2 Main results

If $\mathfrak {B}$ is a Banach algebra, $\mathfrak {B}(b_1,\ldots , b_n)$ denotes the closed subalgebra of $\mathfrak {B}$ generated by the elements $b_1,\ldots , b_n\in \mathfrak {B}$ . We set $\widetilde {\mathfrak {B}}$ to denote the smallest unitization of $\mathfrak {B}$ (it coincides with $\mathfrak {B}$ when $\mathfrak {B}$ is already unital). As usual, if $\mathfrak {B}$ is not unital, the norm of an element $b+\lambda 1\in \widetilde {\mathfrak {B}}$ is given by $\|b+\lambda 1\|_{\widetilde {\mathfrak {B}}}=\|b\|_{\mathfrak {B}}+|\lambda |$ , and we write

$$ \begin{align*}\mathrm{Spec}_{\mathfrak{B}}(b)=\{\lambda\in\mathbb{C}\mid b-\lambda1\text{ is not invertible in }\widetilde{\mathfrak{B}}\} \end{align*} $$

to denote the spectrum of an element $b\in \mathfrak {B}$ .

If $\mathfrak {B}$ has an involution, $\mathfrak {B}_{\mathrm {sa}}$ denotes the set of self-adjoint elements in $\mathfrak {B}$ , that is, of all $b\in \mathfrak {B}$ such that $b^*=b$ . A Banach $^*$ -algebra admitting a $C^*$ -norm is called an $A^*$ -algebra (also called a reduced Banach $^*$ -algebra). If $\mathfrak {B}$ is an $A^*$ -algebra, we use $\mathrm {C^*}(\mathfrak {B})$ to denote the universal $C^*$ -algebra generated by $\mathfrak {B}$ . In such a case, $\mathfrak {B}$ can be identified with a dense subalgebra of $\mathrm {C^*}(\mathfrak {B})$ , and the embedding is contractive.

If $\mathfrak {B}$ is a commutative Banach algebra with spectrum $\Delta _{\mathfrak {B}}$ , then $\hat b\in C_0(\Delta _{\mathfrak {B}})$ denotes the Gelfand transform of $b\in \mathfrak {B}$ . If the Gelfand transform is injective, $\mathfrak {B}$ is called a Banach function algebra.

Definition 2.2 Let $\mathfrak {A}$ be an $A^*$ -algebra and let $\mathfrak {B}$ be a Banach algebra. A continuous monomorphism with dense image $\mathfrak {A}\to \mathfrak {B}$ is called a locally regular inclusion if

(2.1) $$ \begin{align} \mathfrak{A}(b)\text{ is a regular Banach function algebra, }\forall b\in\mathfrak{A}_{\mathrm{sa}}. \end{align} $$

In what follows, we will identify a given Banach algebra with any of its continuous monomorphic images. Under such a view, a subset $\mathfrak {A}$ of the Banach algebra $\mathfrak {B}$ produces a locally regular inclusion if $\mathfrak {A}$ is a dense subalgebra that also admits an involution and a finer norm under which it becomes a Banach $^*$ -algebra satisfying (2.1).

Let $\mathfrak {B}$ be a Banach algebra. A Banach space $\mathcal X$ that is also a $\mathfrak {B}$ -bimodule is called a Banach $\mathfrak {B}$ -bimodule if the maps

$$ \begin{align*}\mathfrak{B}\times \mathcal X\ni(b,\xi)\mapsto b\xi \in\mathcal X \quad\text{ and }\quad \mathcal{X}\times\mathfrak{B}\ni(\xi,b)\mapsto \xi b \in\mathcal X \end{align*} $$

are jointly continuous.

