Chapter 2 Various types of symmetry for discrete groups
2.1 The Main Result
Throughout this thesis, \(\textsf{G}\) will be a discrete group with unit \(\textsf{e}\). As explained in Remark 1.2, there is no loss if we assume it amenable. For constructions involving graded \(C^*\)-algebras we followed (Exel 1997; Fell and Doran 1988) ; we decided not to mention Fell bundles explicitly (with one exception), but they can be perceived in the background.
Definition 2.1 We say that the unital \(C^*\)-algebra \(\mathfrak{C}\) is \(\textsf{G}\)-graded if that there is a family \(\big\{\mathfrak{C}_g\,\vert\,g\in{\textsf{G}}\big\}\) of closed subspaces of \(\mathfrak{C}\) such that the algebraic direct sum \(\,\bigoplus_{g\in{\textsf{G}}}\mathfrak{C_g}\) is dense in \(\mathfrak{C}\) and such that
\(\mathfrak{C}_g\mathfrak{C}_h\subset\mathfrak{C}_{gh}\,\) for every \(g,h\in{\textsf{G}}\) ,
\(\mathfrak{C}_g^*\subset\mathfrak{C}_{g^{-1}}\,\) for every \(g\in{\textsf{G}}\) ,
For every \(h\in{\textsf{G}}\) we denote by \(P_h:\bigoplus_{g\in{\textsf{G}}}\!\mathfrak{C}_g\to\mathfrak{C}_h\) the canonical projection (in general it might not be continuous). If \(P_\textsf{e}\) is continuous, \(\mathfrak{C}\) is a topologically \({\textsf{G}}\)-graded \(C^*\)-algebra. We are going to write \[\begin{equation} \mathfrak{C}=\widetilde{\bigoplus}_{g\in{\textsf{G}}}\mathfrak{C}_g\,. \tag{2.1} \end{equation}\]
For \(\Phi,\Psi\in\bigoplus_{g\in\textsf{G}}\mathfrak{C}_g\) and \(g\in\textsf{G}\) one verifies easily that \[\begin{equation} P_g(\Phi\Psi)=\sum_{hk=g}P_h(\Phi)P_k(\Psi)\,\tag{2.2} \end{equation}\] \[\begin{equation} P_g(\Phi)^*=P_{g^{-1}}(\Phi^*)\,.\tag{2.3} \end{equation}\] When the grading is topological, there are several facilities which will be crucial for our approach. Let us denote by \(\widetilde{P}_\textsf{e}:\mathfrak{C}\to\mathfrak{C_\textsf{e}}\) the extension of \(P_\textsf{e}\) to a linear bounded map. By Tomiyama’s Theorem ((Brown and Ozawa 2008 Thm. 1.5.10), see also (Exel 1997, Thm. 3.3)), it is a positive contractive conditional expectation. Then all the projections \(P_g\) do extend to contractions \(\widetilde{P}_g:\mathfrak{C}\to\mathfrak{C_g}\) . We refer to the proof of Theorem 2.1,(ii) for another use of topological grading, in conjunction with amenability.
::: {.definition #glaro} Let \(\mathfrak{C}\) be a topologically graded \(C^*\)-algebra. On \(\,\bigoplus_{g\in{\textsf{G}}}\!\mathfrak{C}_g\) we can introduce the new norm \[\,\big\Vert\,\!\Phi\!\,\big\Vert\,_{\ell^1(\mathfrak C)}\,:=\sum_{g\in{\textsf{G}}}\,\big\Vert\,\! P_g(\Phi)\!\,\big\Vert\,.\] The completion \(\ell^1(\mathfrak{C})\) of \(\bigoplus_{g\in {\textsf{G}}}\!\mathfrak{C}_g\) in this norm is called the \(\ell^1\)-algebra of the graded \(C^*\)-algebra \(\mathfrak{C}\). :::
Since \(\Vert\!\cdot\!\Vert\,\le\,\Vert\!\cdot\!\Vert_{\ell^1(\mathfrak{C})}\) on \(\bigoplus_{g\in\textsf{G}}\!\mathfrak{C}_g\), one may interpret \(\ell^1(\mathfrak{C})\) as a subspace of \(\mathfrak{C}\): \[\label{interpretatix} \ell^1(\mathfrak{C})=\Big\{\Phi\in\mathfrak{C}\;\Big\vert\,\sum_{g\in\textsf{G}}\,\big\Vert\,\! \widetilde{P}_g(\Phi)\!\,\big\Vert\,\,<\infty\Big\}\,.\] In fact it is a Banach \(^*\)-subalgebra with the algebraic structure borrowed from \(\mathfrak{C}\) and its new norm. Its elements may be written as unconditionally convergent series \(\Phi=\sum_{g\in\textsf{G}}\widetilde{P}_g(\Phi)\) in the norm topology.
The next result, besides being interesting in itself, is basic for our approach.
