B(L^{p}(\mu))$ there is an isometric
isomorphism $\Phi:\overline{\psi(F^{S}(\mathcal{G},\mathcal{L}))L^{p}(\mu)}\to
L^{p}(\overline{\mu})$, where $\overline{\mu}$ is localizable, and then the
formula
$\overline{\psi}(f)=\Phi(\psi(f)|_{\overline{\psi(F^{S}(\mathcal{G},\mathcal{L}))L^{p}(\mu)}})$,
$f\in F(\mathcal{G},\mathcal{L})$, defines a non-degenerate representation
$\overline{\psi}:F^{S}(\mathcal{G},\mathcal{L})\to B(L^{p}(\overline{\mu}))$
such that $\|\overline{\psi}(f)\|=\|\psi(f)\|$ for all $f\in
F^{S}(\mathcal{G},\mathcal{L})$.

###### Proof.

By Lemma 4.13, $F^{S}(\mathcal{G},\mathcal{L})$ and $C_{0}(X)$ have a common
contractive approximate unit $\\{\mu_{i}\\}_{i}$. In particular,
$\overline{\psi(F^{S}(\mathcal{G},\mathcal{L}))L^{p}(\mu)}=\overline{\psi(C_{0}(X))L^{p}(\mu)}$
and as $L^{p}(\mu)$ does not contain an isomorphic copy of $c_{0}$ (see [38,
2.6]) there is a norm one projection
$P:L^{p}(\mu)\to\overline{\psi(C_{0}(X))L^{p}(\mu)}$ by [38, Corollary 3.10].
Thus there is a measure $\overline{\mu}$, which can be chosen to be
localizable, and an isometric isomorphism
$\Phi:\overline{\psi^{S}(F(\mathcal{G},\mathcal{L}))L^{p}(\mu)}\to
L^{p}(\overline{\mu})$ by [75, Theorem 6]. Also for all $f\in
F^{S}(\mathcal{G},\mathcal{L})$ we have $\|\psi(f)\|=\|\psi(f)P\|$ because
$\|\psi(f)\xi\|=\lim_{i}\|\psi(f)\psi(\mu_{i})\xi\|=\lim_{i}\|\psi(f)P\psi(\mu_{i})\xi\|\leq\|\psi(f)P\|\|\xi\|$.
This implies the assertion. ∎

######  Remark 5.8.

Proposition 5.7 fails when $p=\infty$ and $X$ is not compact, see Corollary
2.14. We do not know whether it extends to $C_{0}$ or Lindenstrauss spaces.

###### Proof of Theorem 5.5.

We may assume that $S$ is a wide inverse subsemigroup of $S(\mathcal{L})$.
Indeed, by (4.7) we have a canonical representation $F^{\widetilde{S}\cap
S(\mathcal{L})}(\mathcal{G},\mathcal{L})\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}F^{S}(\mathcal{G},\mathcal{L})$.
Thus composing $\psi$ with this representation, we may replace $S$ with
$\widetilde{S}\cap S(\mathcal{L})$. Then there is a twisted action
$(\alpha,u)$ of $S$ that models $(\mathcal{G},\mathcal{L})$, see Lemma 4.23.
Thus we have $\mathcal{G}=S\ltimes_{h}X$ where
$h:S\to\operatorname{PHomeo}(X)$ is an inverse semigroup action and
$\alpha_{t}:C_{0}(X_{t^{*}})\to C_{0}(X_{t})$; $\alpha_{t}(a)=a\circ
h_{t^{*}}$, for $a\in C_{0}(X_{t^{*}})$, $t\in S$. Also
$F^{S}(\mathcal{G},\mathcal{L})\cong C_{0}(X)\rtimes_{(\alpha,u)}S$ and
$\psi=\pi\rtimes v$ for a covariant representation $(\pi,v)$ of $(\alpha,u)$
in $B(L^{p}(\mu))$ by Theorem 4.27. We consider the following three cases.

(1) Assume that $p<\infty$ and $\psi(C_{0}(X))=\pi(C_{0}(X))$ consists of
multiplication operators, i.e. we have a representation $\pi_{0}:C_{0}(X)\to
L^{\infty}(\mu)$ such that $\pi(a)\xi=\pi_{0}(a)\cdot\xi$ for all $\xi\in
L^{p}(\mu)$, $a\in C_{0}(X)$. Let $f=\sum_{t\in
F}f_{t}\delta_{t}\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, where
$F\subseteq S$ is finite and $f_{t}\in C_{c}(X_{t})$ (see equation (4.10)).
Let $\xi\in L^{p}(\mu)$ and $\eta\in L^{q}(\mu)$. For any $a\in C_{0}(X_{t})$
we have $\int|\pi(a)v_{t}\xi|^{p}d\mu=\int|v_{t}\pi(a\circ
h_{t})\xi|^{p}d\mu\leq\int|\pi(a\circ h_{t})\xi|^{p}d\mu$ because $v_{t}$ is
contractive. Also for any $a\in C_{0}(X)$ and any continuous multiplicative
real function $\varphi$, in particular $\varphi(x)=|x|^{\alpha}$, we have
$\varphi\circ(\pi(a)\xi)=\pi(\varphi\circ a)(\varphi\circ\xi)$ because $\pi$
can be identified with a representation $C_{0}(X)\to L^{\infty}(\mu)$, and the
functional calculus applies (also in the real case [67, 1.1.5]). Thus
$|\pi(a)\xi\cdot\eta|=|\pi(|a|^{1/p})\xi|\,|\pi(|a|^{1/q})\eta|$. Using this
and applying Hölders inequality to the measure $\sum_{t\in F}\mu$, when $p>1$
($q<\infty$), we get

$\begin{split}\left|\int\big{(}\pi\rtimes
v(f)\xi\big{)}\eta\,d\mu\right|&\leq\int\sum_{t\in
F}\big{|}\pi(|f_{t}|^{1/p})v_{t}\xi\big{|}\,\big{|}\pi(|f_{t}|^{1/q})\eta\big{|}\,d\mu\\\
&\leq\left(\sum_{t\in
F}\int\big{|}\pi(|f_{t}|^{1/p})v_{t}\xi\big{|}^{p}\,d\mu\right)^{1/p}\left(\sum_{t\in
F}\int\big{|}\pi(|f_{t}|^{1/q})\eta\big{|}^{q}\,d\mu\right)^{1/q}\\\
&\leq\left(\int\pi\left(\sum_{t\in F}|f_{t}|\circ
h_{t}\right)|\xi|^{p}\,d\mu\right)^{1/p}\left(\int\pi\left(\sum_{t\in
F}|f_{t}|\right)|\eta|^{q}\,d\mu\right)^{1/q}\\\ &\leq\left\|\sum_{t\in
F}|f_{t}|\circ h_{t}\right\|_{\infty}^{1/p}\,\|\xi\|_{p}\,\left\|\sum_{t\in
F}|f_{t}|\right\|_{\infty}^{1/q}\,\|\eta\|_{q}\\\
&=\|f\|_{*d}^{1/p}\,\|f\|_{*r}^{1/q}\,\|\xi\|_{p}\,\|\eta\|_{q}.\end{split}$

This implies that $\|\pi\rtimes v(f)\|\leq\|f\|_{*d}^{1/p}\,\|f\|_{*r}^{1/q}$.
For $p=1$ the proof is even simpler, as then

$\left|\int\pi\rtimes v(f)\xi\,d\mu\right|\leq\sum_{t\in
F}\int\big{|}v_{t}\pi(|f_{t}\circ h_{t}|)\xi\big{|}\,d\mu\leq\sum_{t\in
F}\int\pi(|f_{t}\circ h_{t}|)\big{|}\xi\big{|}\leq\left\|\sum_{t\in
F}|f_{t}|\circ h_{t}\right\|_{\infty}\|\xi\|_{1}.$

(2) Assume that $p=2$. This case is reduced to the previous one by Lemma 2.16.

(3) Assume that $\psi:F^{S}(\mathcal{G},\mathcal{L})\to B(C_{0}(\Omega))$ and
$\psi(C_{0}(X))=\pi(C_{0}(X))$ consists of multiplication operators. The dual
space $E^{\prime}$ of $E:=C_{0}(\Omega)$ is naturally an isomorphic as a
Banach lattice to an $L^{1}$-space $L^{1}(\mu)$. Thus the formula
$\tilde{\psi}(f):=\psi(\widecheck{f})^{\prime}$,
$f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}^{\operatorname{op}})^{\operatorname{op}}$,
defines a representation
$\tilde{\psi}:F^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})^{\operatorname{op}}\to
B(E^{\prime})$ with $\tilde{\psi}|_{C_{0}(X)}$ positive. This representation
reduces to a representation $\tilde{\psi}_{\operatorname{r}}$ on the Banach
sublattice $\tilde{\psi}(C_{0}(X)^{+})L^{1}(\mu)$, which is an AL-space, and
thus it is isomorphic to the Banach lattice $L^{1}(\tilde{\mu})$ for some
localizable $\tilde{\mu}$. Thus
$\tilde{\psi}_{\operatorname{r}}:F^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})^{\operatorname{op}}\to
B(L^{1}(\tilde{\mu}))$ becomes a non-degenerate representation with
$\tilde{\psi}_{\operatorname{r}}|_{C_{0}(X)}$ positive. Hence
$\tilde{\psi}_{\operatorname{r}}|_{C_{0}(X)}$ acts by multiplication
operators, by Theorem 2.12. Also as in the proof of Proposition 5.7 one gets
that $\|\tilde{\psi}(f)\|=\|\tilde{\psi}_{\operatorname{r}}(f)\|$ for
$f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}^{\operatorname{op}})^{\operatorname{op}}$.
In particular,
$\|\psi(f)\|=\|\tilde{\psi}_{\operatorname{r}}(\widecheck{f})\|$ for
$f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$. Therefore applying the case
(1) to $\tilde{\psi}_{\operatorname{r}}$ we get
$\|\psi(f)\|=\|\tilde{\psi}_{\operatorname{r}}(\widecheck{f})\|\leq\|\widecheck{f}\|_{*r}=\|f\|_{*d}$.

(4) Assume that $\mathbb{F}=\mathbb{C}$. By step (2) we may assume that $p\neq
2$. If $p<\infty$ then using Proposition 5.7 we may assume that $\psi$ is non-
degenerate, and then by Theorem 2.12, $\psi$ necessarily acts by
multiplication operators. Hence the assertion follows from step (1). The
estimate $\|\psi(f)\|\leq\|f\|_{*r}$ for $p=\infty$ follows from the estimate
$\|\psi(f)\|\leq\|f\|_{*d}$ for $p=1$ and Corollary 4.18. ∎

###  5.1. Full $L^{p}$-groupoid algebras and spatial covariant representations

The following definition is motivated by Theorem 5.5 and the desire to have a
consistent theory in both complex and real case.

######  Definition 5.9.

Let $p\in[1,+\infty]$. The _full $L^{p}$-operator algebra_ is the completion
$F^{p}(\mathcal{G},\mathcal{L})=\overline{\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{L^{p}}}$
in the norm $\|f\|_{L^{p}}:=\sup_{\psi\in\mathcal{R}}\|\psi(f)\|$, where
$\mathcal{R}$ consists of homomorphisms
$\psi:\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})\to B(E)$ such that:

  1. (R1)

$E=L^{p}(\mu)$ for some measure $\mu$;

  2. (R2)

$\psi$ is $\|\cdot\|_{\max}^{S}$-contractive for some inverse semigroup
$S\subseteq\operatorname{Bis}(\mathcal{G})$ covering $\mathcal{G}$;

  3. (R3)

$\psi|_{C_{0}(X)}$ acts by multiplication operators.

######  Theorem 5.10.

Let $p,q\in[1,\infty]$ with $1/p+1/q=1$.

  1. (1)

If $\mathbb{F}=\mathbb{C}$ or $p=2$, condition (R3) in Definition 5.9 can be
dropped, so $F^{p}(\mathcal{G},\mathcal{L})$ is universal for all
$\|\cdot\|_{\max}^{S}$-contractive representions into $L^{p}$-operator
algebras.

  2. (2)

The norm $\|\cdot\|_{L^{2}}=\|\cdot\|_{C^{*}\max}$ is the largest $C^{*}$-norm
on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, so
$F^{2}(\mathcal{G},\mathcal{L})=C^{*}(\mathcal{G},\mathcal{L})$.

  3. (3)

We have
$F^{1}(\mathcal{G},\mathcal{L})=F^{1}_{\operatorname{r}}(\mathcal{G},\mathcal{L})=F_{d_{*}}(\mathcal{G},\mathcal{L})$
and
$F^{\infty}(\mathcal{G},\mathcal{L})=F^{\infty}_{\operatorname{r}}(\mathcal{G},\mathcal{L})=F_{r_{*}}(\mathcal{G},\mathcal{L})$.

  4. (4)

Condition (R2) in Definition 5.9 can be replaced by
$\|\cdot\|_{I}$-contractiveness of $\psi$ (which alone is stronger than (R2))
as we have
$\|f\|_{L_{p}}\leq\|f\|_{L^{1}}^{1/p}\,\|f\|_{L^{\infty}}^{1/q}\leq\|f\|_{I}$
for $f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$.

  5. (5)

If $p=\infty$ condition (R1) in Definition 5.9 can be replaced by
“$E=C_{0}(\Omega)$ for some locally compact Hausdorff space”, and if
$\mathbb{F}=\mathbb{C}$ we can replace (R1) by “$E$ is a Lindenstrauss space”
(and drop condition (R3)).

  6. (6)

$F^{p}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F^{q}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$ and
$F^{p}(\mathcal{G},\mathcal{L})\stackrel{{\scriptstyle\mathrm{anti}}}{{\cong}}F^{q}(\mathcal{G},\mathcal{L})$
where these isometric maps are induced by the involutions $\widecheck{\,}$ and
$*$ on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, respectively.