Example 2.2 Let $\mathfrak {B}\subset \mathfrak C$ be an inclusion of Banach algebras. Then $\mathfrak C$ is naturally a Banach $\mathfrak {B}$ -bimodule under the actions

$$ \begin{align*}\mathfrak{B}\times\mathfrak C\ni (b,c)\mapsto bc\in \mathfrak C\quad\text{ and }\quad \mathfrak C\times\mathfrak{B}\ni (c,b)\mapsto cb\in \mathfrak C.\end{align*} $$

Definition 2.3 Let $\mathfrak {B}$ be a Banach algebra and let $\mathcal {X}$ be a Banach $\mathfrak {B}$ -bimodule. A linear map $D:\mathfrak {B}\to \mathcal {X}$ is called a derivation if, for each $a,b\in \mathfrak {B}$ , one has

$$ \begin{align*}D(ab)=D(a)b+aD(b).\end{align*} $$

Definition 2.4 Let $\mathfrak {B}$ be a Banach algebra and let $D:\mathfrak {B}\to \mathcal {X}$ be a derivation. Then

$$ \begin{align*}\mathscr I(D)=\{b\in\mathfrak{B}\mid \text{the maps } a\mapsto D(ba)\text{ and } a\mapsto D(a b)\text{ are continuous}\} \end{align*} $$

is the continuity ideal of D.

The following lemma is an immediate application of well-known results by Barnes that can be found in [Reference Barnes6, Theorems 2.1 and 2.2].

Lemma 2.3 Let $\mathfrak {A}\subset \mathfrak {B}$ be a locally regular inclusion and let J be a closed two-sided ideal of $\mathrm { C^*}(\mathfrak {A})$ . Then

$$ \begin{align*}\mathrm{Spec}_{\mathfrak{A}/(J\cap\mathfrak{A})}(a+J\cap\mathfrak{A})=\mathrm{Spec}_{\mathrm{ C^*}(\mathfrak{A})/J}(a+J), \end{align*} $$

for all $a\in \mathfrak {A}_{\mathrm {sa}}$ .

Proof For simplicity, set $J=\overline {\mathscr I(D)\cap \mathfrak {A}}^{\mathrm {C^*}(\mathfrak {A})}\subset \mathrm {C^*}(\mathfrak {A})$ and $K= J\cap \mathfrak {A}\subset \mathfrak {A}$ . Note that K is a closed $^*$ -ideal. Let $\pi :\mathrm {C^*}(\mathfrak {A})\to \mathrm {C^*}(\mathfrak {A})/J$ and $\pi ': \mathfrak {A}\to \mathfrak {A}/K$ be the quotient maps and $\iota : \mathfrak {A}\to \mathrm { C^*}(\mathfrak {A})$ the canonical inclusion. Then there exists a continuous $^*$ -homomorphism $\varphi : \mathfrak {A}/K\to \mathrm {C^*}(\mathfrak {A})/J$ that makes the following diagram commute:

Since $\varphi $ maps $\mathfrak {A}/K$ injectively into the $C^*$ -algebra $\mathrm {C^*}(\mathfrak {A})/J$ , it is necessary for $\mathfrak {A}/K$ to be an $A^*$ -algebra.

Now, for the sake of contradiction, we will assume that $\mathfrak {A}/K$ is infinite-dimensional. That makes $\mathfrak {A}/K$ an infinite-dimensional $A^*$ -algebra and so, by [Reference Aupetit1, Corollary 5.4.3], there exists $a\in \mathfrak {A}_{\mathrm { sa}}$ such that the spectrum $\Sigma =\mathrm {Spec}_{\mathfrak {A}/K}(\pi '(a))$ is infinite and we have $\Sigma =\mathrm {Spec}_{\mathrm {C^*}(\mathfrak {A})/J}(\pi (a))\subset \mathbb {R}$ , due to Lemma 2.3. Using the regularity of $\mathfrak {A}(a)$ , we can find a sequence of functions $f_n\in C(\mathbb {R})$ that are nonzero in $\Sigma $ , have disjoint supports, and such that the sequence $a_n=f_n(a)$ belongs to $\mathfrak {A}$ and satisfies

$$ \begin{align*}a_na_m=f_n(a)f_m(a)=(f_n\cdot f_m)(a)=0,\quad \text{ for }n\not=m. \end{align*} $$

It also satisfies

$$ \begin{align*}a_n^2\not\in K,\quad \text{ for all }n\in\mathbb{N}, \end{align*} $$

since

$$ \begin{align*}\pi'(a_n^2)=\pi'(f_n^2(a))=\varphi^{-1}(f_n^2(\pi(a)))\not=0. \end{align*} $$