Proposition 2.1 Let \(\mathfrak{C}=\widetilde{\bigoplus}_{g\in{\textsf{G}}}\mathfrak{C}_g\) be a topologically graded \(C^*\)-algebra over a discrete group \({\textsf{G}}\). There exists a \(C^*\)-algebra \(\mathcal{B}\) and an isometric \(^*\)-morphism \[\label{guyana} T:\ell^1(\mathfrak{C})\hookrightarrow \ell^1({\textsf{G}};\mathcal{B})\equiv\ell^1({\textsf{G}})\otimes \mathcal{B}\,.\]
Proof. Let \(\pi:\mathfrak{C}\to\mathbb{B}(\mathcal{H})\) be a faithful representation. For any \(g\in\textsf{G}\) we denote by \(\pi_g\) the restriction of \(\pi\) to the closed subspace \(\mathfrak{C}_g\); it is a linear isometry. One has \[\label{primika} \pi_g(\Phi)\pi_h(\Psi)=\pi_{gh}(\Phi\Psi)\,,\quad\forall\,g,h\in\textsf{G}\,,\ \Phi\in\mathfrak{C}_g\,,\ \Psi\in\mathfrak{C}_h\,,\] \[\label{secundika} \pi_g(\Phi^*)=\pi_{g^{-1}}(\Phi)^*,\quad\forall\,g\in\textsf{G}\,,\ \Phi\in\mathfrak{C}_g\,.\]
Then we set \[\label{tronc} \big[T(\Phi)\big](g):=\pi_g\big[\widetilde P_g(\Phi)\big]\,,\quad\Phi\in\ell^1(\mathfrak{C})\,,\ g\in\textsf{G}\,.\] Clearly \(T\) is a well-defined linear isometry from \(\ell^1(\mathfrak C)\) to \(\ell^1\big(\textsf{G},\mathbb B(\mathcal{H})\big)\) : \[\,\big\Vert\,\!T(\Phi)\!\,\big\Vert\,_{\ell^1(\textsf{G},\mathbb B(\mathcal{H}))}\,=\sum_{g\in\textsf{G}}\,\big\Vert\,\!\pi_g\big[\widetilde P_g(\Phi)\big]\!\,\big\Vert\,_{\mathbb B(\mathcal{H})}\,=\sum_{g\in\textsf{G}}\,\big\Vert\,\!\widetilde P_g(\Phi)\!\,\big\Vert\,\,=\,\,\big\Vert\,\!\Phi\!\,\big\Vert\,_{\ell^1(\mathfrak C)}.\] To check that \(T\) respects the algebraic operations, it is enough to work with elements of the algebraic direct sum. By taking \(\Phi,\Psi\in\bigoplus_g\!\mathfrak{C}_g\), one can write \[\begin{aligned} \big[(T\Phi)\star(T\Psi)\big](g)&=\sum_{h\in \textsf{G}}(T\Phi)(h)(T\Psi)(h^{-1}g) \\ &=\sum_{h\in \textsf{G}}\pi_h\left[P_h(\Phi)\right]\,\pi_{h^{-1}g}\big[P_{h^{-1}g}(\Psi)\big] \\ &\overset{\eqref{primika}}{=}\sum_{h\in \textsf{G}}\pi_g\big[P_h(\Phi)P_{h^{-1}g}(\Psi)\big]\\ &=\pi_g\Big(\sum_{h\in \textsf{G}}P_h(\Phi)P_{h^{-1}g}(\Psi)\Big)\\ &\overset{\eqref{multiplix}\text{\@ref(eq:multiplix)}}=\pi_g\big[P_g(\Phi\Psi)\big]=\big[T(\Phi\Psi)\big](g)\,, \end{aligned}\] showing that \(T\) is an algebraic morphism. Finally we treat the involution: \[\begin{aligned} \left[T(\Phi)\right]^\star\!(g) & =\pi_{g^{-1}}\!\left[P_{g^{-1}}(\Phi)\right]^*\overset{\eqref{secundika}}{=} \pi_g\big[P_{g^{-1}}(\Phi)^*\big]\\ & \overset{\eqref{multiplix}}{=} \pi_g[P_g(\Phi^*)]= \big[T(\Phi^*)\big](g)\,. \end{aligned}\]
With Proposition 2.1 in hand, we prove now our main abstract result. Recall that a Banach \(^*\)-algebra is called reduced if its universal \(C^*\)-seminorm is in fact a norm. In this case, its enveloping \(C^*\)-algebra is simply its completion in this \(C^*\)-norm (no quotient is needed).
Theorem 2.1 Let \(\mathfrak C=\widetilde{\bigoplus}_{g\in{\textsf{G}}}\mathfrak C_g\) be topologically graded over a rigidly symmetric discrete group \({\textsf{G}}\) .
(i). \(\ell^1(\mathfrak C)\) is a symmetric Banach \(^*\)-algebra.
(ii). \(\ell^1(\mathfrak C)\) is an inverse closed subalgebra of \(\,\mathfrak C\) .
(iii). Let \(\Pi:\mathfrak C\to \mathbb{B}(\mathcal{H})\) be a faithful representation. Then \(\Pi\left[\ell^1(\mathfrak C)\right]\) is inverse closed in \(\mathbb{B}(\mathcal{H})\) .
Proof. (i). By (Palmer 2001, 2:Th.11.4.2), a closed \(^*\)-algebra of a symmetric Banach \(^*\)-algebra is also symmetric. Proposition 2.1 and the fact that \(\ell^1(\textsf{G};\mathcal B)\) was assumed symmetric proves the result.
(ii). It is known (Palmer 2001, 2:11.4) that a reduced Banach \(^*\)-algebra is symmetric if and only if it is inverse closed in its enveloping \(C^*\)-algebra. If we show that we are in such a case, we then apply (i). Our \(\ell^1(\mathfrak C)\) is clearly reduced, being given as a \(^*\)-subalgebra of \(\mathfrak C\) . Actually, we claim that \(\mathfrak C\) can be seen as (a copy of) the enveloping \(C^*\)-algebra of \(\ell^1(\mathfrak C)\) . In (Samei and Wiersma 2020, Corol. 4.8) it is shown that a symmetric group (as our rigidly symmetric \(\textsf{G}\)) is surely amenable. For such groups, one may identify \(\mathfrak C\) with both the full and the reduced \(C^*\)-algebras of the Fell bundle associated to the topological grading (Exel 1997, Prop. 4.2, Thm. 4.7) and then the assertion about enveloping follows (Fell and Doran 1988, 1:VIII.17).
(iii). Is a consequence of (ii), since any \(C^*\)-algebra is inverse closed in a larger \(C^*\)-algebra.
In the context of group actions on \(C^*\)-algebras one deals with convolution dominated kernels and operators (Fendler and Leinert 2016; Fendler, Gröchenig, and Leinert 2008; Gröchenig 2010; Gröchenig and Leinert 2006; Mantoiu 2015). We briefly indicate an extension for the topologically graded case, leading to another form of a symmetric Banach \(^*\)-algebra at the level of kernels. Representations by integral operators and their inverse closedness properties are left to the reader.