###### Proof.

We get (2) by Lemma 2.16 and Corollary 4.29, which also implies (1) for $p=2$.
For $p\in[1,\infty)\setminus\\{2\\}$ and $\mathbb{F}=\mathbb{C}$ we get (1) by
Proposition 5.7 and Theorem 2.12. Then (1) for $p=\infty$ follows from case
$p=1$ and Corollary 4.18. (3) and (4) follow from Theorem 5.5 combined with
Proposition 5.1(1). (5) follows from Theorem 5.5, and for
$\mathbb{F}=\mathbb{C}$ from Corollary 4.29. The isomorphisms in (6) for
$p=1,\infty$ and $q=\infty,1$ follow from (3) and Proposition 5.1(2). Assuming
$p\in(1,\infty)$, $F^{p}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F^{q}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$ holds by Corollary 4.16.
In fact, if $\mathbb{F}=\mathbb{R}$ we need to modify the proof of Proposition
4.15 by considering only representations under where $\psi(C_{0}(X))$ act by
multiplication operators, but under the standard isomorphism
$L^{p}(\mu)^{\prime}\cong L^{q}(\mu)$, an operator $a\in B(L^{p}(\mu))$ is a
multiplication operator if and only if its dual $a^{\prime}\in B(L^{q}(\mu))$
is a multiplication operator, and so the proof works. The isomorphism
$F^{p}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F^{q}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$ translates to anti-
isomorphism
$F^{p}(\mathcal{G},\mathcal{L})\stackrel{{\scriptstyle\mathrm{anti}}}{{\cong}}F^{q}(\mathcal{G},\mathcal{L})$,
see [17, Remark 3.2]. ∎

We may view representations of $F^{p}(\mathcal{G},\mathcal{L})$ as
representations of $F^{S}(\mathcal{G},\mathcal{L})$ for any inverse semigroup
$S\subseteq\operatorname{Bis}(\mathcal{G})$ covering $\mathcal{G}$. In
particular, we may use our disintegration theorem to treat them as covariant
representations for an inverse semigroup action. To this end, until the end of
this section

_we fix a twisted action $(\alpha,u)$ that models $(\mathcal{G},\mathcal{L})$_

as in Lemma 4.23. Thus we have $\mathcal{G}=S\ltimes_{h}X$ where
$h:S\to\operatorname{PHomeo}(X)$ is an inverse semigroup action and
$\alpha_{t}:C_{0}(X_{t^{*}})\to C_{0}(X_{t^{*}})$; $\alpha_{t}(a)=a\circ
h_{t^{*}}$, for $a\in C_{0}(X_{t})$, $t\in S$. We also fix for a while a
localizable measure space $(\Omega,\Sigma,\mu)$ and discuss covariant
representations of $\alpha$ that are given by data coming from the space
$(\Omega,\Sigma,\mu)$.

######  Definition 5.11.

A covariant representation of $(\alpha,u)$ on $L^{p}(\mu)$, $p\in[1,\infty]$,
as in Definition 3.18, is called _spatial_ if $\pi:C_{0}(X)\to B(L^{p}(\mu))$
acts by multiplication operators on $L^{p}(\mu)$ and
$v:S\to\operatorname{SPIso}(L^{p}(\mu))$ takes values in the inverse semigroup
of spatial partial isometries (Definition 2.25).

######  Proposition 5.12.

If $p\in(1,\infty)$ and $(\pi,v)$ is a covariant representation of
$(\alpha,u)$ on $L^{p}(\mu)$ where $\pi$ is given by multiplication operators,
then $(\pi,v)$ is spatial. If $\mathbb{F}=\mathbb{C}$ and
$p\in(1,\infty)\setminus\\{2\\}$ then every non-degenerate covariant
representation $(\pi,v)$ of $(\alpha,u)$ on $L^{p}(\mu)$ is spatial.

###### Proof.

By Lemma 3.22, for $e\in E(S)$ we have $v_{e}=B_{*}\mathchar
45\relax\lim\pi(\mu_{i}^{e})$, where $\\{\mu_{i}^{e}\\}_{i}$ is an approximate
unit in $C_{0}(X_{e})$ and
$B_{*}:=L^{q}(\mu)\mathbin{\widehat{\otimes}}L^{p}(\mu)$. This means that
$\int\eta(v_{e}\xi)\,d\mu=\lim_{i}\int\eta\pi(\mu_{i}^{e})\xi\,d\mu$ for all
$(\eta,\xi)\in L^{q}(\mu)\times L^{p}(\mu)$. Using that $\pi(\mu_{i}^{e})$ are
operators of multiplication, the latter condition readily implies that $v_{e}$
commutes with projections given by characteristic functions. Thus by [8,
Proposition 4.9], $v_{e}$ is an $L^{p}$-projection. Hence by Theorem 2.27, the
map $v$ takes values in $\operatorname{SPIso}(L^{p}(\mu))$, and so $(\pi,v)$
is spatial. The second part now follows from Theorem 2.12. ∎

In what follows we use the notation of Subsection 2.4.

######  Definition 5.13.

A _spatial covariant triple_ $(\pi_{0},\Phi,\omega)$ for $(\alpha,u)$ and
$\mu$ consists of a representation $\pi_{0}:C_{0}(X)\to L^{\infty}(\mu)$, an
inverse semigroup $\Phi=\\{[\Phi_{s}]\\}_{s\in
S}\subseteq\operatorname{PAut}([\Sigma])$ of partial set automorphisms
$\Phi_{s}:\Sigma_{D_{s}}\to\Sigma_{D_{s^{*}}}$, $D_{s},D_{s^{*}}\in\Sigma$,
$s\in S$, satisfying $[\Phi_{s}]\circ[\Phi_{t}]=[\Phi_{st}]$ for $s,t\in S$,
and a family (cocycle) $\omega=\\{\omega_{s}\\}_{s\in S}$ of partial unimodal
maps $\omega_{s}\in UL^{\infty}(\mu|_{D_{s^{*}}})$, $s\in S$. These must
satisfy:

  1. (SCT1)

$T_{\Phi_{t}}(\pi_{0}(a))=\pi_{0}(\alpha_{t}(a))$ for all $a\in
C_{0}(X_{t*})$, $t\in S$;

  2. (SCT2)

$\pi_{0}(\mu_{i}^{e})1_{D}\nearrow 1_{D}$, for every measurable $D\subseteq
D_{e}$ with $\mu(D)<\infty$, every positive approximate unit
$\\{\mu_{i}^{e}\\}_{i}\subseteq C_{0}(X_{e})$ and all $e\in E(S)$;

  3. (SCT3)

$\pi_{0}(a)\omega_{s}T_{\Phi_{s}}(\omega_{t})=\pi_{0}(au(s,t))\omega_{st}$
$\mu$-almost everywhere, for $s,t\in S$, $a\in C_{0}(X_{st})$.

We say that $(\pi_{0},\Phi,\omega)$ is _non-degenerate_ if
$\pi_{0}(\mu_{i})1_{D}\nearrow 1_{D}$, for every measurable $D$ with
$\mu(D)<\infty$ and a positive contractive approximate unit
$\\{\mu_{i}\\}_{i}\subseteq C_{0}(X)$.

######  Proposition 5.14.

For each $p\in[1,\infty)$ we have a bijective correspondence between spatial
covariant representations $(\pi,v)$ of $(\alpha,u)$ on $L^{p}(\mu)$ and
spatial covariant triples $(\pi_{0},\Phi,\omega)$ for $(\alpha,u)$ and $\mu$.
This correspondence is given by

$\big{(}\pi(a)\xi\big{)}(\omega)=\pi_{0}(a)(\omega)\xi(\omega),\quad
v_{t}:=\omega_{t}\left(\frac{d\mu\circ\Phi_{t*}}{d\mu|_{D_{\Phi_{t^{*}}}}}\right)^{\frac{1}{p}}T_{\Phi_{t}},\quad
a\in C_{0}(X),\ \xi\in L^{p}(\mu),\ t\in S.$

The representation $(\pi,v)$ is non-degenerate if and only if the
corresponding triple $(\pi_{0},\Phi,\omega)$ is non-degenerate.

###### Proof.

A representation $\pi:C_{0}(X)\to B(L^{p}(\mu))$ acting by multiplication
operators is equivalent to a homomorphism $\pi_{0}:C_{0}(X)\to
L^{\infty}(\mu)$. By Proposition 2.24, every map
$v:S\to\operatorname{SPIso}(L^{p}(\mu))$ is given by
$v_{t}=\omega_{t}(\frac{d\mu\circ\Phi^{*}}{d\mu|_{D_{\Phi^{*}}}})^{\frac{1}{p}}T_{\Phi_{t}}$,
$t\in S$, where $\Phi=\\{[\Phi_{s}]\\}_{s\in
S}\subseteq\operatorname{PAut}([\Sigma])$ and $\omega_{t}\in
UL^{\infty}(\mu|_{D_{t^{*}}})$, $t\in S$. The final statement of Proposition
2.24 (and uniqueness of the presentation in the Banach–Lamperti theorem, see
Proposition 2.4) tells us that $\pi(a)v_{s}v_{t}=\pi(au(s,t))v_{st}$ holds for
all $s,t\in S$, $a\in C_{0}(X_{st})$ if and only if (SCT3) in Definition 5.13
holds and $\Phi=\\{[\Phi_{s}]\\}_{s\in
S}\subseteq\operatorname{PAut}([\Sigma])$ is an inverse semigroup. Condition
(SCT2) in Definition 5.13 is equivalent to the equality
$\overline{\pi(C_{0}(X_{e}))L^{p}(\mu)}=L^{p}(\mu|_{D_{e}})$ for all $e\in
E(S)$. Assuming this, condition (SCT1) in Definition 5.13 is equivalent to
$v_{t}\pi(a)v_{t^{*}}=\pi(\alpha_{t}(a))$ for all $a\in C_{0}(X_{t^{*}})$,
$t\in S$. This gives the assertion. ∎

######  Theorem 5.15.

Let $(\mathcal{G},\mathcal{L})$ be a twisted groupoid and let $(\alpha,u)$ be
any twisted inverse semigroup action that models $(\mathcal{G},\mathcal{L})$.

  1. (1)

For each $p\in[1,\infty]$ there is an isometric representaion $\pi\rtimes
v:F^{p}(\mathcal{G},\mathcal{L})\to B(L^{p}(\mu))$ where $(\pi,v)$ is a
spatial covariant representation of $\alpha$ on $L^{p}(\mu)$, for some
localizable measure $\mu$. If $p<\infty$ or when $X$ is compact, then $\pi$
can be chosen to be non-degenerate.

  2. (2)

For $p\in(1,\infty)$ every representation of $F^{p}(\mathcal{G},\mathcal{L})$
on $L^{p}(\mu)$ where $C_{0}(X)$ acts by multiplication operators, and $\mu$
is localizable, is of the form $\pi\rtimes v$ for a spatial covariant
representation $(\pi,v)$ of $(\alpha,u)$. This gives a bijective
correspondence between such representations of
$F^{p}(\mathcal{G},\mathcal{L})$ on $L^{p}(\mu)$ and spatial covariant triples
for $(\alpha,u)$ and $\mu$.

  3. (3)

For any $p,p^{\prime}\in(1,\infty)$ we have a bijective correspondence between
representations of $F^{p}(\mathcal{G},\mathcal{L})$ and
$F^{p^{\prime}}(\mathcal{G},\mathcal{L})$ on $L^{p}$ and
$L^{p^{\prime}}$-spaces such that $C_{0}(X)$ acts by multiplication operators.
This correspondence matches $\pi_{p}:F^{p}(\mathcal{G},\mathcal{L})\to
B(L^{p}(\mu))$ and
$\pi_{p^{\prime}}:F^{p^{\prime}}(\mathcal{G},\mathcal{L})\to
B(L^{p^{\prime}}(\mu))$, given by

$\pi_{p}(a_{t}\delta_{t})=\pi_{0}(a_{t})\omega_{t}\left(\frac{d\mu\circ\Phi_{t^{*}}}{d\mu|_{D_{\Phi_{t^{*}}}}}\right)^{\frac{1}{p}}T_{\Phi_{t}},\qquad\pi_{p^{\prime}}(a_{t}\delta_{t})=\pi_{0}(a_{t})\omega_{t}\left(\frac{d\mu\circ\Phi_{t^{*}}}{d\mu|_{D_{\Phi_{t^{*}}}}}\right)^{\frac{1}{p^{\prime}}}T_{\Phi_{t}},$

where $a_{t}\in C_{c}(X_{t}),t\in S$, and $(\pi_{0},\Phi,\omega)$ is a non-
degenerate spatial covariant triple for $(\alpha,u)$ and a localizable $\mu$.

If all domains $X_{t}$, $t\in S$, are compact, then statements (2) and (3)
hold for all $p,p^{\prime}\in[1,\infty]$.

###### Proof.

(1). For $p\in\\{1,\infty\\}$, the assertion follows from Theorem 5.10(3). For
$p\in(1,\infty)$ one gets the assertion by taking the $\ell^{p}$-direct sum
over a class of representations $\\{\psi_{a,k}\\}_{a\in
F^{p}(\mathcal{G},\mathcal{L}),k\in\mathbb{N}}$ such that
$\psi_{a,k}:F^{p}(\mathcal{G},\mathcal{L})\to B(L^{p}(\mu))$ where $\mu$ is
localizable, $\psi_{a,k}(C_{0}(X))$ consists of multiplication operators, and
$\|a\|_{F^{p}(\mathcal{G},\mathcal{L})}\leq\|\psi_{a,k}(a)\|+1/k$.

(2). If $p\in(1,\infty)$, Theorem 4.27(3) implies that every representation of
$F^{p}(\mathcal{G},\mathcal{L})$ on $L^{p}(\mu)$ is of the form $\pi\rtimes v$
for a covariant represenation $(\pi,v)$ on the space $L^{p}(\mu)$. If all
$X_{t}$, $t\in S$, are compact, the same holds for any $p\in[1,\infty]$ by the
last part of Theorem 4.27. Assume $\mu$ is localizable. Then $(\pi,v)$ is
necessarily spatial by Proposition 5.12, and so it is given by a spatial
covariant triple by Proposition 5.14. This proves (2). Item (3) now follows
readily. ∎

######  Remark 5.16.

If $\mathbb{F}=\mathbb{C}$, then Theorem 5.15 combined with Theorem 2.12 gives
a description of all non-degenerate representations of
$F^{p}(\mathcal{G},\mathcal{L})$ for $p\in(1,\infty)\setminus\\{2\\}$, and a
bijective correspondence between non-degenerate representations of
$F^{p}(\mathcal{G},\mathcal{L})$ and $F^{p^{\prime}}(\mathcal{G},\mathcal{L})$
for all $p,p^{\prime}\in(1,\infty)\setminus\\{2\\}$. This is all the more
striking, as usually there are no non-zero continuous homomorphisms between
$F^{p}(\mathcal{G},\mathcal{L})$ and $F^{p^{\prime}}(\mathcal{G},\mathcal{L})$
for $p\neq p^{\prime}$.