Replacing the $a_n$ ’s by appropriate scalar multiples, we can assume that . Note that $a_n^2\not \in \mathscr I(D)$ , as the opposite situation would mean that $a_n^2\in \mathscr I(D)\cap \mathfrak {A}\subset K$ . Thus, one can find $b_n\in \mathfrak {B}$ such that

where $M>0$ satisfies . Now consider the element ${c=\sum _{n\in \mathbb {N}} a_nb_n\in \mathfrak {B}}$ . It satisfies and $a_nc=a_n^2b_n$ , for all $n\in \mathbb {N}$ . Hence

which is absurd. Hence, the quotient $\mathfrak {A}/K$ has to be finite-dimensional.

Proof An application of Lemma 2.4 tells us that the ideal $\overline {\mathscr I(D)\cap \mathfrak {A}}^{\mathrm {C^*}(\mathfrak {A})}\cap \mathfrak {A}$ has finite codimension in $\mathfrak {A}$ . From this, it is easy to observe that $\overline {\mathscr I(D)\cap \mathfrak {A}}^{\mathrm {C^*}(\mathfrak {A})}$ has finite codimension in $\mathrm { C^*}(\mathfrak {A})$ , and because of the assumptions, it follows that $1\in \overline {\mathscr I(D)\cap \mathfrak {A}}^{\mathrm { C^*}(\mathfrak {A})}$ . Appealing to basic spectral theory, we note that $\mathscr I(D)\cap \mathfrak {A}$ must contain an element a that is invertible in $\mathrm {C^*}(\mathfrak {A})$ . However, by invoking Lemma 2.3, we see that $a^{-1}$ necessarily lies in $\mathfrak {A}$ . Therefore, $1\in \mathscr I(D)$ and the map

$$ \begin{align*}\mathfrak{B}\ni b\mapsto D(b1)=D(b)\in\mathcal{X} \end{align*} $$

is continuous.

The following lemma is an immediate application of a classic result of Barnes on the uniqueness of $C^*$ -norms (see, e.g., [Reference Barnes5, Theorem 4.2]).

Theorem 2.8 Let $\mathfrak {A}$ be an $A^*$ -algebra and let $\mathfrak {B}_1,\mathfrak {B}_2$ be Banach algebras such that there are continuous, injective homomorphisms $\mathfrak {A}\to \mathfrak {B}_i$ and $\mathfrak {B}_i\to \mathrm {C^*}(\mathfrak {A})$ that make the diagrams

commute. Furthermore, suppose that $\mathfrak {A}\to \mathfrak {B}_1$ is a locally regular inclusion. Then every derivation $D:\mathfrak {B}_1\to \mathfrak {B}_2$ is continuous.

Proof Again, for the sake of simplicity, set $J=\overline {\mathscr I(D)\cap \mathfrak {A}}^{\mathrm {C^*}(\mathfrak {A})}$ and $K=J\cap \mathfrak {A}$ . Because of Lemma 2.4, we have that K has finite codimension in $\mathfrak {A}$ and J has finite codimension in $\mathrm {C^*}(\mathfrak {A})$ . Now consider the set

$$ \begin{align*}S=\{c\in\mathfrak{B}_2\mid ct=0,\,\forall t\in \mathscr I(D)\}. \end{align*} $$

If we show that $S=\{0\}$ , then the theorem is proven, and the reason is the following. Suppose that $S=\{0\}$ and, by means of the closed graph theorem, let us show that D is continuous. In order to achieve that purpose, take a sequence $a_n\in \mathfrak {B}_1$ and an element $c\in \mathfrak {B}_2$ such that

Observe that $D(a_nt)\to 0$ for all $t\in \mathscr I(D)$ . On the other hand, one sees that

$$ \begin{align*}D(a_nt)=a_nD(t)+D(a_n)t\to ct. \end{align*} $$

Hence $ct=0$ , for all $t\in \mathscr I(D)$ . It follows that $c\in S=\{0\}$ , which proves the continuity of D.