Definition 2.2 A convolution dominated kernel (or matrix) is a function \(K:{\textsf{G}}\times\textsf{G}\to \mathfrak{C}\) such that \(K(g,h)\in\mathfrak{C}_{gh^{-1}}\) for every \(g,h\in{\textsf{G}}\) and such that the norm \[\label{cambio} \,\big\Vert\,\!K\!\,\big\Vert\,_\mathscr{K}\,:=\inf\big\{\!\,\big\Vert\,\!\kappa\!\,\big\Vert\,_{\ell^1({\textsf{G}})}\,\big\vert\,\,\big\Vert\,\!K(g,h)\!\,\big\Vert\,_\mathfrak{C}\,\le|\kappa(gh^{-1})|\,,\,\forall\,g,h\in{\textsf{G}}\big\}\] is finite. We denote by \(\mathscr K_{{\textsf{G}}}(\mathfrak{C})\) the vector space of all these convolution dominated kernels. An element \(K\) of \(\mathscr K_{{\textsf{G}}}(\mathfrak{C})\) is called covariant, and we write \(K\in\mathscr{K}^{\textrm{ co}}_{{\textsf{G}}}(\mathfrak{C})\) , if \[\label{pusimic} K(gk,hk)=K(g,h)\,,\quad\forall\,g,h,k\in{{\textsf{G}}}\,.\]
From the definition above, \(K\) is covariant if and only if for all \(g,h\in G\) we have \(K(g, h)= k_0(gh^{-1})\) for some function \(k_0:G\rightarrow \mathfrak{C}\) such that \(k_0(k)\in \mathfrak{C}_k\) for all \(g\in G\) and \(\| k_0(\cdot ) \|_{\mathfrak C}\in \ell^1(G)\). Thus the isomorphism between \(\mathscr K^{\textrm{ co}}_\textsf{G}(\mathfrak C)\) and \(\ell^1(\mathfrak C)\) is straightforward.
It is straightforward to show that endowed with the norm \(\Vert\!\cdot\!\Vert_\mathscr K\) , the multiplication \[\label{greta} (K\bullet L)(g,h):=\sum_{k\in\textsf{G}}K(g,k)L(k,h)\] and the involution \(K^\bullet(g,h):=K(h,g)^*\), the space \(\mathscr K_{{\textsf{G}}}(\mathfrak C)\) is a Banch \(^*\)-algebra.
Corollary 2.1 The subspace \(\mathscr K^{\textrm{ co}}_{{\textsf{G}}}(\mathfrak C)\) is a symmetric Banach \(^*\)-algebra.
Proof. It is easy to check that \(\mathscr K^{\textrm{ co}}_\textsf{G}(\mathfrak C)\) is a closed \(^*\)-subalgebra of \(\mathscr K_\textsf{G}(\mathfrak C)\) . Let us define \[\begin{equation} \big[\Upsilon(\Phi)\big](g,h):=\widetilde P_{gh^{-1}}(\Phi)\,,\quad\forall\,g,h\in\textsf{G}\,. \tag{2.4} \end{equation}\] Then \(\Upsilon:\ell^1(\mathfrak C)\to\mathscr K_\textsf{G}(\mathfrak C)\) is an isometric \(^*\)-morphism with range \(\mathscr K^{\textrm{ co}}_\textsf{G}(\mathfrak C)\). For the algebraic relations use (2.2) and (2.3) (extended). On this range, the inverse reads \[\begin{equation} \big[\Upsilon^{-1}(K)\big]_g:=K(g,\textsf{e})\,,\quad\forall\,g\in\textsf{G}\,. \tag{2.5} \end{equation}\] Since \(\mathscr K^{\textrm{ co}}_\textsf{G}(\mathfrak C)\) and \(\ell^1(\mathfrak C)\) are isomorphic, the result follows from Theorem 2.1 (i).
Remark 2.1 In (Raeburn 2016) it is shown how to deform Fell bundles by a cohomological twisting. We adapt this here to our problem. Let \(\omega:{\textsf{G}}\!\times\!{\textsf{G}}\to\mathbb{T}\) be a \(2\)-cocycle, bound to satisfy for every \(g,h,k\in{\textsf{G}}\) the axioms \[\label{bundix} \omega(g,h)\omega(gh,k)=\omega(h,k)\omega(g,hk)\,,\quad\omega(g,\textsf{e})=1=\omega(\textsf{e},g)\,.\] For a topologically graded \(C^*\)-algebra \(\mathfrak C=\widetilde{\bigoplus}_{g\in{\textsf{G}}}\mathfrak C_g\) , it is possible to construct a new one \(\mathfrak C\{\omega\}\) in the following way: On the (same) algebraic direct sum we modify the algebraic structure, setting for finite sums \(\Phi:=\sum_g\!\Phi_g\) and \(\Psi:=\sum_g\!\Psi_g\) \[\label{finitex} \Phi\cdot_\omega\!\Psi:=\sum_{g,h}\omega(g,h)\Phi_g\Psi_h\,,\quad\Phi^{*_\omega}\!:=\sum_g\overline{\omega(g,g^{-1})}\Phi_g\,.\]
An argument using Fell bundles and the amenability of our discrete group certifies the existence of the topological graded \(C^*\)-algebra \(\mathfrak C\{\omega\}=\widetilde{\bigoplus}^\omega_{g\in{\textsf{G}}}\mathfrak C_g\) . The norm is also different (on the common algebraic sum), having in view the way the universal norm (identical here with the reduced one, by amenability) is constructed in the setting of Fell bundles. In the case where \(\omega\) is not trivially equal to \(1\), the Banach \(^*\)-algebra \(\ell^1\big(\mathfrak C\{\omega\}\big)\) is different from \(\ell^1(\mathfrak C)\) and it is symmetric and inverse closed in \(\mathfrak C\{\omega\}\).