##  6\. Banach algebras associated to inverse semigroups

Let $S$ be an inverse semigroup. The algebra $\mathbb{F}S$ of formal finite
$\mathbb{F}$-linear combinations of elements in $S$ and multiplication induced
from $S$ is the universal algebra for semigroup homomorphisms of $S$. Since
$S$ is more than just a semigroup, we will consider here completions of
$\mathbb{F}S$ such that the commutative subalgebra $\mathbb{F}E$, generated by
the idempotents $E:=E(S)$, is a uniform Banach algebra. This boils down to an
assumption of _joint contractiveness_ that we introduce in Definition 6.4
below. When applied to tight representations this will give us a useful
description of Banach algebras associated to ample groupoids. Some of our
results here are already new and might be interesting for $C^{*}$-algebraists.

We start by recalling the groupoid models for $\mathbb{F}S$ and $\mathbb{F}E$
from [73]. Recall that the semigroup $E$ is a semilattice: the ‘meet’ and
‘preorder’ are defined by $e\wedge f:=ef$ and $e\leq f$ if and only if $ef=e$.
We write $e<f$ if $e\leq f$ and $e\neq f$. If $E$ has a zero, we say $e$ and
$f$ are orthogonal if $ef=0$. The set $\widehat{E}\subseteq\\{0,1\\}^{E}$ of
all non-zero homomorphisms $\phi:E\to\\{0,1\\}$ is a totally disconnected
locally compact Hausdorff space in the product topology of the Cantor space
$\\{0,1\\}^{E}$ (which is the topology of pointwise convergence). In fact,
putting $Z(e):=\\{\phi\in\widehat{E}:\phi(e)=1\\}$, the sets
$Z(e)\setminus\bigcup_{f\in F}Z(f)$, where $e\in E$ and $F\subseteq eE$ is
finite, form a basis of compact-open sets for $\widehat{E}$. We call
$\widehat{E}$ the _spectrum_ of $E$. A map $\phi:E\to\\{0,1\\}$ is in
$\widehat{E}$ if and only if its support $\operatorname{supp}(\phi):=\\{e\in
E:\phi(e)=1\\}$ is a _filter_ of $E$, i.e. a non-empty, upward-closed and
downward directed and subset of $E$. There is a natural action
$h:S\to\operatorname{PHomeo}(\widehat{E})$ given by

$h_{t}:Z(t^{*}t)\to Z(tt^{*});\ h_{t}(\phi)(e):=\phi(t^{*}et),\qquad t\in S,\
e\in E,\ \phi\in\widehat{E}$

(see [31, Proposition 10.3]). The corresponding transformation groupoid
$\mathcal{G}(S):=S\ltimes_{h}\widehat{E}$ is ample. It is covered by compact
open bisections $U_{t}=\\{[t,\phi]:\phi\in Z(t^{*}t)\\}$, $t\in S$. The
associated Steinberg algebra is

$A(\mathcal{G}(S)):=\operatorname{span}\\{1_{U}:\text{$U\in\operatorname{Bis}(\mathcal{G}(S))$
compact open}\\}=\operatorname{span}\\{1_{U_{t}}:t\in S\\}\subseteq
C_{c}(\mathcal{G}(S)).$

By a homomorphism from the semigroup $S$ to an algebra $B$ we mean a semigroup
homomorphism $v:S\to B$ into the multiplicative semigroup of $B$.

######  Proposition 6.1 ([73, Theorem 5.1]).

The map $t\mapsto 1_{U_{t}}$ determines an isomorphism $\mathbb{F}S\cong
A(\mathcal{G}(S))$. Thus for any $\mathbb{F}$-algebra $B$, the relation
$\psi(1_{U_{t}})=v_{t}$, $t\in S$, establishes a bijective correspondence
between algebra homomorphisms $\psi:A(\mathcal{G}(S))\to B$ and homomorphisms
$v:S\to B$.

We have the following commutative subalgebra of $A(\mathcal{G}(S))$:

$A(\widehat{E}):=\operatorname{span}\\{1_{U}:\text{$U\subseteq\widehat{E}$
compact open}\\}=\operatorname{span}\\{1_{Z(e)}:e\in E\\}\subseteq
C_{c}(\widehat{E})\subseteq C_{c}(\mathcal{G}(S)).$

Proposition 6.1 applied to $E$ gives a bijective correspondence between
algebra homomorphisms $\pi:A(\widehat{E})\to B$ and homomorphisms $v:E\to B$.
We now aim to characterise injectivity of $\pi$ in terms of $v$. For finite
$F\subseteq E$ we write $\bigwedge{F}:=\bigwedge_{e\in F}e=\prod_{e\in F}e$.

######  Lemma 6.2 (Inclusion-exclusion principle).

Let $v:E\to B$ be a homomorphism into an algebra $B$. For any finite
$F\subseteq E$ and $F_{0}\subseteq F$ put

(6.11) $P_{F_{0}}^{F}:=\prod_{f\in F\setminus
F_{0}}(v_{\bigwedge{F_{0}}}-v_{\bigwedge{F_{0}}\wedge f})=\sum_{F_{0}\subseteq
G\subseteq F}(-1)^{|G\setminus F_{0}|}v_{\bigwedge{G}},$

with the convention $P_{F}^{F}=v_{\bigwedge{F}}$ and $P_{\emptyset}^{F}=0$.
Then $\\{P_{F_{0}}^{F}\\}_{F_{0}\subseteq F}$ are mutually orthogonal
idempotents such that $v_{e}=\sum_{e\in F_{0}\subseteq F}P_{F_{0}}^{F}$, $e\in
F$.

###### Proof.

This can be proved by induction on $|F|$. If $|F|=1$ the assertion is trivial.
Suppose the assertion holds whenever $|F|<n$, and fix $F$ with $|F|=n$. Fix
also $e_{0}\in F$ and let $H=F\setminus\\{e_{0}\\}$. For any $F_{0}\subseteq
F$ we then have

$P_{F_{0}}^{F}=\begin{cases}P_{F_{0}}^{H}-P_{F_{0}}^{H}v_{e_{0}},&e_{0}\not\in
F_{0},\\\ P_{F_{0}\setminus\\{e_{0}\\}}^{H}v_{e_{0}},&e_{0}\in F_{0},\
F_{0}\neq\\{e_{0}\\},\\\ v_{e_{0}}-\sum_{H_{0}\subseteq
H}P_{H_{0}}^{H},&F_{0}\neq\\{e_{0}\\}.\end{cases}$

By the inductive hypothesis $\\{P_{H_{0}}^{H}\\}_{H_{0}\subseteq H}$ are
mutually orthogonal and $v_{e}=\sum_{e\in H_{0}\subseteq H}P_{H_{0}}^{H}$,
$e\in H$. Using this and the above relations it is readily checked that
$\\{P_{F_{0}}^{F}\\}_{F_{0}\subseteq F}$ are also mutually orthogonal, and
$v_{e}=\sum_{e\in F_{0}\subseteq F}P_{F_{0}}^{F}$, $e\in F$. ∎

######  Corollary 6.3.

Let $\pi:A(\widehat{E})\to B$ be an algebra homomorphism and let $v:E\to B$ be
the corresponding semigroup homomorphism. Then $\pi$ is injective if and only
if $\prod_{f\in F}(v_{e}-v_{f})\neq 0$ for every $e\in E\setminus\\{0\\}$ and
finite $F\subseteq eE\setminus\\{e\\}$.

###### Proof.

Idempotents (6.11) corresponding to idempotents $\\{1_{Z(f)}\\}_{f\in
F}\subseteq A(E)$ are characteristic functions of basic cylinder sets
$Z\big{(}\bigwedge F_{0}\big{)}\setminus\bigcup_{f\in F\setminus
F_{0}}Z\big{(}\bigwedge F_{0}\wedge f\big{)}$. Hence, by Lemma 6.2, every
element in $A(E)$ is a finite linear combination of mutually orthogonal
idempotents of the form $1_{Z(e)\setminus\bigcup_{f\in F}Z(f)}$, where $e\in
E\setminus\\{0\\}$ and $F\subseteq eE\setminus\\{e\\}$, which are mapped by
$\pi$ to mutually orthogonal idempotents of the form $\prod_{f\in
F}(v_{e}-v_{f})$. Moreover, $1_{Z(e)\setminus\bigcup_{f\in F}Z(f)}\neq 0$
because $\operatorname{supp}\phi:=\\{f\in E:f\geq e\\}$ is a filter that
defines $\phi$ that is in $Z(e)\setminus\bigcup_{f\in F}Z(f)$. Hence $\pi$ is
injective if and only if it is non-zero on elements of the form
$1_{Z(e)\setminus\bigcup_{f\in F}Z(f)}$. ∎

For representations in Banach algebras we need an analytic condition.

######  Definition 6.4.

Let $B$ be a Banach algebra over $\mathbb{F}$ and let $M\subseteq\mathbb{F}$
be a set. We say that a collection of mutually orthogonal idempotents
$\\{P_{i}\\}_{i\in F}\subseteq B$ is _jointly $M$-contractive_ if

$\|\sum_{i\in F}a_{i}{P_{i}}\|\leq\max_{i\in F}|a_{i}|,\qquad a_{i}\in M,\
i\in F.$

######  Remark 6.5.

The above concept embraces a number of other notions existing in the
literature. For instance, if $B$ is a unital Banach algebra then $\\{P,1-P\\}$
are jointly $\\{0,1\\}$-contractive if and only if $\\{P,1-P\\}\subseteq
B_{1}$, that is $P$ is bicontractive [18], [9]. The projection in Example 2.13
is bicontractive but not an $L^{p}$-projection. If $\mathbb{F}=\mathbb{C}$ and
$\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$, then $\\{P,1-P\\}$ are jointly
$\mathbb{T}$-contractive if and only if $zP+w(1-P)$ are isometries
(contractive invertible elements with contractive inverses), for all
$z,w\in\mathbb{T}$, which by definition means that $P$ is _bicircular_ , see
[72]. Every bicircular projection is bicontractive but not conversely, see
[72]. In fact, a projection $P$ is bicircular if and only if it is hermitian
[45], and so bicontractive proections in Example 2.13 are not bicircular by
Theorem 2.27. For any $p\in[1,\infty]$ and any Banach space $E$, mutually
orthogonal $L^{p}$-projections $\\{P_{i}\\}_{i\in F}\subseteq B(E)$ are
jointly $\mathbb{F}$-contractive (the Banach algebra they generate embeds
isometrically into $c_{0}(F)$).

######  Definition 6.6.

A _representation of the inverse semigroup $S$ in a Banach algebra_ $B$ is a
semigroup homomorphism $v:S\to B_{1}$ such that for every finite $F\subseteq
E$ the idempotents $\\{P_{F_{0}}^{F}\\}_{F_{0}\subseteq F}$ given by (6.11)
are jointly $\mathbb{F}$-contractive.

######  Lemma 6.7.

Let $v:S\to B_{1}$ be a semigroup homomorphism into the contractive elements
in a Banach algebra $B$, and let
$A:=\operatorname{\overline{span}}\\{v_{e}:e\in E\\}\subseteq B$. The
following are equivalent:

  1. (1)

$v$ is a representation of $S$ in $B$;

  2. (2)

the map $1_{Z(e)}\mapsto v_{e}$ extends to a representation
$\pi:C_{0}(\widehat{E})\to A\subseteq B$;

  3. (3)

$A\cong C_{0}(X)$ for a closed subset $X\subseteq\widehat{E}$;

  4. (4)

$A$ embeds isometrically into a $C^{*}$-algebra.

The above equivalent conditions always hold when $B=B(K)$ and $v|_{E}$ takes
values in the $L^{p}$-projections on a Banach space $K$, for some fixed
$p\in[1,\infty]\setminus\\{2\\}$. In particular, any homomorphism
$v:S\to\operatorname{SPIso}(L^{p}(\mu))\subseteq B(L^{p}(\mu))$, where
$p\in[1,\infty]$, is a representation of $S$, and any $*$-homomorphism $v:S\to
B_{1}$ where $B$ is a $C^{*}$-algebra is a representation of $S$.

###### Proof.

(1)$\implies$(2). By Proposition 6.1, the map $1_{Z(e)}\mapsto v_{e}$ extends
to an algebra homomorphism $\pi:A(\widehat{E})\to A\subseteq B$ given by
$\pi(\sum_{e\in F}a_{e}1_{Z(e)})=\sum_{e\in F}a_{e}P_{e}$, for finite
$F\subseteq E$ and $a_{e}\in\mathbb{F}$, $e\in F$. Let
$\\{P_{F_{0}}^{F}\\}_{F_{0}\subseteq F}$ be the orthogonal idempotents
associated to $F$ in Lemma 6.2. The corresponding idempotents associated to
$e\mapsto 1_{Z(e)}$ are $\\{1_{Z(\bigwedge F_{0})\setminus\bigcup_{f\in
F\setminus F_{0}}Z(\bigwedge F_{0}\wedge f)}\\}_{F_{0}\subseteq F}$. So there
are $b_{F_{0}}\in\mathbb{F}$, $F_{0}\subseteq F$, such that

$\sum_{e\in F}a_{e}1_{Z(e)}=\sum_{F_{0}\subseteq
F}b_{F_{0}}1_{Z\big{(}\bigwedge F_{0}\big{)}\setminus\bigcup_{f\in F\setminus
F_{0}}Z\big{(}\bigwedge F_{0}\wedge f\big{)}}\quad\text{ and }\quad\sum_{e\in
F}a_{e}P_{e}=\sum_{F_{0}\subseteq F}b_{F_{0}}P_{F_{0}}^{F}.$

The joint $\mathbb{F}$-contractiveness of $\\{P_{F_{0}}^{F}\\}_{F_{0}\subseteq
F}$ implies

$\Big{\|}\sum_{e\in F}a_{e}P_{e}\Big{\|}\leq\max_{P_{F_{0}}^{F}\neq
0}|b_{F_{0}}|\leq\max_{Z\big{(}\bigwedge F_{0}\big{)}\setminus\bigcup_{f\in
F\setminus F_{0}}Z\big{(}\bigwedge F_{0}\wedge
f\big{)}\neq\emptyset}|b_{F_{0}}|=\Big{\|}\sum_{e\in
F}a_{e}1_{Z(e)}\Big{\|}_{\infty}.$

Hence the map $\pi:A(\widehat{E})\to A\subseteq B$ is contractive and so it
extends (uniquely) to a representation $\pi:C_{0}(\widehat{E})\to A\subseteq
B$, because $A(\widehat{E})$ is dense in $C_{0}(\widehat{E})$.

