Proof. Clearly, for any function $\xi : G\to C_u(X)$, formula (2.3) defines a surjective $C_0(X)$-module isometry $\psi : F(\alpha,\,u)\to F(\alpha,\,v)$ that respects the $G$-grading. Since

\begin{align*} \psi(a_s\delta_s \cdot a_t\delta_t)& =a_{s}\alpha_{s}(a_{t})u(s,t)\xi(st)\delta_{st}, \\ \psi(a_s\delta_s) \cdot \psi(a_t\delta_t) & =a_s\alpha_s(a_t)v(s,t)\alpha_s(\xi(t))\xi(s)\delta_{st}, \end{align*}

we see that $\psi$ is multiplicative if and only if $u(s,\,t)\xi (st)=v(s,\,t)\alpha _s(\xi (t))\xi (s)$ for all $s,\,t\in G$. Thus, if $\xi$ with $u=vd^{2}(\xi )$ exists, then $\psi$ given by (2.3) is an isometric isomorphism. It is the identity on $C_0(X)$ because for any $a\in C_0(X)$, using that $u(1,\,1)=v(1,\,1)\xi (1)$, we get

\[ \psi(a)=\psi(au(1,1)^*\delta_1)=au(1,1)^*\xi(1) \delta_1=av(1,1)^*\delta_1 =a. \]

Using also that $u(t^{-1},\,t) =v(t^{-1},\,t)\alpha _{t^{-1}}(\xi (t))\xi (1)^{*}\xi (t^{-1})$ we get $\psi ((a_t\delta _t)^*)=\psi (a_t\delta _t)^*$, for any $a_t\in C_0(X)$, $t\in G$. Thus, $\psi$ preserves the involution. Moreover, for any $\xi ': G\to C_u(X)$ we have $u=vd^{2}(\xi ')$ if and only if $d^2(\xi '\xi ^{-1})=1$ if and only if $\xi '=\xi \omega$ where $\omega \in Z^1(G,\,C_u(X))$. Hence, isomorphisms given by (2.3) are parametrized by elements of $Z^1(G,\,C_u(X))$.

Let us now assume that $\psi : F(\alpha,\,u) \to F(\alpha,\,v)$ is any isometric isomorphism respecting the $G$-gradings and $\psi |_{C_0(X)}=\text {id}_{C_0(X)}$. It suffices to show that $\psi$ satisfies (2.3) for some $\xi :G\to C_u(X)$. The assumption $\psi |_{C_0(X)}=\text {id}_{C_0(X)}$ means that $\psi$ is a $C_0(X)$-bimodule map. For each $t\in G$, we have a left $C_0(X)$-module map $E_t: F(\alpha,\,v) \to C_0(X)$ given by $E_t(b):=b(t)$, $b\in F(\alpha,\,v)$. Hence, the formula

\[ \xi(t)(a):=E_t\Big(\psi\big(a\delta_t\big)\Big), \qquad a\in C_0(X), \]

defines a left $C_0(X)$-module map on $C_0(X)$. This means that $\xi (t)$ is multiplier of $C_0(X)$. Hence, we may identify $\xi (t)$ with a function in $C_b(X)$, so that $\xi (t)(a)$ becomes $\xi (t) \cdot a$. This gives a map $\xi :G\to C_u(X)$. Since $\psi$ preserves the grading, for any $t\in G$ and $a_t\in C_0(X)$ we have $\psi (a_t\delta _t)=E_t(\psi (a_t\delta _t))\delta _t= (\xi (t) a_t) \delta _t,\,$ as desired.

Proof. That $\pi \rtimes v(b):=\sum _{t\in G}\pi (b(t))v_t$ defines a non-degenerate representation of $F(\alpha,\,u)$, for a covariant representation $(\pi,\, v)$, is straightforward. Let $\psi :F(\alpha,\,u)\to B(E)$ be any non-degenerate representation. Then $\pi :=\psi |_{C_0(X)}$ is a non-denenerate representation of $C_0(X)$. For any $t\in G$ and any contractive approximate unit $(\mu _i)$ in $C_0(X)$ the strong limit $v_t$ exists and we write $v_t= \text {s-}\lim _{i}\psi (\mu _i\delta _t)$. Indeed, by the Cohen–Hewitt factorization theorem, any element in $E$ has the form $\pi (a)\xi$ for some $\xi \in E$, $a\in C_0(X)$, and then $\lim _{i}\psi (\mu _i\delta _t)\pi (a)\xi =\lim _{i}\psi (\mu _i \alpha _t(a)\delta _t)\xi =\pi (\alpha _t(a)\delta _t)\xi$. This calculation also implies that $v_t\pi (a)=\pi (\alpha _t(a))v_t$ and $v_t$ does not depend on the choice of an approximate unit. For $s,\,t\in G$ we get

\begin{align*} v_{s}v_{t}& =\text{s-}\lim_{i}\text{s-}\lim_{j} \psi(\mu_j\delta_{s} \mu_i\delta_{t})=\text{s-}\lim_{i}\text{s-}\lim_{j} \psi(\alpha_{s}(\mu_i)u(s,t)\mu_j\delta_{st}) \\ & =\text{s-}\lim_{i}\pi(\alpha_{s}(\mu_i)u(s,t))v_{st}=\overline{\pi}(u(s,t))v_{st}. \end{align*}

In particular, $v_{t^{-1}} v_{t}=\overline {\pi }(u(t^{-1},\,t))$ is an invertible element. Thus, $v_t$ is in $\operatorname {Iso}(E)$, as an invertible contraction with a contractive inverse. Hence, $(\pi,\, v)$ is a covariant representation. Clearly, $\psi =\pi \rtimes v$ and $(\pi,\, v)$ is uniquely determined by this equality.