2.2 Topologically graded algebras from dual actions and partial crossed products
Given a \(C^*\)-algebra, it is often not obvious whether it has an interesting grading or not. In this subsection we are going to review some ways to solve this problem, following especially results of Exel, Quigg and Raeburn. This will lead to two reformulations of Theorem 2.5 that can be useful when studying examples and which will also provide the reader with a more extended picture. We will take advantage of the fact that our group \(\textsf{G}\) is discrete and amenable; this will allow for some simplifications.
The most general statement is that in this setting, topologically graded \(C^*\)-algebras over \(\textsf{G}\) are equivalent to coactions of \(\,\textsf{G}\) , cf. (Quigg 1996) and (Raeburn 2016 App. A). In this direction we only indicate the case of Abelian groups and the dual of an Abelian group is a group, making the constructions more transparent.
So we fix for a while a discrete Abelian group \(\textsf{G}\) . Its Pontryagin dual \(\widehat{\textsf{G}}\) is a compact Abelian group with normalized Haar measure \(d\chi\) . Let \(\big(\mathfrak C,\alpha,\widehat{G}\,\big)\) be a (usual, full) action of \(\widehat{G}\) on a \(C^*\)-algebra. For every \(g\in\textsf{G}\) let \[\begin{equation} \mathfrak C_g:=\big\{\Phi\in\mathfrak C\,\big\vert\,\alpha_\chi(\Phi)=\chi(g)\Phi\,,\,\forall\,\chi\in\widehat{G}\big\} \tag{2.6} \end{equation}\] be the \(g\)’th spectral subspace of the action. It is easy to see that one gets a grading, as in ((2.1)). We have now the concrete projections \[\begin{equation} \widetilde P_g:\mathfrak C\to\mathfrak C_g\,,\quad \widetilde P_g(\Phi):=\int_{\widehat{G}}\,\overline{\chi(g)}\,\alpha_\chi(\Phi)\,d\chi\,, \tag{2.7} \end{equation}\] which are obviously contractive; then \(\mathfrak C\) is a topological grading. We can reformulate Theorem 2.1 in this Abelian setting; the \(\ell^1\)-algebra will deserve a special notation, to recall the action \(\alpha\) and the type of decay.
Theorem 2.2 Let \(\big(\mathfrak C,\alpha,{\widehat{G}}\,\big)\) be a continuous action of a compact Abelian group \({\widehat{G}}\) on a unital \(C^*\)-algebra. Denoting by \({\textsf{G}}\) the dual of \(\,{\widehat{G}}\) , \[\begin{equation} \ell^1(\mathfrak C)\equiv\bigoplus_{g\in{\textsf{G}}}^{1,\alpha}\mathfrak C_g=\Big\{\Phi\in\mathfrak C\;\Big\vert\,\sum_{g\in{\textsf{G}}}\,\big\Vert\,\! \widetilde P_g(\Phi)\!\,\big\Vert\,\,<\infty\Big\} \tag{2.8} \end{equation}\] is a reduced and symmetric Banach \(^*\)-algebra, which is inverse closed in \(\mathfrak C\) .
Remark 2.2 Using normal subgroups of \({\textsf{G}}\) one can increase the family of symmetric subalgebras. If, as above, \(\big(\mathfrak C,\alpha,\widehat{{\textsf{G}}}\,\big)\) is a continuous action of the dual of the Abelian discrete group \({\textsf{G}}\) and \(\textsf{N}\) is a normal subgroup, the dual \(\widehat{{\textsf{G}}/\textsf{N}}\) of the quotient is isomorphic to the closed subgroup \[\begin{equation} \textrm{ Ann}\big(\textsf{N}\!\mid\!\widehat{G}\big):=\big\{\chi\in\widehat{G}\,\big\vert\,\chi|_{\textsf{N}}=1\big\}\,. \tag{2.9} \end{equation}\] Thus from \(\alpha\) we deduce a continuous action \(\alpha^\textsf{N}\) of \(\widehat{{\textsf{G}}/\textsf{N}}\) on \(\mathfrak C\) , first by restricting \(\alpha\) to \(\textrm{ Ann}\big(\textsf{N}\!\mid\!\widehat{G}\big)\) and then composing with the isomorphism. In this way we obtain the new symmetric Banach \(^*\)-algebra \[\begin{equation} \ell^1\big(\mathfrak{C}^{{\textsf{G}}/\textsf{N}}\big)\equiv\bigoplus_{\gamma\in{\textsf{G}}/\textsf{N}}^{1,\alpha^{\textsf{N}}}\mathfrak{C}_{\gamma}\,. %=\Big\{\Phi\in\mathfrak C\;\Big\vert\,\sum_{g\in{\textsf{G}}}\,\big\Vert\,\! \widehat P_g(\Phi)\!\,\big\Vert\,\,<\infty\Big\} \tag{2.10} \end{equation}\] Extreme cases are \(\ell^1\big(\mathfrak C^{{\textsf{G}}/\{\textsf{e}\}}\big)=\ell^1(\mathfrak C)\) and \(\ell^1\big(\mathfrak C^{{\textsf{G}}/{\textsf{G}}}\big)=\mathfrak C\).
Crossed products assigned to partial actions of (discrete) groups \(\textsf{G}\) on (unital) \(C^*\)-algebras are among the most important examples of graded \(C^*\)-algebras, and they were our initial motivation. In (Exel 2017, Thm. 27.11) they are characterized among the larger class, while (Quigg and Raeburn 1997, Thm. 4.1) gives a Landstad-type description of them in terms of the associated coaction. In many cases what we are actually given is the partial action; the grading comes afterwards. Now the discrete group \(\textsf{G}\) is no longer Abelian.