Now, it remains to show that $S=\{0\}$ . In order to do so, first note that S is a two-sided ideal of $\mathfrak {B}_2$ . Note also that $I=\overline {S}^{\mathrm {C^*}(\mathfrak {A})}$ and J are two-sided ideals of $\mathrm {C^*}(\mathfrak {A})$ and hence $C^*$ -algebras; in particular, they contain bounded approximate identities. It follows that $IJ=I\cap J$ (one can appeal to [Reference Bonsall and Duncan8, Theorem 11.10]). But then, one can easily check that $IJ=\{0\}$ and hence

$$ \begin{align*}(I\cap \mathfrak{A})\cap K\subset I\cap J=\{0\}. \end{align*} $$

Since K has finite codimension, this forces $I\cap \mathfrak {A}$ to be finite-dimensional. But $I\cap \mathfrak {A}$ is a closed $^*$ -ideal of $\mathfrak {A}$ and therefore a finite-dimensional $A^*$ -algebra.

In order to achieve a contradiction, suppose now that $I\cap \mathfrak {A}\not =\{0\}$ . Then, up to renorming, $I\cap \mathfrak {A}$ must be a nontrivial finite-dimensional $C^*$ -algebra and so it must contain a unit $p\in I\cap \mathfrak {A}$ . It is easy to see that p is a central element in $\mathfrak {A}$ and that it satisfies $I\cap \mathfrak {A}=\mathfrak {A} p$ . Using this, we note that the map

$$ \begin{align*}\mathfrak{A}\ni a\mapsto ap\in I\cap \mathfrak{A} \end{align*} $$

has finite-dimensional range and factorizes the map

$$ \begin{align*}\mathfrak{B}_1\ni a\mapsto D(ap)\in \mathfrak{B}_2, \end{align*} $$

making the latter continuous, so we have $p\in \mathscr I(D)\cap \mathfrak {A}\subset K$ . It then follows that ${p\in (I\cap \mathfrak {A})\cap K=\{0\}}$ , a contradiction. Hence $I\cap \mathfrak {A}=\{0\}$ .

Finally, Lemma 2.7 allows us to conclude that $S\subset I=\{0\}$ from the fact that ${I\cap \mathfrak {A}=\{0\}}$ , so the theorem is proven.

3 Applications to $L^p$ crossed products

As mentioned before, the purpose of this section is to apply the previous results to $L^p$ -crossed products and symmetrized $L^p$ -crossed products. For that purpose, we will fix a $C^*$ -dynamical system $(G,C_0(X),\alpha )$ , where G is an amenable locally compact group, X is a locally compact Hausdorff space, and $\alpha $ is a continuous action of G on X. In what follows, we consider that G is always endowed with a Haar measure $\mathrm {d} x$ .

We shall abuse the notation and call $\alpha $ both the action on X and the induced action on $C_0(X)$ . That is

$$ \begin{align*}\alpha_x(f)(z)=f\big(\alpha_{x^{-1}}(z)), \end{align*} $$

where $x\in G,z\in X, f\in C_0(X)$ .

Then the generalized convolution algebra $L^1_{\alpha }(G,C_0(X))$ is given by all Bochner integrable functions $\Phi :G\to C_0(X)$ , endowed with the product

$$ \begin{align*} \Phi*\Psi(x)=\int_G \Phi(y)\alpha_y[\Psi(y^{-1}x)]\mathrm{d} y \end{align*} $$

and the involution

$$ \begin{align*} \Phi^*(x)=\Delta(x^{-1})\alpha_x[\Phi(x^{-1})^*]. \end{align*} $$

With these operations and the norm , we obtain a Banach $^*$ -algebra structure for $L^1_{\alpha }(G,C_0(X))$ .

It is well-known that the algebra $L^1_{\alpha }(G,C_0(X))$ admits $C^*$ -norms, so it is an $A^*$ -algebra. Its enveloping $C^*$ -algebra is denoted $C_0(X)\rtimes _{\alpha }G$ and we call it the crossed product of $C_0(X)$ by G. We now proceed to introduce $L^p$ -crossed products.