Implication (2)$\implies$(3) follows from Proposition 2.10, and
(3)$\implies$(4) is trivial. Condition (4) allows us to assume that
$A\subseteq B(H)$ for a Hilbert space $H$. Then $v_{e}$, $e\in E$, are
orthogonal projections on $H$ (as they are contractive idempotents), so the
idempotents (6.11) are orthogonal projections (as they are self-adjoint
idempotents). In particular, idempotents (6.11) are mutually orthogonal
$L^{2}$-projections, and so they are jointly $\mathbb{F}$-contractive. This
shows (4)$\implies$(1).

When $v|_{E}$ takes values in the $L^{p}$-projections on a Banach space $K$,
for $p\in[1,\infty]\setminus\\{2\\}$, then $v|_{E}$ takes values in the
$p$-Cunningham algebra
$C_{p}(E):=\operatorname{\overline{span}}\mathbb{P}_{p}(E)$ which is
isometrically isomorphic to $C_{0}(X)$, see [8], and hence the equivalent
conditions hold. ∎

######  Definition 6.8.

We define the _universal inverse semigroup Banach algebra_ $F(S)$ as the
completion of $\mathbb{F}S$ in the norm

$\|{\textstyle\sum_{t\in F}}a_{t}t\|_{F(S)}:=\sup\\{\|{\textstyle\sum_{t\in
F}}a_{t}v_{t}\|:\text{$v:S\to B$ is a representation in a Banach algebra
$B$}\\}.$

For $p\in[1,\infty]$ we define the $L^{p}$-operator algebra $F^{p}(S)$ in a
similar way, as the completion of $\mathbb{F}S$ in the maximal norm for
homomorphisms into spatial partial isometries on $L^{p}$-spaces.

######  Theorem 6.9.

Let $\mathcal{G}(S):=S\ltimes_{h}\widehat{E}$ be the associated universal
groupoid for an inverse semigroup $S$ and identify $t\in S$ with
$U_{t}\subseteq\operatorname{Bis}(\mathcal{G})$. We have an isometric
isomorphism $F(S)\cong F^{S}(\mathcal{G}(S))$. In particular, the relations

(6.12) $\psi(1_{U_{t}})=v_{t},\qquad t\in S,$

establish a bijective correspondence between representations
$\psi:F^{S}(\mathcal{G}(S))\to B$ and $v:S\to B_{1}$ in a Banach algebra $B$.
Moreover, $\pi:=\psi|_{C_{0}(\widehat{E})}$ is isometric if and only if for
every $e\in E$ and every finite $F\subseteq eE\setminus\\{e\\}$ we have
$\prod_{f\in F}(v_{e}-v_{f})\neq 0$.

###### Proof.

By the last part of Theorem 4.27 every representation of
$F^{S}(\mathcal{G}(S))$ in $B$ is of the form $\pi\rtimes v$ where
$\pi:C_{0}(\widehat{E})\to B$ is a representation and $v:S\to B_{1}$ is a
semigroup homomorphism such that $\pi(1_{Z(e)})=v_{e}$, $e\in E$, and
$v_{t}\pi(a)v_{t^{*}}=\pi(a\circ h_{t^{*}})$ for all $a\in C(Z(t^{*}t))$,
$t\in S$. In particular, Lemma 6.7 implies that $v:S\to B_{1}$ is a
representation of the inverse semigroup $S$. Conversely, if $v:S\to B_{1}$ is
a representation of the inverse semigroup $S$, then by Proposition 6.1 there
is a unique algebra homomorphism $\psi:A(\mathcal{G}(S))\to B$ such that
$\psi(1_{U_{t}})=v_{t}$, $t\in S$. Putting $\pi:=\psi|_{A(\widehat{E})}$, we
have $v_{t}\pi(a)v_{t^{*}}=\pi(a\circ h_{t^{*}})$ for all $a\in
1_{Z(t^{*}t)}A(\widehat{E})$, $t\in S$. By Lemma 6.7, $\pi:A(\widehat{E})\to
B$ extends to a representation $\pi:C_{0}(\widehat{E})\to B$ and hence the
latter condition holds for all $a\in
C(Z(t^{*}t))=\overline{1_{Z(t^{*}t)}A(\widehat{E})}$. Thus $\psi$ extends to a
representation $\psi=\pi\rtimes v:F^{S}(\mathcal{G}(S))\to B$. This gives the
first part of the assertion.

For the second part recall that basic non-empty open sets are of the form
$Z(e)\setminus\bigcup_{f\in F}Z(f)$ for $e\in E$ and finite $F\subseteq
eE\setminus\\{e\\}$. Moreover, for the corresponding representations we have
$\prod_{f\in F}(v_{e}-v_{f})=\pi(1_{Z(e)\setminus\bigcup_{f\in F}Z(f)})$. Thus
if $\pi$ is injective then $\prod_{f\in F}(P_{e}-P_{f})\neq 0$, see also
Corollary 6.3. If $\pi$ is not isometric then $\ker\pi=C_{0}(U)$ for a non-
empty open set $U\subseteq\widehat{E}$. Hence $\pi$ vanishes on a non-empty
set of the form $1_{Z(e)\setminus\bigcup_{f\in F}Z(f)}$, which implies that
$\prod_{f\in F}(P_{e}-P_{f})=0$. ∎

######  Corollary 6.10.

Let $p\in[1,\infty]$. The isomorphism from Theorem 6.9 descends to an
isometric isomorphism $F^{p}(S)\cong F^{p}(\mathcal{G}(S))$. If
$\mathbb{F}=\mathbb{C}$ then $F^{p}(S)$ is universal for all representations
of $S$ in $L^{p}$-operator algebras, and if $p\neq 2$, then every non-
degenerate representation $\psi:F^{p}(S)\to B(L^{p}(\mu))$, where $\mu$ is
localizable, is given by a semigroup homomorphism
$V:S\to\operatorname{SPIso}(L^{p}(\mu))$ into spatial partial isometries.

###### Proof.

By Theorem 5.15, $F^{p}(\mathcal{G}(S))$ is universal for spatial
representations of $F^{S}(\mathcal{G}(S))$ on $L^{p}(\mu)$, which via Theorem
6.9 correspond to homomorphisms
$v:S\to\operatorname{SPIso}(L^{p}(\mu))\subseteq B(L^{p}(\mu))$, cf. the last
part of Lemma 6.7. This gives $F^{p}(S)\cong F^{p}(\mathcal{G}(S))$. The
remaining part follows from Theorems 5.10(1) and 2.12 ∎

######  Remark 6.11.

When $S$ has a zero it is natural to consider zero-preserving homomorphisms.
The corresponding _contracted algebras_ $\mathbb{F}_{0}S$, $F_{0}^{p}(S)$,
$p\in[1,\infty]$, and $F_{0}(S)$, could be defined as quotients of
$\mathbb{F}S$, $F^{p}(S)$, $p\in[1,\infty]$, and $F(S)$ respectively by the
one-dimensional space generated by $0\in S$. The _contracted groupoid_
$\mathcal{G}_{0}(S)$ modeling these algebras is the transformation groupoid
for the $S$-action restricted to the closed $S$-invariant set
$\widehat{E}\setminus\\{1\\}$.

We now pass to representations of $S$ that exploit the Boolean ring structure
of the range of the semilattice $E$. Here by a _Boolean ring_ (a _generalized
Boolean algebra_ in the nomenclature of [73]) we mean a possibly non-unital
strucutre, so it is a quintuple $(R,0,\vee,\wedge,\setminus)$ where
$\setminus$ is a binary operation (relative complement). We generalize here
Exel’s theory of tight representations [31], or rather its ‘non-unital
version’ [27], to the Banach algebra context. We also characterise injectivity
of the relevant representations, which seems to be a new results even in the
$C^{*}$-algebraic setting.

To this end, we fix again a semilatice $E$ of idempotents in an inverse
semigroup $S$.

######  Definition 6.12.

A _cover_ of $e\in E$ is a finite set $F\subseteq eE$ such that for every
nonzero $z\leq e$ we have $zf\neq 0$ for some $f\in F$. We say that a map
$v:E\to R$ into a Boolean ring $R$ is _tight_ (or _cover-to-join_) if the
element $v_{e}-\bigvee_{f\in F}v_{f}=\bigwedge_{f\in F}(v_{e}\setminus v_{f})$
is zero whenever $F$ covers $e\in E$. A semigroup homomorphism $v:S\to B$ into
an $\mathbb{F}$-algebra $B$ is _tight_ if $\prod_{f\in F}(v_{e}-v_{f})=0$ for
every cover $F$ of $e\in F$ (this is equivalent to saying that $v|_{E}$ is
tight as a map into a Boolean ring of idempotents in
$\operatorname{span}\\{v_{e}:e\in E\\}\subseteq B$).

######  Remark 6.13.

If $E$ has a zero, then $\emptyset$ covers $0$ and so $v_{0}=0$ for every
tight map $v:E\to R$ into a Boolean ring $R$. If $E$ is a Boolean ring, then a
semigroup homomorphism $v:E\to R$ is a Boolean ring homomorphism if and only
if $v$ is tight.

By [31, Theorem 12.9], the set

$\widehat{E}_{\operatorname{tight}}:=\\{\phi:E\to\\{0,1\\}\ \text{is a tight
homomorphism}\\}$

is a closed subset of $\widehat{E}$. In fact,
$\widehat{E}_{{\operatorname{tight}}}$ is the closure of the set of
$\phi\in\widehat{E}$ whose support $\operatorname{supp}(\phi)=\\{e\in
E:\varphi(e)=1\\}$ is an ultrafilter. We call
$\widehat{E}_{{\operatorname{tight}}}$ the _tight spectrum_ of $E$. When
$E=E(S)$ is the semilattice of idempotents in an inverse semigroup $S$ then
$\widehat{E}_{{\operatorname{tight}}}$ is invariant for the $S$-action on
$\widehat{E}$, and hence the corresponding tranformation groupoid
$\mathcal{G}_{\operatorname{tight}}(S):=S\ltimes\widehat{E}_{{\operatorname{tight}}}$,
called the _tight groupoid of $S$_, is the restriction of $\mathcal{G}(S)$ to
the set $\widehat{E}_{{\operatorname{tight}}}$. By [74, Corollary 2.14] the
isomorphism from Proposition 6.1 factors to the isomorphism

(6.13)
$A(\mathcal{G}_{\operatorname{tight}}(S))\cong\mathbb{F}S/\Big{\langle}\prod_{f\in
F}e-f:\text{$F$ covers $e\in E$}\Big{\rangle},$

and this algebra is universal for tight semigroup homomorphisms.

######  Definition 6.14.

The _tight Banach algebra_ $F_{\operatorname{tight}}(S)$ is the completion of
the algebra (6.13) in the maximal norm induced by tight representations
$v:S\to B$ in Banach algebras $B$. Similarly, for $p\in[1,\infty]$, we define
the _tight $L^{p}$-Banach algebra_ $F^{p}_{\operatorname{tight}}(S)$, using
tight representations in spatial partial isometries on $L^{p}$-spaces.

######  Theorem 6.15.

Let $\mathcal{G}_{{\operatorname{tight}}}(S)$ be the tight groupoid for an
inverse semigroup $S$ and identify $t\in S$ with
$U_{t}\subseteq\operatorname{Bis}(\mathcal{G})$. The isomorphism (6.13)
extends to an isometric isomorphism

$F_{\operatorname{tight}}(S)\cong
F^{S}(\mathcal{G}_{\operatorname{tight}}(S))$

and (6.12) gives a bijective correspondence between representations
$\psi:F^{S}(\mathcal{G}_{\operatorname{tight}}(S))\to B$ and tight
representations $v:S\to B_{1}$ in a Banach algebra $B$. Moreover, $\psi$ is
isometric on $C_{0}(\widehat{E}_{\operatorname{tight}})$ if and only if
$\prod_{f\in F}(v_{e}-v_{f})\neq 0$ for every $e\in E$ and finite $F\subseteq
eE$ that does not cover $e$.

###### Proof.

One may mimic the proof of Theorem 6.9, using that the algebras in (6.13) are
universal for cover-to-join maps. In particular, for the second part note that
basic non-empty open sets in $\widehat{E}_{\operatorname{tight}}$ are of the
form $Z(e)\setminus\bigcup_{f\in F}Z(f)\cap\widehat{E}_{\operatorname{tight}}$
for $e\in E$ and finite $F\subseteq eE\setminus\\{e\\}$. Such a set is empty
if and only if $F$ is a cover of $F$. Indeed, if $F$ is not a cover, then
there is $0\neq z\leq e$ with $zf=0$ for all $f\in F$, and taking any
ultrafilter containing $z$ the corresponding
$\phi\in\widehat{E}_{\operatorname{tight}}$ is in $Z(e)\setminus\bigcup_{f\in
F}Z(f)$. ∎

######  Corollary 6.16.

Let $p\in[1,\infty]$. The isomorphism (6.13) extends to an isometric
isomorphism $F^{p}_{\operatorname{tight}}(S)\cong
F^{p}(\mathcal{G}_{\operatorname{tight}}(S)).$ If $\mathbb{F}=\mathbb{C}$,
then $F^{p}_{\operatorname{tight}}(S)$ is a universal for all tight
representations of $S$ in $L^{p}$-operator algebras, and if $p\neq 2$, then
every non-degenerate representation $\psi:F^{p}(S)\to B(L^{p}(\mu))$, where
$\mu$ is localizable, is given by a tight homomorphism
$V:S\to\operatorname{SPIso}(L^{p}(\mu))$ into spatial partial isometries.