Definition 2.3 A partial action of \({\textsf{G}}\) on the \(C^*\)-algebra \(\mathcal{A}\) is a pair \(\big(\{\mathcal{A}_g\}_{g\in {\textsf{G}}},\{\theta_g\}_{g\in{\textsf{G}}}\}\big)\) , where for each \(g\in {\textsf{G}}\) , \(\mathcal{A}_g\) is a closed two-sided ideal in \(\mathcal{A}\) , \(\theta_g\) is a \(^*\)-isomorphism from \(\mathcal{A}_{g^{-1}}\) onto \(\mathcal{A}_g\), satisfying the following conditions, for all \(g,h\in{\textsf{G}}\) :
\(\mathcal{A}_\textsf{e}=\mathcal{A}\) and \(\theta_\textsf{e}\) is the identity automorphism of \(\mathcal{A}\) ,
\(\theta_g\!\left(\mathcal{A}_{g^{-1}}\!\cap \mathcal{A}_h\right)\subset\mathcal{A}_{gh}\) ,
\(\theta_g\big(\theta_h(a)\big)=\theta_{gh}(a)\) , \(\forall\,a\in\mathcal{A}_{h^{-1}}\!\cap\mathcal{A}_{(gh)^{-1}}\) .
We also denote by \((\mathcal{A},\theta,\textsf{G})\) the partial dynamical system. It follows easily that we have \(\theta_{g^{-1}}\!=\theta_g^{-1}\). Our main object is the following vector space: \[\begin{equation} \ell^1_\theta(\textsf{G};\mathcal{A}):= \left\{\Phi\in\ell^1(\textsf{G};\mathcal{A})\mid\Phi(g)\in \mathcal{A}_g \,,\,\forall\,g\in \textsf{G}\right\}, (eq:thetika) \end{equation}\] which is obviously closed. For \(\Phi,\Psi\in\ell^1_\theta(\textsf{G};\mathcal{A})\) we define the product \[\begin{equation} \left( \Phi\star_{\theta} \Psi\right)(g) =\sum_{h\in \textsf{G}} \theta_h\!\left[\theta_{h}^{-1}\big(\Phi(h)\big)\Psi\big(h^{-1}g\big)\right] (eq:strugure) \end{equation}\] and the involution \(\Phi^{\star_\theta}(g) = \theta_g\!\left[\Phi\big(g^{-1}\big)\right]^*.\) Then \(\big(\ell^1_\theta(\textsf{G};\mathcal{A}),\star_{\theta},^{\star_{\theta}}\!,\|\cdot\|_{\ell^1_\theta(\textsf{G};\mathcal{A})}\big)\) is a Banach \(^*\)-algebra. We will refer to it as the \(\ell^1\)-partial crossed product. Its enveloping \(C^*\)-algebra is denoted by \(\mathcal{A}\!\rightthreetimes_\theta\!\textsf{G}\) and called the partial crossed product associated to \((\mathcal{A},\theta,\textsf{G})\) . We used the fancy symbol \(\rightthreetimes\) instead of the more usual \(\rtimes\) to stress that our crossed product is assigned to a partial action.
We indicate the particularization of Theorem 2.1 to partial actions. Since \(\textsf{G}\) must be amenable, by (Exel 1997) the full and the reduced partial crossed products are identical and possess a (faithful) positive contractive conditional expectation.
Theorem 2.3 Let \((\mathcal{A},\theta,{\textsf{G}})\) be a partial \(C^*\)-dynamical system with rigidly symmetric discrete group \({\textsf{G}}\) and unital \(C^*\)-algebra \(\mathcal{A}\). Then \(\ell^1_\theta({\textsf{G}};\mathcal{A})\) is symmetric and inverse closed in \(\mathcal{A}\rightthreetimes_\theta\!{\textsf{G}}\). If \(\,\Pi:\mathcal{A}\rightthreetimes_{\theta}\!{\textsf{G}}\to \mathbb{B}(\mathcal{H})\) is a faithful representation, \(\Pi\left[\ell^1_\theta({\textsf{G}};\mathcal{A})\right]\) is inverse closed in \(\mathbb{B}(\mathcal{H})\).
Proof. The crossed product \(\mathfrak C:=\mathcal{A}\!\rightthreetimes_\theta\!\textsf{G}\) is a topologically graded \(C^*\)-algebra (Exel 2017), where \[\begin{equation} \mathfrak C_g=\big\{\Phi\in\ell^1_\theta(\textsf{ G};\mathcal{A})\,\big\vert\,\textrm{ supp}(\Phi)\subset\{g\}\big\}\,,\quad\forall\,g\in\textsf{G}\,. (eq:verita) \end{equation}\] Its \(\ell^1\)-algebra is \[\begin{equation} \ell^1(\mathfrak C)\equiv\ell^1\big(\mathcal{A}\!\rightthreetimes_\theta\!\textsf{G}\big)=\ell^1_\theta(\textsf{G};\mathcal{A})\,, (eq:veveriz) \end{equation}\] reinterpreting \(\rightthreetimes_\theta\) as the partial crossed product between group \(G\) and the \(C^*\)-algebra \(A\). Thus we can apply Theorem 2.1.
Remark 2.3 The result can also be obtained directly, by means specific to partial action. To prove the isometric linear embedding \(\ell^1_\theta(\textsf{ G};\mathcal{A})\hookrightarrow\ell^1\big(\textsf{G};\mathbb B(\mathcal{H}))\) one uses the map \(\big[T(\Phi)\big](g):=\rho\big[\Phi(g)\big]u_g\) , where \((\rho,u)\) is a covariant representation of \((\mathcal{A},\theta,\textsf{G})\) in the Hilbert space \(\mathcal{H}\) , as in (Exel 2017 Sect. 13). Here \(\rho\) should be a faithful representation of \(\mathcal{A}\) and \(u\) a partial representation of \(\textsf{G}\), connected by an equivariance condition.
2.3 Inverse closedness modulo ideals.
Definition 2.4 Let \(\mathfrak{K}\) be an ideal of the unital C\(^*\)-algebra \(\mathfrak{C}\) . The \(^*\)-subalgebra \(\mathfrak{B}\) of \(\mathfrak{C}\) is called \(\mathfrak{K}\)-inverse closed if \(\mathfrak{B}/(\mathfrak{B}\cap\mathfrak{K})\) is inverse closed in \(\mathfrak{C}/\mathfrak{K}\).