Proof. Routine calculations show that the formulas, for $a\in C_0(X)$, $s,\,t\in G$ and $x\in X$,

\[ \pi(a)\xi(x,t)=a(x)\xi(x,t), \quad v_s \xi(x,t)=u(s,s^{{-}1}t)(x)\xi(\varphi_{s^{{-}1}}(x),s^{{-}1}t) \]

define a covariant representation of $(\alpha,\,u)$ on $\ell ^p(X\times G)$ or $c_0(X\times G)$. For instance,

\begin{align*} v_s v_t\xi(x,r)& =u(s,s^{{-}1}r)(x)u(t,(st)^{{-}1}r)(\varphi_{s^{{-}1}}(x))\xi(\varphi_{(st)^{{-}1}}(x),(st)^{{-}1}r) \\ & \stackrel{(2.1)}=u(s,t)(x) u(st,(st)^{{-}1}r)(\varphi_{(st)^{{-}1}}(x))\xi(\varphi_{(st)^{{-}1}}(x),(st)^{{-}1}r) \\ & =\overline{\pi}(u(s,t))v_{st}\xi(x,r). \end{align*}

Clearly, $\Lambda _p=\pi \rtimes v$. To see that $\Lambda _p$ is injective, let $\{\mathbb {1}_{x,t}\}_{(x,t)\in X\times G}$ be the standard Schauder basis in $\ell ^p(X\times G)$, and note that, for $a\in C_0(X)$, we have

\[ \pi(a)\mathbb{1}_{x,t}= a(x) \mathbb{1}_{x,t} \quad \text{ and }\quad v_s \mathbb{1}_{x,t}=u(s,t)(\varphi_s(x))\mathbb{1}_{\varphi_s(x),st}. \]

If $\Lambda _p(\sum \limits _{s\in G} a_s\delta _s)=0$, then for all $(x,\,t)\in X\times G$ we have

\[ \Lambda_{p}\Big(\sum\limits_{s\in G} a_s\delta_s\Big)\mathbb{1}_{x,t}=\sum\limits_{s\in G} a_s(\varphi_s(x))u(s,t)(\varphi_s(x))\mathbb{1}_{\varphi_s(x),st}=0. \]

Since the elements $\{\mathbb {1}_{\varphi (x),st}\}_{s\in G}$ are linearly independent and $u(s,\,t)(\varphi _s(x))$ are non-zero numbers, this implies that $a_s(\varphi _s(x))=0$ for all $s\in G$ and $x\in X$. Hence, $\sum \limits _{s\in G}a_s\delta _s=0$.

Proof. Let $b\in F(\alpha,\,u)$. Since $\Lambda _p\cong \operatorname {Ind}_p(m_p)$ we only need to show that $\|\operatorname {Ind}_p(\pi )(b)\| \leq \|\Lambda _p(b)\|$ for any non-degenerate $\pi$ on an $L^p$-space or $C_0$-space when $p=\infty$. Let us first consider the case when $p=\infty$, and let $\pi :C_0(X)\to B(C_0(\Omega ))$ be a non-degenerate representation where $\Omega$ is some locally compact Hausdorff space. By [Reference Bardadyn, Kwaśniewski and Mckee6, Theorem 2.16] there is a continuous map $\Phi :\Omega \to X$ such that $\pi (a)\xi (y)=a(\Phi (y))\xi (y)$ for $\xi \in C_0(\Omega )$, $y\in \Omega$ and $a\in C_0(X)$. Using this and (2.4), for $\xi \in C_0(G,\, C_0(\Omega ))$, we get

\begin{align*} \| \operatorname{Ind}_{\infty}(\pi)(b)\xi\|& = \max_{y\in \Omega, t\in G} \left|\sum_{s\in G}[b(s)u(s,s^{{-}1}t)](\varphi_{t}(\Phi(y))) \xi(s^{{-}1}t)(y)\right| \\ & \leq \max_{x\in X} \sum_{s\in G} \left| [b(s)u(s,s^{{-}1}t)](\varphi_t(x))\right| \max_{y\in \Omega, t\in G} \left| \xi(t)(y)\right|= \|\Lambda_{\infty}(b)\| \|\xi\|, \end{align*}

and so $\|\operatorname {Ind}_{\infty }(\pi )(b)\| \leq \|\Lambda _{\infty }(b)\|$. For $p=2$ the assertion is a well-known fact, see [Reference Zeller-Meier43, Theorem 4.11]. Hence, it suffices to consider the case when $p\in [1,\,\infty )\setminus \{2\}$. Let us first consider a probability Borel measure $\mu$ on $X$ and a representation $m_p^{\mu }:C_0(X)\to B(L^p(\mu ))$ given by multiplication $m_p^{\mu }(a)\xi (x):=a(x)\xi (x)$, $\xi \in L^p(\mu )$. Denote by $\operatorname {Ind}_p(\mu )=\operatorname {Ind}_p(m_p^{\mu })$ the corresponding representation of $F(\alpha,\,u)$. Then $\operatorname {Ind}_p(\delta _x)$, where $\delta _x$ is the point mass measure at $x\in X$, can be treated as a representation on $\ell ^{p}(G)$. Moreover, $\Lambda _p\cong \operatorname {Ind}_p(m_p)\cong \bigoplus _{x\in X/G} \operatorname {Ind}_p(\delta _x)$ where $X/G$ is the orbit space for the action $\varphi$. Hence, for any $\xi \in \ell ^{p}(X,\,L^p(\mu ))$, putting $\xi ^x(t)=\xi (t)(x)$ for $x\in X$, we get

\begin{align*} \| \operatorname{Ind}_p(\mu)(b)\xi\|^p& = \int_{X}\sum_{t\in G} \left|\sum_{s\in G}[b(s)u(s,s^{{-}1}t)](\varphi_{t}(x))\xi(s^{{-}1}t)(x)\right|^{p} d\mu \\ & = \int_{X} \|\operatorname{Ind}_p(\delta_x)(b)\xi^x\|_{\ell^{p}(G)}^p\, d\mu \\ & \leq \sup_{x\in X} \|\operatorname{Ind}_p(\delta_x)(b)\|^p \int_{X} \|\xi^x\|_{\ell^{p}(G)}^p\, d\mu =\|\Lambda_p(b)\|^p \|\xi\|^p. \end{align*}