Definition 3.1 Let $p \in [1, \infty ],$ and let $(G,C_0(X),\alpha )$ be a $C^*$ -dynamical system as above. Let $(Y,\mathcal B,\mu )$ be a measure space. A covariant representation of $(G,C_0(X),\alpha )$ on $L^p (Y, \mu )$ is a pair $(v, \pi )$ consisting of a map $g \mapsto v_g$ from G to the invertible isometries on $L^p (Y, \mu )$ that is continuous in the sense that $g\mapsto v_g\xi $ is continuous for all $\xi \in L^p(Y,\mu )$ , and a non-degenerate, contractive representation $\pi : C_0(X) \to \mathbb B(L^p (Y, \mu )),$ such that

$$ \begin{align*}v_xv_y=v_{xy}, \qquad v_x\pi(f)v_{x}^{-1}=\pi(\alpha_x(f)), \end{align*} $$

for all $x,y \in G$ and $f \in C_0(X)$ .

Given a covariant representation $(v, \pi )$ on $L^p (Y, \mu )$ , we define the associated integrated representation $\pi \rtimes v : L^1_{\alpha }(G,C_0(X))\to \mathbb B(L^p (Y, \mu ))$ by

$$ \begin{align*}(\pi\rtimes v)(\Phi)(\xi)=\int_G \pi(\Phi(x))(v_x(\xi))\mathrm{d} x, \end{align*} $$

for all $\Phi \in L^1_{\alpha }(G,C_0(X))$ and all $\xi \in L^p (Y, \mu )$ .

With these definitions at hand, we can introduce the following norm on the algebra $L^1_{\alpha }(G,C_0(X))$ :

$$ \begin{align*}\|\Phi\|_{p,\mathrm{max}}=\sup_{(v,\pi)}\|(\pi\rtimes v)(\Phi)\|, \end{align*} $$

where the supremum is taken over all covariant representations $(v,\pi )$ on some space $L^p (Y, \mu )$ .

The completion $F^p(G,X,\alpha ):=\overline {L^1_{\alpha }(G,C_0(X))}^{\|\cdot \|_{p,\mathrm {max}}}$ is usually called the full $L^p$ -crossed product associated with $(G,C_0(X),\alpha )$ . It is easily seen to be a Banach algebra. In the case where the action is trivial, we set $F^p(G):=F^p(G,\{\mathrm {pt}\},1)$ and name the resulting algebra the group $L^p$ -operator algebra associated with G.

These algebras have appeared in [Reference Choi, Gardella and Thiel9, Reference Gardella20]. See also [Reference Bardadyn and Kwaśniewski4, Reference Delfín, Farsi and Packer13, Reference Dirksen, de Jeu and Wortel14] for generalizations. For the groupoid case, we recommend [Reference Austad and Ortega2].

It is observed in [Reference Bardadyn and Kwaśniewski4, Lemma 2.11] that the formulas

$$ \begin{align*}\pi(f)\xi(x,t)=f(x)\xi(x,t), \quad v_y \xi(x,t)=\xi(\alpha_{y^{-1}}(x),y^{-1}t), \end{align*} $$

valid for $f\in C_0(X)$ , $x,y,t\in G$ , and $\xi \in L^p(X\times G)\equiv \ell ^p(X,L^p(G))$ , define a covariant representation of $L^1_{\alpha }(G,C_0(X))$ on $L^p(X\times G)$ . Its integrated form satisfies the formula (note the special notation)

$$ \begin{align*}\Lambda_p(\Phi)\xi(x,t)=\int_{G} \Phi(y)(x)\xi(\alpha_{y^{-1}}(x),y^{-1}t)\mathrm{d} y. \end{align*} $$

Using this representation, one may define the reduced $L^p$ -crossed product as the completion $\overline {\Lambda _p\big (L^1_{\alpha }(G,C_0(X))\big )}\subset \mathbb B(L^p(X\times G))$ . However, in our setting (the groups are amenable), this construction is known to coincide with the full crossed product (see [Reference Bardadyn and Kwaśniewski4, Theorem 3.5]).

In any case, the representation $\Lambda _p$ allows us to introduce a “symmetrized” version of the $L^p$ -crossed product. This construction appears explicitly in [Reference Bardadyn and Kwaśniewski4]; similar constructions also appear in [Reference Austad and Ortega2, Reference Delfín, Farsi and Packer13]. In any case, consider the norm

where $p,q\in [1,\infty ]$ are Hölder duals. We define $F^q_*(G,X,\alpha )$ to be the completion of $L^1_{\alpha }(G,C_0(X))$ under the norm . As before, we define the symmetrized group algebra by setting $F^q_*(G):=F^q_*(G,\{\mathrm {pt}\},1)$ .