Example 2.1 If \(\mathfrak{K}=\{0\}\), the notion coincides with that introduced in Definition (1.1). However the main motivating case is as follows: Let \(\mathfrak C\) be a \(C^*\)-algebra of bounded operators in the Hilbert space \(\mathcal{H}\) , containing the ideal \(\mathbb K(\mathcal{H})\) of all the compact operators. We recall that \(T\in\mathbb B(\mathcal{H})\) is called Fredholm if its range is closed and its kernel and its cokernel are both finite-dimensional. By Atkinson’s Theorem, this happens exactly if the image of \(T\) in the Calkin algebra \(\mathbb B(\mathcal{H})/\mathbb K(\mathcal{H})\) is invertible. A \(^*\)-subalgebra \(\mathfrak B\) of \(\mathbb B(\mathcal{H})\) will be called Fredholm inverse closed if the situation in Definition (2.4) holds with \(\mathfrak K=\mathbb K(\mathcal{H})\) . For \(\mathfrak C\) we can take any \(C^*\)-algebra of \(\mathbb B(\mathcal{H})\) containing \(\mathfrak B\).
Definition 2.5 Let \(\mathfrak C=\widetilde{\bigoplus}_{g\in{\textsf{G}}}\mathfrak C_g\) be a graded \(C^*\)-algebra over the discrete group \(\textsf{G}\) . The ideal \(\mathfrak K\) of \(\mathfrak C\) is a graded ideal if the ideal generated by \(\mathfrak K\cap\mathfrak C_\textsf{e}\) coincides with \(\mathfrak K\) (or, equivalently, that \(\bigoplus_{g\in{\textsf{G}}}\mathfrak K\cap\mathfrak C_g\) is dense in \(\mathfrak K\)).
Theorem 2.4 Let \(\mathfrak C=\widetilde{\bigoplus}_{g\in{\textsf{G}}}\mathfrak C_g\) be a topologically graded \(C^*\)-algebra over the discrete rigidly symmetric group \({\textsf{G}}\) and \(\mathfrak K\) a graded ideal. Then \(\ell^1(\mathfrak C)\) is \(\mathfrak K\)-inverse closed in \(\mathfrak C\) .
Proof. One needs to show that \(\ell^1(\mathfrak C)/\ell^1(\mathfrak C)\cap\mathfrak K\) is inverse closed in \(\mathfrak C/\mathfrak K\) . Let us denote by \(\kappa:\mathfrak C\to\mathfrak C/\mathfrak K\) the quotient map. By (Exel 2017, Prop. 23.10) (or by (Exel 1997, Prop. 3.11)), under the stated conditions upon the graded algebra and the ideal, \(\mathfrak C/\mathfrak K\) is topologically graded by the subspaces \[\big\{(\mathfrak C/\mathfrak K)_g:=\kappa(\mathfrak C_g)\,\big\vert\,g\in\textsf{G}\big\}\,.\] The corresponding linear contraction \(\widetilde Q_g:\mathfrak C/\mathfrak K\to(\mathfrak C/\mathfrak K)_g\) satisfies \[\label{traznau} \widetilde Q_g\circ\kappa=\kappa\circ\widetilde P_g\,,\quad\forall\,g\in\textsf{G}\,.\] Using this and the form of the \(\ell^1\)-norms, one shows immediately that \(\kappa\) sends \(\ell^1(\mathfrak C)\) into \(\ell^1(\mathfrak C/\mathfrak K)\) contractively. Its kernel is clearly \(\ell^1(\mathfrak C)\cap\mathfrak K\) . Let us show its surjectivity, starting from \[\varphi:=\sum_{g\in\textsf{G}}\varphi_g\equiv\sum_{g\in\textsf{G}}\widetilde Q_g(\varphi)\in\ell^1(\mathfrak C/\mathfrak K)\] (unconditional convergence in \(\mathfrak C/\mathfrak K\)) . Let \(\{\beta_g\!\mid\!g\in\textsf{G}\}\) be a summable family of positive numbers. For each \(g\), there is an element \(\Phi_g\in\mathfrak C_g\) such that \(\kappa(\Phi_g)=\varphi_g\) and \(\Vert\!\Phi_g\!\Vert_\mathfrak C\,\le\,\Vert\!\varphi_g\!\Vert_{\mathfrak C/\mathfrak K}+\beta_g\) . Then \(\Phi:=\sum_{g}\Phi_g\in\ell^1(\mathfrak C)\) and \(\kappa(\Phi)=\varphi\) . So \(\ell^1(\mathfrak C)/\ell^1(\mathfrak C)\cap\mathfrak K\) may be identified with \(\ell^1(\mathfrak C/\mathfrak K)\) . We apply Theorem 2.1 (ii) to the topologically graded quotient algebra \(\mathfrak C/\mathfrak K\) and finish the proof. ◻
The particular case of partial actions is worth mentioning. Suppose that \((\mathcal{A},\theta,\textsf{G})\) is a partial dynamical system with discrete group \(\textsf{G}\) and that \(\mathcal{K}\) is an ideal of \(\mathcal{A}\) that is \(\theta\)-invariant: \[\theta_g\big(\mathcal{A}_{g^{-1}}\!\cap\mathcal{K}\big)\subset \mathcal{K}\,,\quad\forall\,g\in \textsf{G}\,.\] We denote by the same letter \(\theta\) the action of \(\textsf{G}\) by partial automorphisms of \(\mathcal{K}\) defined by restrictions. One gets the C\(^*\)-partial dynamical system \((\mathcal{K},\theta,\textsf{G})\) . The partial crossed product \(\mathcal{K}\rightthreetimes_{\theta}\textsf{G}\) may be identified with an ideal of \(\mathcal{A}\rightthreetimes_{\theta}\textsf{G}\) . Under this identification, \(\ell^1_{\theta}(\textsf{G};\mathcal{K})\) becomes an ideal of \(\ell^1_{\theta}(\textsf{G};\mathcal{A})\) in the natural way: the \(\ell^1\)-function \(\varphi:\textsf{G}\rightarrow \mathcal{K}\) is taken to be \(\mathcal{A}\)-valued. One can use the exactness of the partial crossed product construction (Exel, Laca, and Quigg 2002, Sect. 3) to prove
Theorem 2.5 Assume that the discrete group \({\textsf{G}}\) is rigidly symmetric. The Banach \(^*\)-algebra \(\ell^1_{\theta}({\textsf{G}};\mathcal{A})\) is \(\mathcal{K}\!\rightthreetimes_{\theta}\!{\textsf{G}}\)-inverse closed in the partial crossed product \(\mathcal{A}\!\rightthreetimes_{\theta}\!{\textsf{G}}\) for any \(\theta\)-invariant ideal \(\mathcal{K}\) of \(\mathcal{A}\) .