###### Proof.

Follow the proof of Corollary 6.10 using Theorem 6.15. ∎

En passant we improve the main result of [32] by removing the second-
countability assumption, and replacing the assumption that the semigroup forms
a basis for the topology by the requirement that it is wide and separates the
points (this slight generalization is useful in applications).

######  Corollary 6.17.

Let $X$ be a totally disconnected locally compact Hausdorff space and let $E$
be a semilattice consisting of compact open sets that separate the points of
$X$ and cover $X$. The pairing $\phi(e)=1_{e}(x)$, for $x\in X$,
$\phi\in\widehat{E}_{{\operatorname{tight}}}$, $e\in E$, gives a homeomorphism
$X\cong\widehat{E}_{{\operatorname{tight}}}$.

###### Proof.

The map $E\ni e\mapsto 1_{e}\in C_{0}(X)$ is a tight representation, and so it
gives rise to a representation
$\pi:C_{0}(\widehat{E}_{{\operatorname{tight}}})\to C_{0}(X)$ which is
surjective by the Stone–Weierstrass Theorem. To see that $\pi$ is isometric
note that
$\pi(1_{Z(e)\setminus\bigcup_{f}Z(f)}|_{\widehat{E}_{{\operatorname{tight}}}})=\prod_{f\in
F}(1_{e}-1_{f})=1_{e\setminus\bigcup F}$ for every $e\in E$ and finite
$F\subseteq eE$. So if $\prod_{f\in F}(1_{e}-1_{f})=0$, then $e=\bigcup F$
which means that $F$ is a cover of $e$. Hence $\pi$ is isometric by Theorem
6.15. It is immediate that the homeomorphism dual to the isomorphism
$C_{0}(\widehat{E}_{{\operatorname{tight}}})\cong C_{0}(X)$ is of the
described form. ∎

######  Corollary 6.18.

Let $\mathcal{G}$ be an ample groupoid, and let
$S\subseteq\operatorname{Bis}(\mathcal{G})$ be a wide inverse semigroup of
compact open bisections that separate points in $X$. We have a natural
isomorphism $\mathcal{G}\cong\mathcal{G}_{\operatorname{tight}}(S)$ of
topological groupoids, and so we have a natural isomorphism
$F_{\operatorname{tight}}(S)\cong F^{S}(\mathcal{G})$.

###### Proof.

Recall that $\mathcal{G}_{\operatorname{tight}}(S)$ is the transformation
groupoid for the action $\hat{h}$ where
$\hat{h}_{t}:\hat{Z}(t^{*}t)\to\hat{Z}(tt^{*})$,
$\hat{Z}(t^{*}t):=\\{\phi\in\widehat{E}_{\operatorname{tight}}:\phi(t^{*}t)=1\\}$
and $\hat{h}_{t}(\phi)(e):=\phi(t^{*}et)$ for $t\in S$, $e\in E$,
$\phi\in\hat{Z}(t^{*}t)$. Similarly $\mathcal{G}$ is the transformation
groupoid for the action

$h:S\to\operatorname{PHomeo}(X);\ h_{t}=r\circ d|_{t}^{-1}:t^{*}t\to
tt^{*},\qquad t\in S.$

Note that for $e\leq t^{*}t$ we have $h_{t}(e)=tet^{*}$. Let
$\Psi:\widehat{E}_{{\operatorname{tight}}}\to X$ be the homeomorphism from
Corollary 6.17. For $\phi\in\widehat{E}_{{\operatorname{tight}}}$,
$\phi(t^{*}t)=1$ if and only if $1_{t^{*}t}(\Psi(\phi))=1$. Thus
$\Psi:\hat{Z}(t^{*}t)\to t^{*}t$. Moreover, for $\phi\in\hat{Z}(t^{*}t)$ and
$e\in E$, we have

$1_{e}\Big{(}\Psi\big{(}\hat{h}_{t}(\phi)\big{)}\Big{)}=\hat{h}_{t}(\phi)(e)=\phi(t^{*}et)=\phi\big{(}h_{t}^{-1}(e)\big{)}=1_{h_{t}^{-1}(e)}(\phi)=1_{e}\Big{(}h_{t}\big{(}\Psi(\phi)\big{)}\Big{)}.$

Thus $\Psi$ intertwines the $S$-actions and so the transformation groupoids
are isomorphic. ∎

##  7\. Further examples and applications

###  7.1. Crossed products by twisted partial group actions

Twisted partial actions of a discrete group on $C^{*}$-algebras and their
crossed products were gradually introduced in [28], [56], and [29], and the
literature on that subject is now vast, see [33]. We will consider only
partial actions on commutative Banach algebras $A=C_{0}(X)$, which are
equivalent to partial actions on the space $X$.

######  Definition 7.1.

A _partial action of a discrete group_ $G$ on a locally compact Hausdorff
space $X$ is a map $\theta:G\to\operatorname{PHomeo}(X)$ such that
$\theta_{1}=\mathrm{id}_{X}$, the partial homeomorphism $\theta_{ts}$ extends
$\theta_{t}\circ\theta_{s}$, and $\theta_{t}^{-1}=\theta_{t^{-1}}$ for all
$t,s\in G$.

We fix a partial action $\theta:G\to\operatorname{PHomeo}(X)$ of a group $G$.
Note that in general this is not the same as an action of the group $G$ viewed
as an inverse semigroup. Namely, for each $t\in G$ we have a homeomorphisms
$\theta_{t}:X_{t^{-1}}\to X_{t}$ between two open subsets of $X$ but for all
$s,t\in G$ we only require that $\theta_{t^{-1}}(X_{s^{-1}}\cap
X_{t})=X_{t^{-1}}\cap X_{(st)^{-1}}\subseteq X_{(st)^{-1}}$, so that the
composition $\theta_{s}\circ\theta_{t}$ might be defined only a subset of the
domain of $\theta_{st}$, rather than on all of $X_{(st)^{-1}}$.

######  Definition 7.2.

A _twist of the partial action $\theta$_, or of the associated algebraic
partial action $\alpha:G\to\operatorname{PAut}(C_{0}(X))$, where
$\alpha_{t}(a):=a\circ\theta_{t}^{-1}$, is a family of maps $u(s,t):G\times
G\to C_{u}(X_{s}\cap X_{st})$, $s,t\in G$, satisfying $u(1,t)=u(t,1)=1$ for
$t\in G$, and

$\alpha_{r}\big{(}au(s,t)\big{)}u(r,st)=\alpha_{r}(a)u(r,s)u(rs,t),\qquad a\in
C_{0}(X_{r^{-1}}\cap X_{s}\cap X_{st}),\ r,s,t\in G.$

We fix a twist $u=\\{u(s,t)\\}_{s,t\in G}$ of $\theta$. The space
$\ell^{1}(\alpha,u):=\\{f\in\ell^{1}(G,C_{0}(X)):f(t)\in C_{0}(X_{t}),t\in
G\\}$ with operations

$(f*g)(r):=\sum_{st=r}\alpha_{s}\Big{(}\alpha_{s}^{-1}\big{(}f(s)\big{)}g(t)\Big{)}u(s,t),\qquad
f^{*}(t):=\alpha_{t}\big{(}f(t^{-1})^{*}\big{)}u(t,t^{-1})^{*},$

for $f,g\in\ell^{1}(\alpha,u)$ and $s,t,r\in G$, is a Banach $*$-algebra.

######  Definition 7.3.

We denote the above Banach $*$-algebra by $F(\theta,u)$ and we call it a
_Banach algebra crossed product for $(\theta,u)$_. For $p\in[1,\infty]$ we
define the _$L^{p}$ -crossed product_ for $(\theta,u)$ as the completion
$F^{p}(\theta,u)=\overline{F(\theta,u)}^{\|\cdot\|_{L_{p}}}$ where
$\|f\|_{L^{p}}$ is the supremum of $\|\psi(f)\|$ over representations
$\psi:F(\theta,u)\to B(L^{p}(\mu))$ such that $\psi|_{C_{0}(X)}$ acts by
multiplication operators.

We now use our theory to describe representations of the above introduced
algebras. To this end, we recall that $\theta$ gives rise to an action of the
universal inverse semigroup $S(G)$ introduced in [30]. By definition $S(G)$ is
the universal semigroup generated by elements $[t]$, where $t\in G$, subject
to the relations $[s][t][t^{-1}]=[st][t^{-1}]$, $[s^{-1}][s][t]=[s^{-1}][st]$,
$[t][1]=[t]$ and $[1][t]=[t]$ for all $s,t\in G$. Then $S(G)$ is a unital
inverse semigroup, $G\ni t\mapsto[t]\in S(G)$ is a unital semigroup embedding,
and $[t]^{*}=[t^{-1}]$ for all $t\in G$, see [30]. Moreover, putting
$e_{t}:=[t^{-1}][t]$ for $t\in G$ every $\overline{t}\in S(G)$ has a _standard
form_ :

(7.14) $\overline{t}=e_{t_{1}}\cdots e_{t_{n}}[t_{0}],\qquad
t_{0},\ldots,t_{n}\in G,$

where the $t_{i}$ are mutually distinct and different from $1$ and $t_{0}$ for
$1\leq i<j\neq n\in\mathbb{N}$, and this form is unique up to the order of the
$e_{t_{i}}$s. By [30, Theorem 4.2] we have a bijective correspondence between
partial actions of $G$ and actions of $S(G)$. Thus there is a unique action
$[\theta]:S(G)\to\operatorname{PHomeo}(X)$ such that
$[\theta]_{[t]}=\theta_{t}$ for $t\in G$. Namely, if $\overline{t}$ is of the
form (7.14) we put $X_{\overline{t}}:=\bigcap_{k=0}^{n}X_{t_{k}}$, and then

(7.15)
$[\theta]_{\overline{t}}=\theta_{t_{0}}|_{X_{\overline{t}^{*}}}:X_{\overline{t}^{*}}\to
X_{\overline{t}}.$

The algebraic action $[\alpha]:S(G)\to\operatorname{PAut}(C_{0}(X))$ is
associated to $[\theta]:S(G)\to\operatorname{PHomeo}(X)$ iw the unique action
of $S(G)$ on $C_{0}(X)$ such that $[\alpha]_{[t]}=\alpha_{t}$. Moreover, by
[69, Theorems 4.2, 4.3], there is a unique twist $[u]$ of $[\theta]$ (or
$[\alpha]$) such that

$[u]([s],[t])=u(s,t),\quad[u]([s],e_{t})=1_{X_{st^{*}t}},\quad[u](e_{t},[s])=1_{X_{t^{*}ts}}\quad\text{for
}s,t\in G.$

Namely, if $\overline{t}=e_{t_{1}}\cdots e_{t_{n}}[t_{0}]$ and
$\overline{s}=e_{s_{1}}\cdots e_{s_{m}}[s_{0}]$ are in standard form, then
$[u](\overline{s},\overline{t})$ is the restriction of $u(s_{0},t_{0})$ to
$X_{\overline{s}\overline{t}}\subseteq X_{s_{0}}\cap X_{s_{0}t_{0}}$.

######  Definition 7.4.

The _transformation groupoid for $\theta$_ is
$\mathcal{G}_{\theta}:=\bigsqcup_{t\in G}\\{t\\}\times X_{t^{-1}}$, equipped
with the product topology inherited from $G\times X$ and the algebraic
structure detemined given by

$\big{(}s,\theta_{t}(x)\big{)}\cdot(t,x):=(st,x),\qquad s,t\in G,\ x\in
X_{t^{-1}}\cap X_{t^{-1}s^{-1}},$

$r(t,x):=\theta_{t}(x)$, $d(t,x):=x$, for $x\in X_{t^{-1}}$, $t\in G$. One
checks that $\sigma_{u}((s,\theta_{t}(x)),(t,x)):=u(s,t)(\theta_{st}(x))$ for
$x\in X_{t^{-1}}\cap X_{t^{-1}s^{-1}}$, $s,t\in\mathcal{G},$ defines a
$2$-cocycle on $\mathcal{G}_{\theta}$, cf. Example 4.3. Thus there is a _twist
$\mathcal{L}_{u}=\mathcal{G}_{\theta}\times\mathbb{F}$ associated_ to $u$ with
multiplication and involution given by

$(s,\theta_{t}(x),a)\cdot(t,x,b):=\big{(}st,x,abu(s,t)(\theta_{st}(x))\big{)},\quad(t,x,b)^{*}:=\big{(}t^{-1},\theta_{t}(x),\overline{b}\cdot\overline{u(t,t^{-1})}(x)\big{)}.$

######  Lemma 7.5.

The twisted groupoids $(\mathcal{G}_{\theta},\mathcal{L}_{u})$ and
$(S(G)\ltimes_{[\theta]}X,\mathcal{L}_{[u]})$ associated to $(\theta,u)$ and
$([\theta],[u])$ are naturally isomorphic.