Thus, $\| \operatorname {Ind}_p(\mu )(b)\|\leq \|\Lambda _p(b)\|$. Let now $\pi :C_0(X)\to B(L^{p}(\nu ))$ be any non-degenerate representation on some measure space $(Y,\,\nu )$. By passing to the extension of the action $\alpha$ and representation $\pi$ to the algebra ${\mathbb{C}} 1+C_0(X)\subseteq C_{b}(X)$ we may assume that $X$ is compact. For every $\varepsilon >0$ there is $\xi \in \ell ^{p}(G,\,L^{p}(\nu ))$ such that

\[ \|\xi\|=1\,\, \text{ and }\,\, \|\operatorname{Ind}_p(\pi)(b)\|-\varepsilon\leq \|\operatorname{Ind}_p(\pi)(b)\xi \|. \]

Since the set $\{t\in G: \xi (t)\neq 0\}$ is countable, it generates a countable subgroup $G'$ of $G$. Since $\{s\in G: b(s)\neq 0\}$ is countable, the $C^*$-subalgebra $A$ of $C(X)$ generated by $\{\alpha _t(b(s)): s\in G,\,t\in G'\}\cup \{u(s,\,t): s,\,t\in G'\}$ is separable, unital and the twisted action $(\alpha,\,u)$ of $G$ on $C(X)$ restricts to a twisted action $(\alpha ',\, u')$ of $G'$ on $A$. Write $A\cong C(X')$ for a compact Hausdorff $X'$, and note that $\pi$ restricts to a unital representation $\pi :C(X')\to B( L^{p}(\nu ))$. By the proof of [Reference Phillips36, Proposition 1.25] there is a separable Banach subspace $F\subseteq L^{p}(\nu )$ such that $F$ is an $L^p$-space, $\pi (C(X'))F\subseteq F$ and $\{\xi (t):t\in G\}\subseteq F$. As $F$ is a separable $L^p$-space, there is a standard Borel probability measure $\nu '$ such that $F\cong L^p(\nu ')$. We identify $F=L^p(\nu ')$. Then $\pi$ yields a unital representation $\pi ':C(X')\to B(L^{p}(\nu '))$. By construction $\xi \in \ell ^{p}(G',\,L^{p}(\nu '))\subseteq \ell ^{p}(G,\,L^{p}(\nu ))$, $b\in F(\alpha ',\,u')$ and $\operatorname {Ind}_{p}(\pi )(b)\xi =\operatorname {Ind}_{p}(\pi ')(b)\xi$. Let $\mu$ be a Borel probability measure on $X'$ such that $\mu (a)=\nu '(\pi '(a))$ for $a\in C(X')$. By (the proof of) [Reference Choi, Gardella and Thiel12, Lemma 6.3] we have $\|\operatorname {Ind}_p(\pi ')(b)\|=\|\operatorname {Ind}_p(\mu )(b)\|$. Hence

\[ \|\operatorname{Ind}_{p}(\pi)(b)\xi\|=\|\operatorname{Ind}_{p}(\pi')(b)\xi\|\leq \| \operatorname{Ind}_p(\mu)(b)\|\leq \|\Lambda_p'(b)\| \]

where $\Lambda _p':F(\alpha ',\,u')\to \ell ^{p}(G',\,L^{p}(\nu '))$ is the regular representation (see the previous step). Since $\|\Lambda _p'(b)\|=\sup _{x'\in X'/G'} \| \operatorname {Ind}_p(\delta _{x'})\|$ there is $x' \in X'$ with $\|\Lambda _p'(b)\|-\varepsilon \leq \| \operatorname {Ind}_p(\delta _{x'})(b)\|$. Taking any preimage $x\in X$ of $x'$ under the quotient map $X\twoheadrightarrow X'$, one verifies that $\|\operatorname {Ind}_p(\delta _{x'})(b)\|=\|\operatorname {Ind}_p(\delta _{x})(b)\|$. Consequently, gathering all the pieces together, we get

\[ \|\operatorname{Ind}_p(\pi)(b)\|-\varepsilon\leq \|\operatorname{Ind}_p(\pi)(b)\xi \|\leq \|\Lambda_p'(b)\|\leq \|\Lambda_p(b)\| + \varepsilon, \]

which implies that $\|\operatorname {Ind}_p(\pi )(b)\|\leq \|\Lambda _p(b)\|$.

Proof. If $(f_i)_{i\in I}$ is an approximate invariant mean, then $g_i(t):=f_i(t)^{{1}/{p}}$ and $h_i(t)=f_i(t)^{{1}/{q}}$ are the desired nets. Indeed, using Hölders inequality, and that $|a-b|^p\leq |a^p-b^{p}|$ for $a,\,b\geq 0$, for any $s\in G$ we get

\begin{align*} \Bigg| \sum_{t\in G} \alpha_{s}(g_i(t))h_{i}(st) &-\sum_{t\in G} f_i(t)\Bigg| \leq\sum_{t\in G} \left|\alpha_{s}(f_i(t)^{{1}/{p}})f_{i}(st)^{{1}/{q}} - f_i(st)\right| \\ &\quad =\sum_{t\in G} \left|\alpha_{s}(f_i(t)^{{1}/{p}}) -f_i(st)^{{1}/{p}}\right| f_{i}(st)^{{1}/{q}} \\ & \quad\leq\left(\sum_{t\in G}\left| \alpha_{s}(f_i(t)^{{1}/{p}}) -f_i(st)^{{1}/{p}}\right|^p\right)^{{1}/{p}} \left(\sum_{t\in G} f_{i}(st) \right)^{{1}/{q}} \\ & \quad\leq \left(\sum_{t\in G} \left|\alpha_{s}(f_i(t)) -f_i(st)\right|\right)^{{1}/{p}}\left(\sum_{t\in G} f_{i}(t) \right)^{{1}/{q}}. \end{align*}

As $\sum _{t\in G} f_{i}(t)$ converges compactly to $1$, the above estimate implies that $\sum _{t\in G} \alpha _{s}(g_i(t))h_{i}(st)$ also converges compactly to $1$.