Theorem (@(thm:theoremix)) can also be obtained as a consequence of Theorem (@(thm:ruye)), since partial crossed products are topologically graded, and the ideal \(\mathcal{K}\!\rightthreetimes_{\theta}\!\textsf{G}\) is indeed graded under the stated assumptions.
The situation in Example (2.1) is the most interesting:
Corollary 2.2 Let \((\mathcal{A},\theta,{\textsf{G}})\) be a partial C\(^*\!\)-dynamical system with discrete rigidly symmetric group \({\textsf{G}}\) and \(\mathcal{K}\) a \(\theta\)-invariant ideal in \(\mathcal{A}\) . Let \(\Pi\!:\!\mathcal{A}\rightthreetimes_{\theta}\!{\textsf{G}} \to \mathbb{B}(\mathcal{H})\) be a faithful representation such that \(\Pi\big(\mathcal{K}\rightthreetimes_{\theta}\!{\textsf{G}}\big)=\mathbb{K}(\mathcal{H})\) . Then the Banach \(^*\)-algebra \(\Pi[\ell^1_{\theta}({\textsf{G}};\mathcal{A})]\) is Fredholm inverse closed.
2.4 Other types of decay
::: {.definition #sconx} A weight on the discrete group \({\textsf{G}}\) is a function \(\nu: {\textsf{G}}\to [1,\infty)\) satisfying everywhere \[\label{submultiplicative} \nu(gh)\leq \nu(g)\nu(h)\,,\quad\nu(g^{-1})=\nu(g)\,, \quad \forall g,h\in{\textsf{G}}.\] Let \(\mathfrak C\) be a topologically graded \(C^*\)-algebra. On \(\,\bigoplus_{g\in{\textsf{G}}}\!\mathfrak C_g\) we can introduce the norm \[\begin{equation} \,\big\Vert\,\!\Phi\!\,\big\Vert\,_{\ell^{1,\nu}(\mathfrak C)}\,:=\sum_{g\in{\textsf{G}}}\nu(g)\!\,\big\Vert\,\!P_g(\Phi)\!\,\big\Vert\,. \tag{2.11} \end{equation}\] The completion in this norm, denoted by \(\ell^{1,\nu}(\mathfrak C)\) , can be seen as a Banach \(^*\)-algebra with the algebraic structure inherited from \(\ell^1(\mathfrak C)\) (or from \(\mathfrak C\)) . :::
Theorem 2.6 Let \({\textsf{G}}\) be a rigidly symmetric discrete group and \(\nu\) a weight. Assume that there exists a generating subset \(V\) of \(\,{\textsf{G}}\) containing the unit \(\textsf{e}\) such that
the following uGRS (uniform Gelfand-Raikov-Shilov) condition holds: \[\begin{equation} \lim_{n\rightarrow \infty}\sup_{g_1,\dots,g_n \in V}\nu(g_1\cdots g_n)^{\frac{1}{n}}=1\,, \tag{2.12} \end{equation}\]
for some finite constant \(C\) one has for any \(n\in\mathbb{N}\) \[\begin{equation} \sup_{g\in V^n\setminus V^{n-1}}\nu(g) \leq C\!\inf_{g\in V^n\setminus V^{n-1}}\nu(g)\,.\tag{2.13} \end{equation}\]
Then \(\ell^{1,\nu}(\mathfrak C)\) is a symmetric Banach \(^*\)-algebra for every topologically \({\textsf{G}}\)-graded \(C^*\)-algebra \(\mathfrak C\).
Proof. The problem of the symmetry of a weighted \(\ell^1\)-algebra as a \(^*\)-subalgebra of an unweighted one has been solved in (Fendler, Gröchenig, and Leinert 2008, Th.3) in a more particular context. We are going to check that the arguments can be adapted to the more general case. The main idea is proving that, given an element \(\Phi\in \ell^{1,\nu}(\mathfrak C)\) , its spectral radius is the same as in \(\ell^{1}(\mathfrak C)\) . Then one applies Hulanicki’s Lemma.