###### Proof.

Note that if $\overline{t}=e_{t_{1}}\cdots e_{t_{n}}[t_{0}]$ and
$\overline{s}=e_{s_{1}}\cdots e_{s_{m}}[s_{0}]$ are in a standard form (7.14)
then $\overline{s}\leq\overline{t}$ if and only if $t_{0}=s_{0}$ and
$\\{t_{1},\ldots,t_{n}\\}\subseteq\\{s_{1},\ldots,s_{m}\\}$. Hence in view of
(7.15) we have $[\overline{t},x]=[[t_{0}],x]$ for any $x\in
X_{\overline{t}^{-1}}$. Using this we get that the map
$\mathcal{G}_{\theta}\ni(t,x)\mapsto[[t],x]\in S(G)\ltimes_{[\theta]}X$ is
bijective. Now it is straightforward to see that this is an isomorphism of
topological groupoids, and that it extends to an isomorphism of twists
$\mathcal{L}_{u}\cong\mathcal{L}_{[u]}$. ∎

######  Definition 7.6.

A _covariant representation of the twisted partial action $(\theta,u)$ in a
Banach algebra $B$_ is a pair $(\pi,v)$ where $\pi:C_{0}(X)\to B$ is a
representation and $v:G\to(B^{\prime\prime})_{1}$ is a map sat satisfying

(7.16) $\pi(a)v_{s}v_{t}=\pi(au(s,t))v_{st},\quad\text{ for }\quad a\in
C_{0}(X_{st}),\ s,t\in G.$

and

$v_{t}\pi(a\circ\theta_{t})=\pi(a)v_{t}\in B\quad\text{ and }\quad
v_{t}v_{t^{-1}}\pi(a)=\pi(a),\qquad a\in C_{0}(X_{t}),\ t\in G.$

A _covariant representation of $\theta$ on a Banach space $E$_ is a pair
$(\pi,v)$ where $\pi:A\to B(E)$ is a representation and $v:G\to B(E)_{1}$
satisfies (7.16) and

$v_{t}\pi(a\circ\theta_{t})v_{t^{-1}}=\pi(a),\quad\text{ and }\quad
v_{t}v_{t^{-1}}E=\overline{\pi(C_{0}(X_{t}))E},\qquad a\in C_{0}(X_{t}),\ t\in
G.$

######  Remark 7.7.

In the untwisted case ($u\equiv 1$), condition (7.16) may be replaced by the
assumption that $v$ is a _partial representation_ of the group $G$, i.e. for
all $s,t\in G$,

$v_{s}v_{t}v_{t^{-1}}=v_{st}v_{t^{-1}},\qquad
v_{s^{-1}}v_{s}v_{t}=v_{s^{-1}}v_{st},\qquad v_{t}v_{1}=v_{t},\qquad
v_{1}v_{t}=v_{t}.$

Then $v$ induces a semigroup homomorphism $[v]:S(G)\to B^{\prime\prime}_{1}$,
given by $[v]_{[t]}=v_{t}$. Thus each $v_{t}$ is a partial isometry in $B$,
and if $B$ is a $C^{*}$-algebra, then $v_{t}^{*}=v_{t^{-1}}$ for $t\in G$.

######  Lemma 7.8.

For every covariant representation $(\pi,v)$ of $(\theta,u)$ in $B$ there is a
unique covariant representation $(\pi,[v])$ of $([\alpha],[u])$ in $B$, where
$v_{t}=[v]_{[t]}$, $t\in G$. Every covariant representation of
$([\alpha],[u])$ in $B$ is of this form. This gives a bijective correspondence
between covariant representations of $(\theta,u)$ and $([\alpha],[u])$ on a
Banach space $E$.

###### Proof.

Let $(\pi,v)$ be a covariant representation of $\theta$ in a Banach algebra
$B$. For any $\overline{t}$ in the form (7.14) we put
$[v]_{\overline{t}}:=v_{t_{1}^{-1}}v_{t_{1}}\cdots
v_{t_{n}^{-1}}v_{t_{n}}v_{t_{0}}$. In view of the form of $[u]$ this is a
unique map $[v]:S(G)\to B^{\prime\prime}$ satisfying $v_{t}=[v]_{[t]}$, for
$t\in G$, and (CR2) for all elements in $S(G)$. Every idempotent in $S(G)$ is
of the form $\overline{e}=e_{t_{1}}\cdots e_{t_{n}}$, and for $a\in
C_{0}(X_{\overline{e}})=C_{0}(\bigcap_{k=0}^{n}X_{t_{k}})$, using the
covariance of $(\pi,v)$ $n$-times, we get
$[v]_{\overline{e}}\pi(a)=v_{t_{1}^{-1}}v_{t_{1}}\cdots
v_{t_{n}^{-1}}v_{t_{n}}\pi(a)=\pi(a)$. This gives (CR3). Similarly,
$\pi(a)[v]_{\overline{e}}=\pi(a)$. Thus for any $a\in
C_{0}(X_{\overline{t}})=C_{0}(\bigcap_{k=0}^{n}X_{t_{k}})$ we get

$[v]_{\overline{t}}\pi(a\circ\theta_{t})=[v]_{\overline{e}}v_{t_{0}}\pi(a\circ\theta_{t})=[v]_{\overline{e}}\pi(a)v_{t_{0}}=\pi(a)v_{t_{0}}=\pi(a)[v]_{\overline{e}}v_{t_{0}}=\pi(a\circ\theta_{t})[v]_{\overline{t}}.$

This proves (CR1). Thus $(\pi,[v])$ is a covariant representation of
$([\theta],[u])$. Also if $B=B(E)$ and $(\pi,v)$ is a covariant representation
on $E$, one readily sees that $(\pi,[v])$ is a covariant representation of
$([\theta],[u])$ on $E$.

Conversely, for any covariant representation $(\pi,[v])$ of $([\theta],[u])$
in $B$ the pair $(\pi,v)$, where $v_{t}:=[v]_{[t]}$, $t\in G$, is clearly a
covariant representation of $(\theta,u)$. Also it follows from the above
description of $(\pi,[v])$ in terms of $(\pi,v)$ that if $B=B(E)$, then
$(\pi,[v])$ is a covariant representation on a Banach space $E$ if and only if
$(\pi,v)$ is a covariant representation on $E$. ∎

######  Theorem 7.9.

Let $(\theta,u)$ be a twisted partial action of a group $G$ on $X$. We have
natural isometric isomorphisms

$F(\theta,u)\cong F^{S_{\theta}}(\mathcal{G}_{\theta},\mathcal{L}_{u})\cong
F^{S_{[\theta]}}(S(G)\ltimes_{[\theta]}X,\mathcal{L}_{[u]})\cong
C_{0}(G)\rtimes_{[\alpha],[u]}S(G)$

where $S_{\theta}$ is the inverse semigroup generated by bisections
$\\{t\\}\times X_{t^{-1}}\subseteq\mathcal{G}_{\theta}$, $t\in G$, and
$S_{[\theta]}$ consists of $\\{[\overline{t},x]:x\in
X_{\overline{t}^{*}}\\}\subseteq S(G)\ltimes_{[\theta]}X$, $\overline{t}\in
S(G)$. Moreover,

  1. (1)

Every representation of $F(\theta,u)$ in a Banach algebra $B$ is given by

(7.17) $\pi\rtimes v(a)=\sum_{t\in G}\pi(a(t))v_{t},\qquad a\in F(\theta,u),$

for a covariant representation $(\pi,v)$ of $(\theta,u)$ in $B$. If $B$ is a
dual Banach algebra, then $v$ can be chosen to take values in $B$.

  2. (2)

If $E$ is a reflexive Banach space, then (7.17) gives a bijective
correspondence between representations of $F(\theta,u)$ on $E$ and covariant
representations of $(\theta,u)$ on $E$.

  3. (3)

If each $X_{t}$, $t\in G$, is compact, then (7.17) gives a bijective
correspondence between representations of $F(\theta,u)$ in $B$ and covariant
representations of $(\theta,u)$ in $B$ such that $v_{t}=\pi(1_{X_{t}})v_{t}\in
B$ for all $t\in G$.

  4. (4)

For every $p\in(1,\infty)$ and localizable $\mu$, (7.17) gives a bijective
correspondence between representations $\psi:F^{p}(\theta,u)\to B(L^{p}(\mu))$
where $\psi|_{C_{0}(X)}$ acts by multiplication operators, and covariant
representations $(\pi,v)$ of $(\theta,u)$ on $L^{p}(\mu)$ where $\pi$ acts by
multiplication operators and $v:G\to\operatorname{SPIso}(L^{p}(\mu))$ takes
values in spatial partial isometries. For every $p\in[1,\infty]$,
$F^{p}(\theta,u)$ admits an isometric such representation.

  5. (5)

If $p\in[1,\infty]$ and $\mathbb{F}=\mathbb{C}$, the algebra $F^{p}(\theta,u)$
is universal for representations of $F(\theta,u)$ in all $L^{p}$-operator
algebras (every such representation factors to a representation of
$F^{p}(\theta,u)$).

###### Proof.

The isometric isomorphisms
$F^{S_{\theta}}(\mathcal{G}_{\theta},\mathcal{L}_{u})\cong
F^{S_{[\theta]}}(S(G)\ltimes_{[\theta]}X,\mathcal{L}_{[u]})\cong
C_{0}(G)\rtimes_{[\alpha],[u]}S(G)$ follow from Lemma 7.5 and Corollary 4.28.
It is straightforward to check that the map
$C_{c}(\mathcal{G}_{\theta},\mathcal{L}_{u})\ni\hat{f}\mapsto
f\in\ell^{1}(\theta,u)$ where $f(t)(x):=\hat{f}(t,\theta_{t^{-1}}(x))$ is an
injective $\|\cdot\|_{\max}^{S_{\theta}}$-contractive homomorphism. Thus it
extends to a representation
$\Psi:F^{S_{\theta}}(\mathcal{G}_{\theta},\mathcal{L}_{u})\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}F(\theta,u)=\ell^{1}(\theta,u)$.
It is routine to check that every covariant representation of $(\pi,v)$ of
$([\theta],[u])$ defines a representation of $\ell^{1}(\theta,u)$ via
$\pi\rtimes v(a)=\sum_{t\in G}\pi(a(t))v_{[t]}$. Disintegrating the identity
representation of $F^{S_{\theta}}(\mathcal{G}_{\theta},\mathcal{L}_{u})$ we
get a covariant representation $(\pi,v)$ of $([\theta],[u])$ such that the
associated representation $\pi\rtimes v:F(\theta,u)\to
F^{S_{\theta}}(\mathcal{G}_{\theta},\mathcal{L}_{u})$ is inverse to $\Psi$.
Thus $F(\theta,u)\cong F^{S_{\theta}}(\mathcal{G}_{\theta},\mathcal{L}_{u})$.

Now items (1), (2), (3) follow from Lemma 7.8 and Theorem 4.27. Then items (4)
and (5) follow from Theorem 5.15(1)(2) and Theorem 5.10(1). ∎

######  Remark 7.10.

Combining the above with Theorem 5.15(2) we see that $F^{2}(\theta,u)$ is
equal to the $C^{*}$-algebra
$C^{*}(\theta,u):=\overline{F(\theta,u)}^{\|\cdot\|_{C^{*}\max}}$ where
$\|\cdot\|_{C^{*}\max}$ is the maximal $C^{*}$-norm on $F(\theta,u)$. In
particular, Theorem 7.9 describes representations of the $C^{*}$-algebraic
crossed product $C^{*}(\theta,u)$ in terms of covariant representations, which
seems to be known only in the untwisted (and complex) case, see [62]. Twisted
crossed products in [29] were defined and analyzed in terms of Fell bundles
(without the use covariant representations), cf. a comment below [69, Theorem
4.3].

###  7.2. Banach algebras associated to twisted Deaconu–Renault groupoids

We fix a local homeomorphism $\varphi:\Delta\to X$ between open subsets
$\Delta$, $\varphi(\Delta)\subseteq X$ of a locally compact Hausdorff space
$X$. Thus $(X,\varphi)$ is a singly generated dynamical system (SGDS) in the
sense of [64]. Put $\Delta_{0}=X$ and inductively define
$\Delta_{n}:=\varphi^{-1}(\Delta_{n-1})$, the natural domain of the partial
local homeomorphism $\varphi^{n}$, $n\in\mathbb{N}$. The _Deaconu–Renault
groupoid_ associated to $(X,\varphi)$, see [64], is an étale, amenable,
locally compact Hausdorff groupoid where

$\mathcal{G}(X,\varphi):=\\{(y,n-m,x):n,m\in\mathbb{N}_{0},\ x\in\Delta_{n},\
y\in\Delta_{m},\ \varphi^{n}(x)=\varphi^{m}(y)\\},$

and the groupoid structure is given by

$(z,n,y)(y,m,x):=(z,n+m,x),\qquad(y,n,x)^{-1}:=(x,-n,y),$

and the topology is inherited from $X\times\mathbb{Z}\times X$. The groupoid
$\mathcal{G}(X,\varphi)$ is étale as it has a basis for the topology
consisting of bisections of the form

(7.18) $Z(V,n-m,U):=\\{(y,n-m,x):(y,x)\in V\times U,\
\varphi^{m}(y)=\varphi^{n}(x)\\},$

where $U\subseteq\Delta_{n}$ and $V\subseteq\Delta_{m}$ are open sets such
that $\varphi^{n}|_{U}$ and $\varphi^{m}|_{V}$ are injective and
$\varphi^{n}(U)=\varphi^{m}(V)$. We identify the unit space
$Z(X,0,X)=\\{(x,0,x)\in X\times\mathbb{Z}\times X\\}$ of
$\mathcal{G}(X,\varphi)$ with $X$ via the map $X\ni x\longmapsto(x,0,x)\in
Z(X,0,X)$. Thus the range and domain maps are given by $r(y,n,x)=y$ and
$d(y,n,x)=x$.

######  Lemma 7.11.

The collection $S(X,\varphi)$ of bisections (7.18) forms a wide inverse
semigroup of bisections of $\mathcal{G}(X,\varphi)$ whose canonical action is
given by the formula

$Z(V,n-m,U)\mapsto
h_{Z(V,n-m,U)}=(\varphi|_{V}^{m})^{-1}\circ\varphi|_{U}^{n}.$

In particular, we have a natural isomorphism $S(X,\varphi)\ltimes
X\cong\mathcal{G}(X,\varphi)$. If $X$ is totally disconnected, then the
collection $S_{c}(X,\varphi)$ of bisections (7.18) with compact open $U$, $V$
is also a wide inverse semigroup, and so $S_{c}(X,\varphi)\ltimes
X\cong\mathcal{G}(X,\varphi)$.