Proof. Let $(\pi,\,v)$ be a covariant representation of $(\alpha,\,u)$ on some Banach space $E$ and let $(g_i)_{i\in I}$, $(h_i)_{i\in I}$ be nets in $C_c(G,\,C_c(X)^+)$ as in lemma 3.4. For $\xi \in E$, $\eta \in E'$ we put

\[ \xi_i(t):= v_{t^{{-}1}}\pi(\overline{u(t,t^{{-}1})}g_i(t))\xi, \qquad \eta_i(t):=v_{t}' \pi(h_i(t))'\eta, \qquad t\in G. \]

Then $\xi _i\in \ell ^{p}(G,\,E)$ and $\eta _i\in \ell ^{q}(G,\,E')$. Using the pairing establishing the isomorphism $\ell ^{p}(G,\,E)'\cong \ell ^{q}(G,\,E')$, for any $a\in C_c(X)$ and $s\in G$ we get

\begin{align*} \langle \operatorname{Ind}_p(\pi)(a\delta_{s})\xi_{i}, \eta_{i}\rangle & = \sum_{t\in G} \langle \pi(\alpha_{st}^{{-}1}(au(s,t) )) \xi_{i}(t),\eta_i(st)\rangle \\ & = \sum_{t\in G} \Big\langle \pi(\alpha_{st}^{{-}1}(au(s,t) )) v_{t^{{-}1}} \pi\big(\overline{u(t,t^{{-}1})} g_i(t)\big)\xi, v_{st} '\pi(h_i(st))'\eta\Big\rangle \\ & = \sum_{t\in G} \Big\langle \pi\Big(h_{i}(st) au(s,t) u(st,t^{{-}1}) \alpha_{s}(g_i(t)) \overline{u(t,t^{{-}1})}\Big)v_{s} \xi , \eta\Big\rangle \\ & =\Big\langle \pi\Big( \sum_{t\in G} h_{i}(st) \alpha_{s}(g_i(t)) a\Big)v_{s} \xi , \eta\Big\rangle \longrightarrow \langle \pi(a)v_{s} \xi,\eta\rangle. \end{align*}

By linearity, for every $b\in C_c(G,\,C_c(X))\subseteq F(\alpha,\,u)$ we get $\langle \pi \rtimes v (b) \xi,\,\eta \rangle =\lim _{i} \langle \operatorname {Ind}_p(\pi )(b)\xi _{i},\, \eta _{i}\rangle$. This shows that $\pi \rtimes v (b)\neq 0$ implies $\operatorname {Ind}_p(\pi )(b)\neq 0$ for any $b\in F(\alpha,\,u)$. To get norm estimates we specialize to the case where $E=L^p(\mu )$ for some measure $\mu$ (then $E'=L^{q}(\mu )$). Without changing the Banach space we may assume that $\mu$ is localizable. $E=L^p(\mu )$ for some measure $\mu$ (then $E'=L^{q}(\mu )$). Without changing the Banach space we may assume that $\mu$ is localizable. Also we may assume that $p\neq 2$, as for $p=2$ the assertion is well known, see [Reference Anantharaman-Delaroche1, Theorem 5.3]. Then $\pi$ is given by multiplication operators, so there is a homomorphism $\pi _0:C_0(X)\to L^{\infty }(\mu )$, such that $\pi (a)\xi =\pi _0(a) \cdot \xi$, $\xi \in L^p(\mu )$, see [Reference Bardadyn, Kwaśniewski and Mckee6, Theorem 2.14]. Using this, and that $\sum _{t\in G} g_i(t)^p$ converges compactly to $1$, for any $\xi \in \pi (C_c(X))L^{p}(\mu )$, we get

\begin{align*} \|\xi_i\|^p& =\sum_{t\in G}\|v_{t^{{-}1}}\pi(g_i(t))\xi\|^p=\sum_{t\in G}\int |\pi_0(g_i(t))\xi|^p \, d\mu=\int \pi_0\left(\sum_{t\in G}g_i(t)^p\right)| \xi|^p \, d\mu \\ & \longrightarrow \int |\xi|^p \, d\mu=\|\xi\|^p. \end{align*}

Similarly, for $\eta \in \pi (C_c(X))L^{q}(\mu )$ we get $\|\eta _i\|\longrightarrow \|\eta \|$. Thus, for any $b\in C_c(G,\,C_c(X))\subseteq F(\alpha,\,u)$ we have $|\langle \pi \rtimes v (b) \xi,\,\eta \rangle |=|\lim _{i} \langle \operatorname {Ind}_p(\pi )(b)\xi _{i},\, \eta _{i}\rangle |\leq \| \operatorname {Ind}_p(\pi )(b)\| \|\xi \| \|\eta \|$, which implies $\| \pi \rtimes v (b)\|\leq \|\operatorname {Ind}_p(\pi )(b)\|$. Since $(\pi,\, v)$ was arbitrary, we obtain $\|b\|_{L^p}\leq \|\Lambda _p(b)\|$ by proposition 3.1. Therefore, the full and reduced norms coincide.