As \(\nu(\cdot)\geq 1\) , we have \[\begin{equation*} \,\big\Vert\,\!\Phi\!\,\big\Vert\,_{\ell^{1}(\mathfrak C)}=\sum_{g\in{\textsf{G}}}\!\,\big\Vert\,\!P_g(\Phi)\!\,\big\Vert\,\leq\sum_{g\in{\textsf{G}}}\nu(g)\!\,\big\Vert\,\!P_g(\Phi)\!\,\big\Vert\, = \,\big\Vert\,\!\Phi\!\,\big\Vert\,_{\ell^{1,\nu}(\mathfrak C)}\, \end{equation*}\] and using the spectral radius formula \[\begin{equation*} r_{\ell^{1}(\mathfrak{C})}(\Phi)\leq r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)\,,\quad\forall\,\Phi\in {\ell^{1,\nu}(\mathfrak{C})}\,. \end{equation*}\] To prove the opposite inequality, pick \(V\) a generating set for \(\textsf{G}\) and define on \(\mathbb Z\) the function \[\begin{equation*} v(n):=\sup_{g\in V^{|n|}} \nu(g)\,. \end{equation*}\] Due to the uGRS condition, it is a weight and one has the obvious associated weighted space \(\ell^{1,\nu}(\mathbb{Z})\). By induction, from the equation ((2.11)) we get \[\begin{equation*} \,\big\Vert\,\!\Phi^n \!\,\big\Vert\,_{\ell^{1,\nu}(\mathfrak{C})}\,\leq \sum_{g_1\in\textsf{G}}\cdots\sum_{g_n\in\textsf{G}}\nu(g_1\dots g_n)\,\big\Vert\,\!P_{g_1}(\Phi) \!\,\big\Vert\,\cdots \!\,\big\Vert\,\!P_{g_n}(\Phi) \!\,\big\Vert\,. \end{equation*}\] Since \(\textsf{G}=\bigsqcup_{m\in\mathbb N}(V^m\setminus V^{m-1})\) , where \(V^0=\emptyset\), we may split the sum accordingly. This yields \[\begin{equation} \,\big\Vert\,\!\Phi^n \!\,\big\Vert\,_{\ell^{1,\nu}(\mathfrak{C})}\,\leq\!\sum_{m_1,\dots,m_n=1}^{\infty}\sum_{V^{m_1}\setminus V^{m_1-1}}\cdots\sum_{V^{m_n}\setminus V^{m_n-1}} \nu(g_1\dots g_n)\,\big\Vert\, P_{g_1}(\Phi) \!\,\big\Vert\,\cdots \!\,\big\Vert\, P_{g_n}(\Phi) \!\,\big\Vert\,. \tag{2.14} \end{equation}\] If \(g_j\in V^{m_j}\setminus V^{m_j-1}\), then \(g_1\cdots g_n \in V^{m_1+\cdots +m_n}\) and so the weight is majorized by \[\nu(g_1\cdots g_n)\leq \sup_{h\in V^{m_1+\cdots +m_n}} \nu(h) =v(m_1+\cdots m_n)\,.\] Set \(b_m:=\sum_{g\in V^m\setminus V^{m-1}}\!\,\big\Vert\, P_g(\Phi) \!\,\big\Vert\,\) and \(b=(b_m)_{m\in\mathbb N}\) . Then we have \(\!\,\big\Vert\,\Phi \!\,\big\Vert\,_{\ell^1(\mathfrak C)}\,=\,\,\big\Vert\, b \!\,\big\Vert\,_{\ell^1(\mathbb N)}\) . Also the condition ((2.13)) implies immediately that \[C^{-1}\!\,\big\Vert\, b\!\,\big\Vert\,_{\ell^{1,v}(\mathbb Z)}\,\leq \,\,\big\Vert\,\Phi \!\,\big\Vert\,_{\ell^{1,\nu}(\mathfrak C)}\,\leq\,\,\big\Vert\,\,b \!\,\big\Vert\,_{\ell^{1,v}(\mathbb Z)}.\] Returning to ((2.14)) we obtain \[\!\,\big\Vert\, \Phi^n \!\,\big\Vert\,_{\ell^{1,\nu}(\mathfrak{C})}\,\leq\!\sum_{m_1,\dots,m_n=1}^{\infty}\!v(m_1+\cdots +m_n)\,b_{m_1}\cdots b_{m_n}\!=\;\,\big\Vert\, b^n \!\,\big\Vert\,_{\ell^{1,v}(\mathbb Z)}\,<\infty\,.\] By its definition the weight \(v\) on \(\mathbb Z\) satisfies the GRS-condition, and \(\ell^{1,v}(\mathbb Z)\) is symmetric by (Fendler, Gröchenig, and Leinert 2006). Hence \[\begin{aligned} r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)&=\lim_{n\rightarrow\infty} \!\,\big\Vert\, \Phi^n \!\,\big\Vert\,_{\ell^{1,\nu}(\mathfrak{C})}^{1/n}\leq \lim_{n\rightarrow\infty} \!\,\big\Vert\, b^n \!\,\big\Vert\,_{\ell^{1,\nu}(\mathbb{Z})}^{1/n}\\ &=r_{\ell^{1,\nu}(\mathbb{Z})}(b)=r_{\ell^{1}(\mathbb{Z})}(b)\\ &\le\,\,\big\Vert\, b \!\,\big\Vert\,_{\ell^{1}(\mathbb{Z})}=\,\,\big\Vert\, \Phi \!\,\big\Vert\,_{\ell^{1}(\mathfrak{C})}. \end{aligned}\] So for all \(k\in\mathbb N\) we have \[r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)=r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi^n)^{1/n}\leq\,\,\big\Vert\, \Phi^n \!\,\big\Vert\,_{\ell^{1}(\mathfrak{C})}^{1/n},\] and by letting \(n\rightarrow\infty\) we obtain the required inequality \(\,r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)\leq r_{\ell^{1}(\mathfrak{C})}(\Phi)\) . ◻
Remark 2.4 There is a different way to prove Theorem (2.6). We define the Beurling algebra associated to the weight \(\nu\) and to the \(C^*\)-algebra \(\mathcal B\) as \[\label{sharpix} \ell^{1,\nu}({\textsf{G}};\mathcal B):=\big\{\Psi\, \vert \,\nu\Psi\in \ell^{1}({\textsf{G}};\mathcal B)\big\}\,,\] with norm \(\Vert \cdot \!\Vert_{\ell^{1,\nu}({\textsf{G}},\mathcal B)}\,:=\,\Vert\nu\,\cdot\,\!\Vert_{\ell^{1}({\textsf{G}};\mathcal B)}\). By (Fendler, Gröchenig, and Leinert 2008), the assumptions on the weight imply that \(\ell^{1,\nu}({\textsf{G}};\mathcal B)\) is a symmetric Banach \(^*\)-algebra. One then shows as in Proposition 2.1 an isometric embedding \(\ell^{1,\nu}(\mathfrak C)\hookrightarrow\ell^{1,\nu}({\textsf{G}};\mathcal B)\) , for some \(C^*\)-algebra \(\mathcal B\) , and use this as in Theorem 2.1 to deduce the symmetry of \(\ell^{1,\nu}(\mathfrak C)\) .