###### Proof.

Let $Z(V,N,U)$ and $Z(W,M,Y)$ be bisections of the form (7.18), so $N=n-m$,
$M=k-l$ where $n,m,k,l\in\mathbb{N}_{0}$, $\varphi|_{V}^{m}$,
$\varphi|_{U}^{n}$, $\varphi|_{W}^{l}$, $\varphi|_{Y}^{k}$ are injective and
$\varphi^{m}(V)=\varphi^{n}(U)$, $\varphi^{l}(Y)=\varphi^{k}(W)$. Clearly,
$Z(V,N,U)^{-1}Z(U,-N,V)$ and
$h_{Z(V,N,U)}=(\varphi|_{V}^{m})^{-1}\circ\varphi|_{U}^{n}:U\to V$ is a
homeomorphism. Moreover,

$Z(V,N,U)\cdot Z(W,M,Y)=Z(V^{\prime},N+M,Y^{\prime}),$

where $Y^{\prime}=(\varphi|_{Y}^{k})^{-1}\circ\varphi^{l}(U\cap W)$ and
$V^{\prime}=(\varphi|_{V}^{m})^{-1}\circ\varphi^{n}(U\cap W)$. More
specifically, if $n\geq l$ then $\varphi^{k+n-l}$ is injective on $Y^{\prime}$
and
$h_{Z(V^{\prime},N+M,Y^{\prime})}=(\varphi|_{V^{\prime}}^{m})^{-1}\circ\varphi|_{Y^{\prime}}^{k+n-l}$
maps $Y^{\prime}$ onto $V^{\prime}$. If $n\leq l$, then $\varphi^{m+l-n}$ is
injective on $V^{\prime}$ and
$h_{Z(V^{\prime},N+M,Y^{\prime})}=(\varphi|_{V^{\prime}}^{m+l-n})^{-1}\circ\varphi|_{Y^{\prime}}^{k}$
maps $Y^{\prime}$ onto $V^{\prime}$. Thus $S(X,\varphi)$ is an inverse
semigroup. It is wide because it forms a basis for the topology of
$\mathcal{G}(X,\varphi)$. If the sets $U$, $V$, $W$, $Y$ above are compact
open, then also $Y^{\prime}$ and $V^{\prime}$ are compact open. Hence
$S_{c}(X,\varphi)$ is an inverse semigroup. It forms a basis for the topology
of $\mathcal{G}(X,\varphi)$ when $X$ is totally disconnected. ∎

######  Remark 7.12.

Let $S(\varphi)$ be the inverse subsemigroup of $\operatorname{PHomeo}(X)$
generated by restrictions $\varphi|_{U}\in\operatorname{PHomeo}(X)$ to open
sets $U\subseteq X$ where $\varphi|_{U}$ is injective. Lemma 7.11 implies that
$S(\varphi)$ consists of partial homeomeomorphisms of the form
$(\varphi|_{V}^{m})^{-1}\circ\varphi|_{U}^{n}$, and we have the canonical
semigroup epimorphism $h:S(X,\varphi)\twoheadrightarrow S(\varphi)$. The
groupoid of germs $\textup{Germ}(X,\varphi)$ defined in [64] is canonically
isomorphic to the transformation groupoid $S(\varphi)\ltimes X$. So we have a
canonical groupoid epimorphism $\mathcal{G}(X,\varphi)\cong
S(X,\varphi)\ltimes X\twoheadrightarrow S(\varphi)\ltimes
X\cong\textup{Germ}(X,\varphi)$. By [64, Proposition 2.3] this is an
isomorphism if and only if $\mathcal{G}(X,\varphi)$ is topologically free,
which holds if and only if $\varphi$ is topologically free, i.e. for each
$n\in\mathbb{N}$ the set $\\{x\in\Delta_{n}:\varphi^{n}(x)=x\\}$ has empty
interior.

Twisted groupoids of the form $(\mathcal{G}(X,\varphi),\mathcal{L})$ model,
for instance, $C^{*}$-algebras associated to twisted topological graphs, see
[48]. It seems that $L^{p}$-operator algebras associated to
$\mathcal{G}(X,\varphi)$ have not been considered yet. Moreover, even the
$C^{*}$-algebraic versions of the following results does not seem to appear in
the literature.

######  Proposition 7.13.

Let $\mathcal{L}$ be any twist over $\mathcal{G}(X,\varphi)$. Let $S\subseteq
S(X,\varphi)\cap S(\mathcal{L})$ be any wide inverse semigroup of bisections
where the bundle $\mathcal{L}$ can be trivialized (one may always take
$S=S(X,\varphi)\cap S(\mathcal{L})$) and consider the action

$\alpha:S\to\operatorname{PAut}(C_{0}(X));\
\alpha_{Z(V,n-m,U)}(a):=a\circ(\varphi|_{U}^{n})^{-1}\circ\varphi|_{V}^{m},\qquad
a\in C_{0}(U).$

For $t\in S$ pick any $c_{t}\in C_{u}(t,\mathcal{L})$, unless $t\subseteq X$,
in which case put $c_{t}(x):=(x,1)\subseteq X\times\mathbb{F}$. For $s,t\in
S(\mathcal{L})$ put $u(s,t):=c_{s}*c_{t}*c_{st}^{*}$. Then $(\alpha,u)$ is a
twisted action of $S$ and we have a natural isomorphism

$F^{S}(\mathcal{G}(X,\varphi),\mathcal{L})\cong
C_{0}(X)\rtimes_{(\alpha,u)}S.$

Thus every representation $\psi:F^{S}(\mathcal{G}(X,\varphi),\mathcal{L})\to
B$ is determined by the formula

(7.19) $\psi(a\delta_{Z(V,N,U)})=\pi(a)v_{Z(V,N,U)},\qquad a\in
C_{c}(V),\,\,Z(V,N,U)\in S,$

for a covariant representation $(\pi,v)$ of $(\alpha,u)$.

###### Proof.

Both $S(X,\varphi)$ and $S(\mathcal{L})$ are inverse semigroups that form a
basis for the topology of $\mathcal{G}(X,\varphi)$ which is downward directed
by inclusion, see Lemma 4.2. Hence $S(X,\varphi)\cap S(\mathcal{L})$ has the
same properties and therefore is a wide inverse semigroup. The corresponding
pair $(\alpha,u)$ is an example of a twisted action from Lemma 4.19 which
models $(\mathcal{G}(X,\varphi),\mathcal{L})$ by Lemma 4.23. Hence the
assertion follows from Corollary 4.28. ∎

######  Corollary 7.14.

Let $S\subseteq S(X,\varphi)$ be a wide inverse semigroup of bisections. If
$p\in(1,\infty)$, then (7.19) establishes a bijective correspondence between
representations $\psi:F^{p}(\mathcal{G}(X,\varphi))\to B(L^{p}(\mu))$ where
$\psi|_{C_{0}(X)}$ acts by multiplication operators, and pairs $(\pi,v)$ where
$\pi:C_{0}(X)\to B(L^{p}(\mu))$ is a representation acting by multiplication
operators, and $v:S\to\operatorname{SPIso}(L^{p}(\mu))$ is a semigroup
homomorphism satisfying

$v_{Z(V,n-m,U)}\pi(a)=\pi(a\circ(\varphi|_{U}^{n})^{-1}\circ\varphi|_{V}^{m})v_{Z(V,n-m,U)},\qquad
a\in C_{0}(U),$

and $v_{Z(U,0,U)}$ is a projection onto $\overline{\pi(C_{0}(U))L^{p}(\mu)}$
for any $U\in S$ with $U\subseteq X$. For every $p\in[1,\infty]$ there is a
pair $(\pi,v)$ as above such that (7.19) defines an isometric representation
of $F^{p}(\mathcal{G}(X,\varphi))$.

###### Proof.

Follows from Proposition 7.13 and Theorem 5.15(1)(2), modulo explicit
description of spatial covariant representations of the action
$\alpha:S\to\operatorname{PAut}(C_{0}(X))$. ∎

######  Corollary 7.15.

If $X$ is totally disconnected and $S\subseteq S(X,\varphi)$ is a wide inverse
semigroup consisting of compact open bisections that separate points of $X$,
then

$F_{\operatorname{tight}}(S)\cong
F^{S}(\mathcal{G}_{\operatorname{tight}}(S))\cong
F^{S}(\mathcal{G}(X,\varphi))$

and so the relation $\psi(\delta_{t})=v_{t}$, $t\in S$, yields a bijective
correspondence between representations $\psi:F^{S}(\mathcal{G}(X,\varphi))\to
B$ and tight representations $v:S\to B$ of $S$ in a Banach algebra $B$.

###### Proof.

By Corollary 6.18, we have a canonical homeomorphism
$\mathcal{G}_{\operatorname{tight}}(S)\cong\mathcal{G}(X,\varphi)$ and so the
assertion follows from Theorem 6.15. ∎

###  7.3. Banach algebras associated to directed graphs

Let $Q=(Q^{0},Q^{1},r_{Q},s_{Q})$ be a _directed graph_ (sometimes also called
a _quiver_). So $Q^{0}$ is the set of vertices, $Q^{1}$ is the set of edges
and $r_{Q},s_{Q}:Q^{1}\to Q^{0}$ are the range and source maps. The vertices
in $Q_{\textup{reg}}:=\\{v\in Q^{0}:1<|r_{Q}^{-1}(v)|<\infty\\}$ are called
_regular_ and the remaining $Q_{\textup{sing}}:=Q^{0}\setminus
Q_{\textup{reg}}$ are called _singular_. We denote by $Q^{n}$, $n>0$, the set
of finite paths $\mu=\mu_{1}\cdots\mu_{n}$, where
$s_{Q}(\mu_{i})=r_{Q}(\mu_{i+1})$ for all $i=1,\ldots,n-1$. Then $|\mu|=n$
stands for the length of $\mu$ and $Q^{*}=\bigcup_{n=0}^{\infty}Q^{n}$ is the
set of all finite paths (vertices are treated as paths of length zero). We
denote by $Q^{\infty}$ the set of infinite paths and put
$Q^{\leq\infty}:=Q^{*}\cup Q^{\infty}$. The maps $r_{Q},s_{Q}$ extend
naturally to $Q^{*}$ and $r_{Q}$ extends also to $Q^{\infty}$. We recall the
defintion of the Leavitt path algebra associated to $Q$, see [2].

######  Definition 7.16.

A _$Q$ -family in an algebra $B$_ (over $\mathbb{F}=\mathbb{R},\mathbb{C}$) is
a pair $(P,T)$ where $P=\\{P_{v}\\}_{v\in Q^{0}}\subseteq B$ consists of
pairwise orthogonal idempotents and $T=\\{T_{e},T_{e}^{*}\\}_{e\in
Q^{1}}\subseteq B$ consists of elements where $T_{e}^{*}$ is a generalized
inverse of $T_{e}$ ($e\in Q^{1}$) satisfying

  1. (CK1)

$T_{e}^{*}T_{e}=P_{s(e)}$ and $T_{e}T_{e}^{*}\leq P_{r(e)}$ for all $e\in
Q^{1}$;

  2. (CK2)

$P_{v}=\sum_{e\in r^{-1}(v)}T_{e}T_{e}^{*}$ for all $v\in
Q^{0}_{\operatorname{reg}}$.

Let $(P,T)$ be a $Q$-family in $B$. For $\mu=\mu_{1}\cdots\mu_{n}\in Q^{*}$ we
put $T_{\mu}:=T_{\mu_{1}}\cdots T_{\mu_{n}}$ and
$T_{\mu}^{*}:=T_{\mu_{n}}^{*}\cdots T_{\mu_{1}}^{*}$. Then the elements
$T_{\mu}T_{\nu}^{*}$ form an inverse semigroup with multiplication from $B$
and so $\operatorname{span}\\{T_{\mu}T_{\nu}^{*}:\mu,\nu\in Q^{*}\\}$ is a
subalgebra of $B$ generated by $P\cup T$. In fact, it is a $*$-algebra with
the involution determined by $(T_{\mu}T_{\nu}^{*})^{*}=T_{\nu}T_{\mu}^{*}$,
$\mu,\nu\in Q^{*}$. By definition the _Leavitt path algebra_ $\mathcal{L}(Q)$
is the algebra generated by a universal $Q$-family $(p,t)$. Thus

$\mathcal{L}(Q)=\operatorname{span}\\{t_{\mu}t_{\nu}^{*}:\mu,\nu\in Q^{*}\\},$

and for any $Q$-family $(P,T)$ in $B$ there is a unique algebra homomorphism
$\psi_{(P,T)}:\mathcal{L}(Q)\to B$ such that $\psi_{(P,T)}(p_{v})=P_{v}$,
$\psi_{(P,T)}(t_{e})=T_{e}$ and $\psi_{(P,T)}(t_{e}^{*})=T_{e}^{*}$, for $v\in
Q^{0}$, $e\in Q^{1}$. We refer to [2] for more details. The elements
$\\{t_{\mu}t_{\nu}^{*}:\mu,\nu\in Q^{*}\\}$ form an inverse semigroup that can
be described in terms of $Q$ as follows. Put $S_{Q}:=\\{(\mu,\nu)\in
Q^{*}\times Q^{*}:s_{Q}(\mu)=s_{Q}(\nu)\\}\cup\\{0\\}$ and define
multiplication in $S_{Q}$ by the formula

$(\mu,\nu)(\alpha,\beta):=\begin{cases}(\mu\alpha^{\prime},\beta)&\text{if
$\alpha=\nu\alpha^{\prime}$},\\\ (\mu,\beta\nu^{\prime})&\text{if
$\nu=\alpha\nu^{\prime}$},\\\ 0&\text{otherwise.}\end{cases}$

Then the involution is given by $(\mu,\nu)^{*}=(\nu,\mu)$. It is known to
experts (and not hard to see) that tight representations of $S_{Q}$ in are in
bijective correspondence with $Q$-families. We deduce this from our results.
The groupoid model for $\mathcal{L}(Q)$ (isomorphic to the tight groupoid
$\mathcal{G}_{T}(S_{Q})$) can be described as follows [21, Examples 2.1, 3.2].
We put $Q^{*}_{\mathrm{sing}}:=\\{\mu\in Q^{*}:s_{Q}(\mu)\in
Q_{\mathrm{sing}}\\}$. For any $\eta\in Q^{*}\setminus Q^{0}$ let
${\eta}Q^{\leq\infty}:=\\{\mu=\mu_{1}\cdots\in
Q^{\leq\infty}:\mu_{1}\cdots\mu_{|\eta|}=\eta\\}$ and for $v\in Q^{0}$ put
${v}Q^{\leq\infty}:=\\{\mu\in Q^{\leq\infty}:r_{Q}(\mu)=v\\}$. The _boundary
space_ of $Q$ [77, Section 2] is the set