Proof. We may assume that $F_{\mathcal {R}}(\alpha,\,u)\subseteq B(E)$ acts in a non-degenerate way on some Banach space $E$ (one may take $E=F_{\mathcal {R}}(\alpha,\,u)$). By lemma 2.8, the inclusion map disintegrates to a covariant representation $(\text {id}_{C_0(X)},\, v)$. In particular, $b=\sum _{t\in G} b(t)v_t$ for $b\in F(\alpha,\,u)\subseteq F_{\mathcal {R}}(\alpha,\,u)$. The invertible isometries $v_t$, $t\in G$, are multipliers of $C_c(G,\,C_0(X))\subseteq F(\alpha,\,u)\subseteq F_{\mathcal {R}}(\alpha,\,u)$, and so also of $F_{\mathcal {R}}(\alpha,\,u)$. Thus, the formula $E_{t}^{\mathcal {R}}(b):=E_{1}^{\mathcal {R}}(b v_{t}^{-1})$ yields a well-defined contractive map $E_{t}^{\mathcal {R}}:F_{\mathcal {R}}(\alpha,\,u)\to C_0(X)$. Simple verification, using that $v_{t}^{-1}=\overline {u(t^{-1},\,t)} v_{t^{-1}}= v_{t^{-1}}\overline {u(t,\,t^{-1})}$, shows that $E_{t}^{\mathcal {R}}$ extends $E_t$ and we also have $E_{t}^{\mathcal {R}}(b)=\alpha _t(E_{1}^{\mathcal {R}}(v_{t}^{-1}b))$, $b\in F_{\mathcal {R}}(\alpha,\,u)$. Since $E_{1}^{\mathcal {R}}$ is $C_0(X)$-bimodule map, this readily implies that

(4.1)\begin{equation} E_{t}^{\mathcal{R}}(ab)=a E_{t}^{\mathcal{R}}(b), \qquad E_{t}^{\mathcal{R}}(ba)=E_{t}^{\mathcal{R}}(b)\alpha_t(a), \end{equation}

for all $a\in C_0(X)$, $b\in F_{\mathcal{R}}(\alpha,u)$. Similarly, using that $v_{t}^{-1}v_{s}^{-1}=v_{st}^{-1}\overline {u(s,\,t)}$ for any $s,\,t\in G$ and $b\in F_{\mathcal {R}}(\alpha,\,u)$ we have

\[ E_{t}^{\mathcal{R}}(v_s^{{-}1}b)=E_{st}^{\mathcal{R}}(b) \alpha_t \big(\overline{u(s,t)}\big), \qquad E_{s}^{\mathcal{R}}(bv_t^{{-}1})=E_{st}^{\mathcal{R}}(b) \overline{u(s,t)}. \]

Exploiting these relations we see that $\bigcap _{t\in G}\ker (E_t^{\mathcal {R}})$ is an ideal.

Proof. Assume that $p<\infty$. We identify $F^{p}_{\operatorname {r}}(\alpha,\,u)$ with $\overline {\Lambda _p(F(\alpha,\,u))}\subseteq B(\ell ^p (X\times G))$ and use the notation from the proof of lemma 2.11. For each $(x,\,t)\in X\times G$ we let $P_{x,t}(\xi ):=\xi (x,\,t) \mathbb {1}_{x,t}$ be a contractive projection in $B(\ell ^p(X\times G))$ onto the one-dimensional space $\mathbb {C} \mathbb {1}_{x,t}$. For any $b\in B(\ell ^p(X\times G))$ the series

\[ \mathbb{E}_1(b):=\sum\limits_{(x,t)\in X\times G}P_{x,t} b P_{x,t} \]

is strongly convergent and $\|\mathbb {E}_1(b)\|\leq \|b\|$, as for any $\xi \in \ell ^p(X\times G)$ we get

\[ \|E_1(b)\xi\|^{p}\!=\!\sum\limits_{(x,t)\in X\times G} \|P_{x,t} b \xi(x,t) \mathbb{1}_{x,t}\|^p \leq \|b\|^{p} \sum\limits_{(x,t)\in X\times G} \|\xi(x,t) \mathbb{1}_{x,t}\|^p\!=\!\|b\|^{p} \|\xi\|^{p} \]

(these sums are in fact countable, as $\xi (x,\,t)\neq 0$ only for a countable set of pairs $(x,\,t)$). Also if $b\in F(\alpha,\,u)\subseteq F^{p}_{\operatorname {r}}(\alpha,\,u)\subseteq B(\ell ^p(X\times G))$, then clearly $\mathbb {E}_1(b)=E_1(b)=b(1)$. Thus $\mathbb {E}_1$ is a contractive extension of $E_1$. For any $t\in G$, the formula $\mathbb {E}_t(b):=\mathbb {E}_1(b v_{t}^{-1})$ defines a contractive map $\mathbb {E}_t:F^{p}_{\operatorname {r}}(\alpha,\,u)\to C_0(X)$ that extends $E_t$. Hence $\{\mathbb {E}_t\}_{t\in G}$ is the Fourier decomposition of $F^{p}_{\operatorname {r}}(\alpha,\,u)$. For any $s,\,t\in G$, $x\in X$ we have

(4.2)\begin{equation} \mathbb{E}_s(b)\mathbb{1}_{x,t}:=P_{x,t} b \overline{\pi}(u(s,s^{{-}1}t)^{{-}1}) \mathbb{1}_{\varphi_s^{{-}1}(x),s^{{-}1}t} . \end{equation}

The operators of the form $av_{s}$, $a\in C_0(X)$, $s\in G$, and hence also every $b\in F^{p}_{\operatorname {r}}(\alpha,\,u)$, maps $\mathbb {1}_{x,t}$ to the closed linear span of elements $\mathbb {1}_{\varphi _s(x),st}$, $s\in G$. Therefore, if $b\neq 0$ there are $s,\,t\in G$ and $x\in X$ such that $P_{\varphi _s(x),st} b \mathbb {1}_{x,t}\neq 0$. By (4.2), the latter is equivalent to $\mathbb {E}_s(b)\mathbb {1}_{\varphi _s(x),st}\neq 0$. Hence, $b\neq 0$ implies $\mathbb {E}_{s}(b)\neq 0$ for some $s\in G$. Consequently, $\bigcap _{t\in G}\ker (\mathbb {E}_t)=\{0\}$.