The point is that $F^q_*(G,X,\alpha )$ is naturally a Banach $^*$ -algebra that sits between $L^1_{\alpha }(G,C_0(X))$ and $C_0(X)\rtimes _{\alpha }G$ , so it fits the setting of Theorem 2.8 perfectly. Indeed, the fact that this algebra has a continuous involution arises from the fact that, for all $\Phi \in L^1_{\alpha }(G,C_0(X)),\xi \in L^p(X\times G),\eta \in L^q(X\times G)$ , one has

$$ \begin{align*}\langle \Lambda_p(\Phi)\xi,\eta\rangle=\langle\xi,\Lambda_q(\Phi^*)\eta\rangle, \end{align*} $$

where $\langle \cdot ,\cdot \rangle $ denotes the duality pairing between $L^p(X\times G)$ and $L^q(X\times G)$ (cf. [Reference Austad and Ortega2]). On the other hand, the inclusions $L^1_{\alpha }(G,C_0(X))\subset F^q_*(G,X,\alpha )\subset C_0(X)\rtimes _{\alpha }G$ follow from complex interpolation. Indeed, note that $F^q_*(G,X,\alpha )$ acts on both ${L^p(X\times G)}$ and $L^q(X\times G)$ , so an immediate application of the Riesz–Thorin interpolation theorem yields the existence of $\theta \in [0,1]$ such that

We refer the reader to [Reference Austad and Ortega2, Section 3] or [Reference Bardadyn and Kwaśniewski4, Sections 2 and 3] for a careful discussion of the algebra $F^q_*(G,X,\alpha )$ and the properties that we have just mentioned. The interpolation argument can be explicitly found in [Reference Austad and Ortega2] or in [Reference Elkiær15]. In [Reference Bardadyn and Kwaśniewski4], a different argument is presented.

In any case, let us describe our source of locally regular inclusions that involve $L^p$ -crossed products.

Definition 3.2 A weight on the locally compact group G is a measurable, locally bounded function $w: G\to [1,\infty )$ satisfying

$$ \begin{align*} w(xy)\leq w(x)w(y)\,,\quad w(x^{-1})=w(x)\,,\quad\forall\,x,y\in G. \end{align*} $$

During the rest of the section, $\mathfrak E_w$ will denote the subalgebra of $L^1_{\alpha }(G,C_0(X))$ defined using the weight w, as the set of all the functions $\Phi $ such that , where the norm in question is given by

It is not hard to see that $\mathfrak E_w$ is a dense Banach $^*$ -subalgebra of $L^1_{\alpha }(G,C_0(X))$ . Furthermore, the following lemma was proven in [Reference Flores18, Proposition 4.10], using some results from [Reference Flores19]. Note that the algebra $\mathfrak E_w$ was first introduced in [Reference Flores19].

The above lemma implies that, for any Banach algebra completion $\mathfrak {B}$ of $\mathfrak E_w$ , the inclusion $\mathfrak E_w\subset \mathfrak {B}$ is locally regular; thus, our theorems are applicable to $\mathfrak {B}$ . In particular, the $L^p$ -crossed products we just described are Banach algebra completions of $L^1_{\alpha }(G,C_0(X))$ ; hence, they are completions of $\mathfrak E_w$ . The following corollary is then just a simple application of Theorem 2.8.

In order to apply Theorem 2.5, we also need to guarantee that ${\mathrm {C^*}(\kern-1pt \mathfrak E_w\kern-1pt )\kern-1pt =\kern-1pt C_0(X)\kern-1pt \rtimes _{\alpha }\kern-1pt G}$ lacks proper closed two-sided ideals of finite codimension. But this is known to be the case when G is discrete, infinite, and the restriction of $\alpha $ to any invariant subset is topologically free (see [Reference Exel16, Theorem 29.9]).Footnote ¹ As a consequence, we derive the following corollary.