$\partial Q:=Q^{\infty}\cup Q^{*}_{\mathrm{sing}}$

equipped with the topology generated by the ‘cylinders’ $Z(\mu):=\partial
Q\cap{\mu}Q^{\leq\infty}$, $\eta\in Q^{*}$, and their complements. In fact,
the sets $Z(\mu)\setminus\bigcup_{e\in F}Z(\mu e)$, where $\mu\in Q^{*}$ and
$F\subseteq{s(\mu)}Q^{1}$ is a finite set of edges, form a basis of compact-
open sets for the Hausdorff topology on $\partial Q$ [77, Section 2]. The one-
sided _topological Markov shift_ associated to $Q$ is the map $\sigma:\partial
Q\setminus Q^{0}\to\partial Q$ defined, for
$\mu=\mu_{1}\mu_{2}\cdots\in\partial Q\setminus Q^{0}$, by the formulas

$\sigma(\mu):=\mu_{2}\mu_{3}\cdots\ \text{ if $\mu\notin
Q^{1}$},\qquad\sigma(\mu):=s(\mu_{1})\ \text{if $\mu=\mu_{1}\in Q^{1}$}.$

This is a partial local homeomorphism on $\partial Q$. By definition the
_groupoid of the graph_ $Q$ is the Deaconu–Renault grooupoid
$\mathcal{G}_{Q}:=\mathcal{G}(\partial Q,\sigma)$ of $\sigma$. Thus

$\mathcal{G}_{Q}=\\{(\mu x,|\mu|-|\eta|,\eta x):\mu,\eta\in Q^{*},\
x\in\partial Q,\ s_{Q}(\mu)=s_{Q}(\nu)=r_{Q}(x)\\}$

is an ample Hausdorff groupoid with the topology generated by the ‘cylinders’
$Z(\mu,\nu):=\\{(\mu x,|\mu|-|\eta|,\eta x)\in\mathcal{G}_{Q}\\}$, for
$\mu,\eta\in Q^{*}$ with $s_{Q}(\mu)=s_{Q}(\nu)$, and their complements. In
fact, the sets $Z(\mu,\nu)\setminus\bigcup_{\alpha\in F}Z(\mu\alpha)$, where
$\mu,\eta\in Q^{*}$ with $s_{Q}(\mu)=s_{Q}(\nu)$ and $F\subseteq{s(\mu)}Q^{*}$
is finite, form a basis of compact open bisections for $\mathcal{G}_{Q}$.
Moreover, every compact open set in $\mathcal{G}_{Q}$ is a finite disjoint
union of these basic sets. The associated Steinberg algebra is

$A(\mathcal{G}_{Q}):=\operatorname{span}\\{1_{Z(\mu,\nu)}:\mu,\eta\in Q^{*},\
s_{Q}(\mu)=s_{Q}(\nu)\\}.$

Note that $A(\mathcal{G}_{Q})\subseteq C_{c}(\mathcal{G}_{Q})$ is a dense
$*$-subalgebra of $F(\mathcal{G}_{Q})$, and $A(\partial
Q):=\operatorname{span}\\{1_{Z(\mu)}:\mu\in Q^{*}\\}\subseteq C_{c}(\partial
Q)$ is a dense $*$-subalgebra in $C_{0}(\partial Q)$.

######  Proposition 7.17 ([21]).

There is an algebra isomorphism $\Psi:\mathcal{L}(Q)\to A(\mathcal{G}_{Q})$
that maps $t_{\mu}t_{\nu}^{*}$ to $1_{Z(\mu,\nu)}$, $\mu,\nu\in Q^{*}$. It is
uniquely determined by $\Psi(p_{v})=1_{Z(v)}$, $\Psi(t_{e})=1_{Z(e,s_{Q}(e))}$
and $\Psi(t_{e}^{*})=1_{Z(s_{Q}(e),e)}$, for $v\in Q^{0}$, $e\in Q^{1}$. In
particular we have an isomorphism $A(\partial
Q)\cong\operatorname{span}\\{t_{\mu}t_{\mu}^{*}:\mu\in Q^{*}\\}$ where
$1_{Z(\mu)}\mapsto t_{\mu}t_{\mu}^{*}$.

######  Lemma 7.18.

For any $Q$-family $(P,T)$ in an algebra $B$ and any finite $F\subseteq Q^{*}$
the elements

$P^{F}_{\mu}:=\prod_{\mu\mu^{\prime}\in
F\setminus\\{\mu\\}}T_{\mu}T_{\mu}^{*}-T_{\mu\mu^{\prime}}T_{\mu\mu^{\prime}}^{*},\qquad\mu\in
F,$

(with the convention $P^{F}_{\mu}:=T_{\mu}T_{\mu}^{*}$ if the product is over
the empty set) are mutually orthogonal idempotents such that
$T_{\mu}T_{\mu}^{*}=\sum_{\mu\mu^{\prime}\in F\setminus\\{\mu\\}}P^{F}_{\mu}$,
so they span the algebra $\operatorname{span}\\{T_{\mu}T_{\mu}^{*}:\mu\in
F\\}$. If $B$ is a Banach algebra, then the map $(A(\partial
Q),\|\cdot\|_{\infty})\rightarrow\operatorname{span}\\{T_{\mu}T_{\mu}^{*}:\mu\in
Q^{*}\\}\subseteq B$; $1_{Z(\mu)}\mapsto t_{\mu}t_{\mu}^{*}$ is contractive if
and only if for every finite $F\subseteq E^{*}$ the idempotents
$\\{P^{F}_{\mu}\\}_{\mu\in F}$ are jointly $\mathbb{F}$-contractive
(Definition 6.4).

###### Proof.

The first part follows from [71, Lemma 3.1], see also [77, Lemma 3.4]. This
corresponds to the fact every element $f\in A(\partial Q)$ can be written in
the form $f=\sum_{\mu\in F}a_{\mu}1_{Z(\mu,F)}$ where
$Z(\mu,F):=Z(\mu)\setminus\bigcup_{\mu^{\prime}\in
F\setminus\\{\mu\\}}Z(\mu\mu^{\prime})$ are pairwise disjoint compact open
sets for all $\mu\in F$ and a fixed finite set $F\subseteq Q^{*}$. Then
$\|f\|_{\infty}=\max_{\mu\in F}|a_{\mu}|$, and the map in the second part of
the assertion sends $f$ to $\sum_{\mu\in F}a_{\mu}{P^{F}_{\mu}}$, so it is
contractive if $\sum_{\mu\in F}a_{\mu}{P^{F}_{\mu}}\leq\max_{\mu\in
F}|a_{\mu}|$ for all choices of $F$ and $a_{\mu}$. ∎

######  Definition 7.19.

A _Banach $Q$-family_ is a $Q$-family $(P,T)$ in a Banach algebra $B$ where
$P\cup T\subseteq B_{1}$ and for any finite $F\subseteq Q^{*}$ the idempotents
$\\{P^{F}_{\mu}\\}_{\mu\in F}$ are jointly $\mathbb{F}$-contractive. A
_spatial $Q$-family on $L^{p}(\mu)$_, where $p\in[1,\infty]$, is a $Q$-family
$(P,T)$ in the algebra $B(L^{p}(\mu))$ where $P\cup
T\subseteq\operatorname{SPIso}(L^{p}(\mu))$ are spatial partial isometries
(then $T_{e}^{*}$ is the adjoint of $T_{e}$ in the sense of the inverse
semigroup $\operatorname{SPIso}(L^{p}(\mu))$). We define the _universal graph
Banach algebra_ $F(Q)$ of $Q$ to be the completion of $\mathcal{L}(Q)$ in

$\|b\|_{F(Q)}:=\sup\\{\psi_{(P,T)}(b):\text{$(P,T)$ is a $Q$-family in a
Banach algebra}\\}.$

For $p\in[1,+\infty]$ we define the _graph $L^{p}$-operator algebra_
$F^{p}(Q)$ to be the completion of $\mathcal{L}(Q)$ in the norm
$\|b\|_{L^{p}}=\sup\\{\psi_{(P,T)}(b):\text{$(P,T)$ is spatial $Q$-family on
an $L^{p}$-space}\\}$.

######  Remark 7.20.

Lemma 6.7 implies that a spatial $Q$-family on $L^{p}(\mu)$ is a Banach
$Q$-family in $B(L^{p}(\mu))$, and any $Q$-family $(P,T)$ in a $C^{*}$-algebra
$B$ consisting of contractive elements is a Banach $Q$-family in $B$.

######  Theorem 7.21.

For any directed graph $Q$ we have natural isometric isomorphisms

(7.20) $F(Q)\cong F^{S_{Q}}(\mathcal{G}_{Q})\cong
F_{\operatorname{tight}}(S_{Q})$

In particular, for any Banach algebra $B$ the relations

(7.21)
$\psi(p_{v})=P_{v}=V_{(v,v)},\qquad\psi(t_{e})=T_{e}=V_{(e,s(e))},\qquad\psi(t_{e}^{*})=T_{e}^{*}=V_{(s(e),e)},$

for $v\in Q^{0}$, $e\in Q^{1}$, establish bijective correspondences between
representations $\psi:F(Q)\to B$, Banach $Q$-families $(P,T)$ in $B$, and
tight Banach algebra representations $V:S_{Q}\to B_{1}$.

###### Proof.

The collection $S:=\\{Z(\mu,\nu):\mu,\eta\in
Q^{*},s_{Q}(\mu)=s_{Q}(\nu)\\}\cup\\{\emptyset\\}$ forms a wide inverse
semigroup of compact open bisections in $\mathcal{G}_{Q}$ that separates the
points in $\mathcal{G}_{Q}$. Thus
$\mathcal{G}_{Q}\cong\mathcal{G}_{\operatorname{tight}}(S)$ is the tight
groupoid of $S$ by Corollary 6.18. We have natural semigroup isomorphisms
$S\cong S_{Q}\cong\\{t_{\mu}t_{\nu}^{*}:\mu,\nu\in Q^{*}\\}$ from Proposition
7.17. Thus by Theorem 6.15 and Lemmas 6.7 and 7.18 we have isometric
isomorphims (7.20), and the relations in the assertion establish bijective
correspondences between the corresponding objects. ∎

######  Corollary 7.22.

For $p\in[1,+\infty]$ the isomorphisms (7.20) descend to $F^{p}(Q)\cong
F^{p}(\mathcal{G}_{Q})\cong F_{\operatorname{tight}}^{p}(S_{Q}),$ and for any
localizable $\mu$, (7.21) establish bijective correspondences between spatial
$Q$-families $(P,T)$ on $L^{p}(\mu)$, tight homomorphisms
$V:S_{Q}\to\operatorname{SPIso}(L^{p}(\mu))$ and representations
$\psi:F^{p}(Q)\to B(L^{p}(\mu))$ such that $\psi(t_{\mu}t_{\mu}^{*})$, $\mu\in
Q^{*}$, are operators of multiplication.

If $\mathbb{F}=\mathbb{C}$, then $F^{p}(Q)$ is universal for $Q$-families in
all $L^{p}$-operator algebras and for $p\neq 2$ every non-degenerate
representation $\psi:F^{p}(Q)\to B(L^{p}(\mu))$ comes from a spatial
$Q$-family $(P,T)$ on $L^{p}(\mu)$.

###### Proof.

Combine Theorem 7.21 and Corollary 6.16. ∎

######  Remark 7.23.

Originally $L^{p}$-Cuntz algebras [59] and graph $L^{p}$-operator algebras
$F^{p}(Q)$ [22] were defined (for $p\in[1,\infty)$ and
$\mathbb{F}=\mathbb{C}$) using spatial representations on standard measure
spaces. As $F^{p}(Q)$ is separable, one can limit only to such measures using
[60, Proposition 1.25]. The last part of Corollary 7.22 says that for
$p\in[1,+\infty]\setminus\\{2\\}$ the use of spatial $Q$-families is
inevitable. The tight semigroup picture for general graph $C^{*}$-algebras
seems to be absent from the literature.

######  Remark 7.24 (Banach–Cuntz algebras).

For any $n>1$ consider the graph $Q_{n}$ with a single vertex $v$ and $n$
edges $e_{1},...,e_{n}$. Then our $F(Q_{n})$ is a Banach algebra analogue of
the Cuntz $C^{*}$-algebra $F_{n}$. Daws and Horváth [25] proposed a different
construction: the _Daws–Horváth algebra_ $\mathcal{DH}_{n}$ is by definition
the quotient of the Banach $*$-algebra $\ell^{1}(S_{Q_{n}}\setminus\\{0\\})$
by an ideal $J$ generated by the element
$\delta_{(v,v)}-\sum_{i=1}^{n}\delta_{(e_{i},e_{i})}$. Thus it is the
universal Banach algebra generated by a contractive $Q_{n}$-family, or
equivalently the completion of $\mathcal{L}(Q_{n})$ in the largest
submutliplicative norm $\|\cdot\|_{DH}$ such that
$\|t_{e_{i}}\|_{DH}=\|t_{e_{i}}^{*}\|_{DH}=\|t_{v}\|_{DH}=1$ for
$i=1,\ldots,n$. As shown in [25],
$\mathcal{DH}_{n}=\overline{\mathcal{L}(Q_{n})}^{\|\cdot\|_{DH}}$ is naturally
a simple purely infinite Banach $*$-algebra. Thus we have an injective
$*$-homomorphism
$\mathcal{DH}_{n}\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}F(Q_{n})$
which is the identity on $\mathcal{L}(Q_{n})$. It seems unlikely that this
homomorphism is surjective. To prove this one could try to show that there are
no bounded traces on $F(Q_{n})$, cf. [25, Section 4].

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