For $p=\infty$ the proof above works with $\ell ^p(X\times G)$ replaced by $c_0(X\times G)$, but also the assertion follows from the case $p=1$ using the anti-isomorphism $F^{\infty }_{\operatorname {r}} (\alpha,\,u)\stackrel {anti}{\cong }F^{1}_{\operatorname {r}} (\alpha,\,u)$.

Proof. Put $E_{p,t}:=\mathbb {E}_t\circ \Lambda _p$, $t\in G$, where $\Lambda _p:F^{p}(\alpha,\,u)\to F^p_{\operatorname {r}}(\alpha,\,u)$ is the regular representation and $\{\mathbb {E}_{t}\}_{t\in G}$ is the Fourier decomposition of $F^p_{ \operatorname {r}}(\alpha,\,u)$ given by lemma 4.8.

Proof. Clearly, $\|\cdot \|_{\mathcal {R}^*}$ is a submultiplicative norm with $\|b\|_{\mathcal {R}^*}\leq \sum _{t\in G} \|b(t)\|_{\infty }$ for $b\in F(\alpha,\,u)$ (because $\|\cdot \|_{\mathcal {R}}$ has these properties). The involution on $F(\alpha,\,u)$ extends to an antimultiplicative antilinear isometry $*$ from $F_{\mathcal {R}^*}(\alpha,\,u)$ onto $F_{\mathcal {R}}(\alpha,\,u)$. In particular, if $F_{\mathcal {R}}(\alpha,\,u)$ admits the FD, then so does $F_{\mathcal {R}^*}(\alpha,\,u)$, as then $\|b(1)\|=\|b(1)^*\|\leq \|b^*\|_{\mathcal {R}}= \|b\|_{\mathcal {R}^*},\,$ for all $b\in F(\alpha,\,u)$. Moreover, if $\{E_{t}^{\mathcal {R}}\}_{t\in G}$ is the FD for $F_{\mathcal {R}}(\alpha,\,u)$ and $\{E_{t}^{\mathcal {R}^*}\}_{t\in G}$ is the FD for $F_{\mathcal {R}^*}(\alpha,\,u)$, then by continuity and the formula for involution on $F(\alpha,\,u)$ we get

\[ E_t^{\mathcal{R}^*}(a)=\alpha_{t}(E_{t^{{-}1}}^{\mathcal{R}}(a^*))^* u(t,t^{{-}1})^*, \qquad t\in G, \, a\in F_{\mathcal{R}^*}(\alpha,u). \]

This readily implies that $F_{\mathcal {R}^*}(\alpha,\,u)$ is reduced if and only if $F_{\mathcal {R}}(\alpha,\,u)$ is.

Clearly, $\|\cdot \|_{\{ \mathcal {R}_i\}_{i\in I}}$ is a submultiplicative norm with $\|b\|_{\{ \mathcal {R}_i\}_{i\in I}}\leq \sum _{t\in G} \|b(t)\|_{\infty }$ for $b\in F(\alpha,\,u)$ (because each $\|\cdot \|_{\mathcal {R}_{i}}$ has these properties). If there is at least one $i_0\in I$ such that $F_{\mathcal {R}_{i_0}}(\alpha,\,u)$ admits the FD, then $F_{\{ \mathcal {R}_i\}_{i\in I}}(\alpha,\,u)$ admits the FD, as then $\|b(1)\|_{\infty }\leq \|b\|_{R_{i_0}}\leq \|b\|_{\{ \mathcal {R}_i\}_{i\in I}}$ for all $b\in F(\alpha,\,u)$. For each $i\in I$, let $\{E_{t}^{\mathcal {R}_i}\}_{t\in G}$ be the FD for $F_{\mathcal {R}_{i}}(\alpha,\,u)$, and denote by $\{\mathcal {E}_{t}\}_{t\in G}$ the FD for $F_{\{ \mathcal {R}_i\}_{i\in I}}(\alpha,\,u)$. Note that $\pi (a):=\prod _{i\in I} a$ is an isometric embedding of $F_{\{ \mathcal {R}_i\}_{i\in I}}(\alpha,\,u)$ into the direct product $\prod _{i\in I} F_{\mathcal {R}_{i}}(\alpha,\,u)$. Moreover, $\prod _{i\in I} \mathcal {E}_{t}= \prod _{i\in I} E_{t}^{\mathcal {R}_i}\circ \pi$ (as maps from $F_{\{ \mathcal {R}_i\}_{i\in I}}(\alpha,\,u)$ to $\prod _{i\in I} C_0(X)$). Thus, $\bigcap _{t\in G}\ker (\mathcal {E}_t)=0$ if $\bigcap _{t\in G}\ker ( E_{t}^{\mathcal {R}_i})=0$ for all $i\in I$. That is, $F_{\{ \mathcal {R}_i\}_{i\in I}}(\alpha,\,u)$ is reduced if all $F_{\mathcal {R}_{i}}(\alpha,\,u)$, $i\in I$, are reduced.

Proof. The formula for the norm follows from proposition 3.1. The statement when $P \subseteq \{1,\,\infty \}$ holds by remark 2.13 and definition 4.3. When $\varphi$ is amenable, then full equals to reduced by theorem 3.5.

Proof. If $\varphi$ is not topologically free, then there is $t\in G\setminus \{1\}$ and a non-zero $a\in C_0(X)$ with $\operatorname {supp}(a):=\overline {\{x:a(x)\neq 0\}}\subseteq \{x\in X:\varphi _t(x)=x\}$. Then $b:=av_t\notin C_0(X)$ and $b$ commutes with every $f\in C_0(X)$ because $b\cdot f= av_t\cdot f= a\alpha _t(f) v_{t}=f a v_t= f \cdot b$. So $C_0(X)$ is not maximal abelian subalgebra of $F_{\mathcal {R}}(\alpha,\,u)$ (here $F_{\mathcal {R}}(\alpha,\,u)$ does not need to be reduced). Conversely, assume that $C_0(X)$ is not a maximal abelian subalgebra of $F_{\mathcal {R}}(\alpha,\,u)$, and let $b\in F_{\mathcal {R}}(\alpha,\,u) \setminus C_0(X)$ commute with all elements in $C_0(X)$. Since $b\not \in C_0(X)$ and $F_{\mathcal {R}}(\alpha,\,u)$ is reduced, there is $t\in G\setminus \{1\}$ such that $E_t(b)\neq 0$. For any $f\in C_0(X)$, using (4.1) and that $b$ commutes with $f$ we get $f E_t(b)= E_t(fb)= E_t(bf)=E_t(b)\alpha _t(f)$, which implies that $\operatorname {supp}(E_t(b))\subseteq \{x\in X:\varphi _t(x)=x\}$. Hence $\varphi$ is not topologically free.

Proof. For $t=1$ this follows by (the proof of) [Reference Exel, Laca and Quigg17, Proposition 2.4]. For general $t\in G$ we may use that $E_t(b)=\lim _{i} E_1(b \cdot (\mu _i \delta _{t})^*)$ for any contractive approximate unit $(\mu _i)$ in $C_0(X)$. Namely, putting $b_i:=b \cdot (\mu _i \delta _{t})^*$ from some point on we have $\|E_t(b)-E_1(b_i)\|\leq \varepsilon /3$, and by the previous remark we may find a norm one $h\in C_c(X)^+$ such that $\|E_1(b_i)\|\leq \| hE_1(b_i)h\|+\varepsilon /3$ and $\|hE_1(b_i)h-hb_ih\|\leq \varepsilon /3.$ Using these three inequalities we get

\begin{gather*} \| E_t(b)\|\leq \varepsilon/3+ \|E_1(b_i)\|\leq \| hE_1(b_i)h\|+2\varepsilon /3 \leq \| hE_t(b)h\|+\varepsilon \\ \|hE_t(b)h-h b_ih\|\leq\varepsilon/3 +\|hE_1(b_i)h-hb_ih\|\leq 2\varepsilon/3.\end{gather*}

Since $\lim _{i}b_ih =b (h\delta _t)^*$ we get the assertion.

Proof. Fix $t\in G$. Take any $b\in F(\alpha,\,u)\subseteq F_{\mathcal {R}}(\alpha,\,u)$ and $\varepsilon >0$. Let $h\in C_c(X)^+$ be as in lemma 5.3. Using that $\psi$ is isometric on $C_0(X)$, by lemma 5.4, we get

\begin{align*} \|E_t^{\mathcal{R}}(b)\| & = \|E_t(b)\|\leq \| h E_t(b)h\|+\varepsilon=\|\psi(h E_t(b)h)\|+\varepsilon \\ & \leq\|\psi(h b (h\delta_t)^*)\|+2\varepsilon \leq\|\psi(b)\|+2\varepsilon. \end{align*}

This shows that $\|E_t^{\mathcal {R}}(b)\|\leq \|\psi (b)\|$. Hence, the formula $E^{\psi }_{t}(\psi (b)):=E_t^{\mathcal {R}}(b)$, $b\in F(\alpha,\,u)$, yields a well-defined linear contraction $\psi (F(\alpha,\,u))\to C_0(X)$ which extends to the contraction $E^{\psi }_{t}:B\to C_0(X)$ satisfying $E^{\psi }_{t}(\psi (b))=E_t^{\mathcal {R}}(b)$ for $b\in F_{\mathcal {R}}(\alpha,\,u)$.

Proof. (1) implies (2) by corollary 5.8 and remark 4.13. (2) implies (3) because $\ker (\Lambda _P^{\operatorname {tr}})\cap C_0(X)=\{0\}$. To show that (3) implies (1) assume that $\varphi$ is not topologically free. Then there are $t\in G\setminus \{1\}$ and an open non-empty set $U\subseteq X$ such that $\varphi _t|_{U}=\text {id}_{U}$. Take any non-zero $a\in C_0(U)\subseteq C_0(X)$. Then $b:=a-\delta _{t^{-1}}a\in F(\alpha )$ is non-zero, and so $\Lambda _P(b)\neq 0$, but (2.5) readily implies that $\Lambda _P^{\operatorname {tr}}(b)=0$. Hence, $\ker (\Lambda _P^{\operatorname {tr}})\not \subseteq \ker (\Lambda _P)$. Thus, (1)–(3) are equivalent. These conditions are independent of $P$ because (1) is. Condition (4) is equivalent to (2) for $P=T$, because $F^T(\alpha )=F^T_{\operatorname {r}}(\alpha )$ by corollary 4.14. Hence, all conditions (1)–(4) are equivalent.

Proof. Since $\ker (\Lambda _P)\cap C_0(X)=\{0\}$, we see that $C_0(X)$ detects ideals in $F^P(\alpha )$ if and only if $\ker (\Lambda _P)=\{0\}$ and condition (2) in theorem 5.9 holds.

Proof. Amenability of $\varphi :G\to \operatorname {Homeo}(X)$ passes to the restricted action $\varphi :H\to \operatorname {Homeo}(X)$. Hence, $F^P (\alpha |_{H},\,u|_{H})=F_{\operatorname {r}}^P (\alpha |_{H},\,u|_{H})$ by corollary 4.14. In particular, $F^P (\alpha |_{H},\,u|_{H})$ embeds into $F^P (\alpha,\,u)$.

Proof. If $J$ is an ideal in $F_{\mathcal {R}}(\alpha,\,u)$, then $J\cap C_0(X)$ is an ideal in $C_0(U)$ and hence $J\cap C_0(X)=C_0(U)$ for an open set $U\subseteq X$. Disintegrating the identity map on $F_{\mathcal {R}}(\alpha,\,u)$, see lemma 2.8, and using covariance relations, for any $a\in C_0(U)$ and $t\in G$ we get $\alpha _t(a)=v_t a v_{t}^{-1}\in J\cap C_0(X)=C_0(U)$. Hence, $U$ is $\varphi$-invariant. Now let $U$ be any $\varphi$-invariant open set. The ideal generated by $C_0(U)$ in $F(\alpha,\,u)= \ell ^1(G,\,C_0(X))$ is $F(\alpha ^U,\,u^U)=\ell ^1(G,\,C_0(U))$, and hence the ideal generated by $C_0(U)$ in $F_{\mathcal {R}}(\alpha,\,u)$ is $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)$. Moreover, $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)\cap C_0(X)=C_0(U)$. Indeed, the inclusion $C_0(U)\subseteq F_{\mathcal {R}^U}(\alpha ^U,\,u^U)\cap C_0(X)$ is obvious and for the opposite inclusion note that $E_1^{\mathcal {R}}$ is the identity on $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)\cap C_0(X)$, and $E_1^{\mathcal {R}}(F_{\mathcal {R}^U}(\alpha ^U,\,u^U))=C_0(U)$ because $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)$ is the closure of $F(\alpha ^U,\,u^U)$ and $E_1(F(\alpha ^U,\,u^U))=C_0(U)$. This proves the first part of the assertion. For the second part, assume that $C_0(U)$ does not detect ideals in $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)$, so that there is a non-zero ideal $I$ in $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)$ with $I\cap C_0(U)=\{0\}$. Since $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)$ has an approximate unit which is an approximate unit in $C_0(U)$, and $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)$ is an ideal in $F_{\mathcal {R}}(\alpha,\,u)$, we conclude that $I$ is an ideal in $F_{\mathcal {R}}(\alpha,\,u)$ and $I\cap C_0(X)=I\cap C_0(U)=\{0\}$. Hence, $C_0(X)$ does not detect ideals in $F_{\mathcal {R}}(\alpha,\,u)$.

Proof. Since $\alpha _{t}=\text {id}$ for all $t\in G$, each $u_x \in Z^2(G,\,\mathbb {T})$, $x\in X$, is a 2-cocycle for $G$. Since $\varphi$ is trivial we have $\|\cdot \|_{L^1,\operatorname {r}}=\|\cdot \|_{L^{\infty },\,\operatorname {r}}$ by (2.4). Hence, we may assume that $P\subseteq [1,\,\infty )$. A simple calculation shows that for any $b\in F(\alpha,\,u)$ and $p\in [1,\,\infty )$ we have

\[ \|b\|_{L^p,\operatorname{r}}=\|\Lambda_{p}(b)\|=\max_{x\in X}\left( \sum_{t\in G}|b(t)(x)|^{p}\right)^{{1}/{p}}. \]

This calculation also shows that $\|\widehat {b}(x)\|_{F_{\operatorname {r}}^p(G,\,u_x)}=( \sum _{t\in G}|b(t)(x)|^{p})^{{1}/{p}}$, for each $x\in X$. This implies that the map $b\mapsto \widehat {b}$ is a well-defined isometry from $F(\alpha,\,u)\subseteq F^P(\alpha,\,u)$ into the space of bounded sections of $\mathcal {B}=\{ F_{\operatorname {r}}^P(G,\,u_x)\}_{x\in X}$ equipped with the supremum norm, as we have

\[ \|b\|=\sup_{s\in P}\max_{x\in X}\left( \sum_{t\in G}|b(t)(x)|^{p}\right)^{{1}/{p}}= \sup_{x\in X}\sup_{s\in P} \left( \sum_{t\in G}|b(t)(x)|^{p}\right)^{{1}/{p}}=\|\widehat{b}\|. \]

Clearly, $b\mapsto \widehat {b}$ is an algebra homomorphism. By Fell's reconstruction theorem [Reference Fell and Doran18, II.13.18] there is a unique topology on $\mathcal {B}$ making sections $\widehat {b}$, $b\in F(\alpha,\,u)$, continuous. Since elements of $C_c(G,\,C_c(X))\subseteq F(\alpha,\,u)$ are mapped into compactly supported sections of $\mathcal {B}$, and any compactly supported section can be approximated by such elements, we conclude that $b\mapsto \widehat {b}$ extends to the isometric isomorphism $F^P(\alpha,\,u)\cong C_0(\mathcal {B})$.

Proof. If all non-trivial ideals in $F_{\mathcal {R}}(\alpha,\,u)$ are contained in $\bigcap _{t\in G}\ker (E_t^{\mathcal {R}})$, then $\bigcap _{t\in G}\ker (E_t^{\mathcal {R}})$ is the hidden ideal for $C_0(X)\subseteq F_{\mathcal {R}}(\alpha,\,u)$, and $\varphi$ is minimal by lemma 5.14. Conversely, assume $\bigcap _{t\in G}\ker (E_t^{\mathcal {R}})$ is the hidden ideal for $C_0(X)\subseteq F_{\mathcal {R}}(\alpha,\,u)$ and let $J$ be an ideal in $F_{\mathcal {R}}(\alpha,\,u)$ which is not contained in $\bigcap _{t\in G}\ker (E_t^{\mathcal {R}})$. Then $J$ intersects $C_0(X)$ non-trivially and hence, by lemma 5.14, $J$ contains $F_{\mathcal {R}^U}(\alpha ^U,\,u^U)$ for some non-empty $\varphi$-invariant open $U\subseteq X$. Thus, if $\varphi$ is minimal, we necessarily have $J=F_{\mathcal {R}}(\alpha,\,u)$, that is $J$ is trivial.

Proof. (1) implies (2) and (4) by lemma 6.2 and corollary 5.8. Implications (2)$\Rightarrow$(3) and (4)$\Rightarrow$(5) are immediate, see remark 4.13 and corollary 4.14. We have (3)$\Rightarrow$(1) and (5)$\Rightarrow$(1) by lemma 6.2 and theorem 5.9. This proves the equivalence of conditions (1)–(5). The final claim now follows from the equivalence (1)$\Leftrightarrow$(5).