Chapter 3 Examples
3.1 Inverse closed Banach *-algebras from topological partial actions
We introduce now certain partial crossed product \(C^*\)-algebras associated to partial topological actions of the discrete group. We do not consider the most general case; in particular, we restrict generality in such a way to have certain natural faithful Hilbert space representations. In this way, transparent results on inverse closedness will be available.
So let \((X,\Theta,\textsf{G})\) be a partial action of the group \(\textsf{G}\) (assumed here to be countable) on the compact Hausdorff space \(X\). In more detail, the partial action is denoted by \(\big\{\Theta_g:X_{g^{-1}}\!\to X_g\,\big\vert\, g\in\textsf{G}\big\}\) , where each \(X_g\) is an open subset of \(X\) and \(\Theta_g\) is a homeomorphism with domain \(X_{g^{-1}}\) and image \(X_g\) . One requires
\(X_\textsf{e}=X\) and \(\Theta_\textsf{e}=\textrm{id}_X\) ,
for every \(g,h\in\textsf{G}\) , the homeomorphism \(\Theta_{gh}\) extends \(\Theta_{g}\circ\Theta_{h}\).
Observe that the composition \(\Theta_g\circ\Theta_h\) is meant to refer to the map whose domain is the set of all elements \(x \in X\) for which \(\Theta_g\circ\Theta_h(x)\) is well defined, therefore the domain of \(\Theta_g\circ\Theta_h\) is the set \(\Theta_h^{-1}( X_h\cap X_{g^{-1}})\).
In fact each locally compact partial dynamical system may be deduced by restricting a global one in an essentially canonical way: there exists a global action \(\tau\) of the group \(G\) on a topological space \(\tilde{X}\) such that \[X_g=\tau_g(\tilde{X})\cap \tilde{X} \quad \quad \forall g\in G\] and \(\tau_g|_{X_{g^{-1}}}=\Theta_g\) (Exel 2017 Ch. 3). But this restriction of the global action might fail in the case the total space \(\tilde X\) is not Hausdorff ((Exel 2017 Prop. 5.6) and (Abadie 2003 Th. 4.44)).
We obtain from \((X,\Theta,\textsf{G})\) a partial \(C^*\)-dynamical system \(\big(C(X),\theta,\textsf{G}\big)\) , where the ideals are \[\label{santodomingo} \mathcal A_g\!:=C\big(X_{g}\big)\equiv\big\{a\in C(X)\,\big\vert\,a(x)=0\,,\,\forall\,x\notin X_{g}\big\}\] and the isomorphisms are \[\label{panama} \theta_g:C\big(X_{g^{-1}}\big)\to C\big(X_{g}\big)\,,\quad\theta_g(a):=a\circ\Theta_{g^{-1}}\,.\] Therefore, one can construct the Banach \(^*\)-algebra \(\ell^1_\theta\big(\textsf{G};C(X)\big)\) and its enveloping partial crossed product \(C(X)\!\rightthreetimes_\theta\!\textsf{G}\) . We deduce from Theorem (2.1)
Proposition 3.1 If \(\,{\textsf{G}}\) is rigidly symmetric, \(\ell^1_\theta\big({\textsf{G}},C(X)\big)\) is reduced, symmetric and inverse closed in the partial crossed product.
Assume now that under the partial action there is an open dense orbit \(Y\) with trivial isotropy. Thus, for some (and then any) \(y\in Y\) one has \[\label{guadelupa} \textsf{G}(y):=\big\{g\in\textsf{G}\,\big\vert\,y\in X_{g^{-1}}\,,\Theta_g(y)=y\big\}=\{\textsf{e}\}\,.\] Then \(X\) will be a compactification of the discrete orbit \(Y\) and for any \(y\in Y\) the orbit map \(g\to\Theta_g(y)\) will be a homeomorphism on its image.
In the Hilbert space \(\ell^2\big(Y\big)\) we introduce the representation by multiplication operators \[\label{mexic} \pi:C(X)\to\mathbb B\big[\ell^2(Y)\big]\,,\quad\big[\pi(a)\xi\big](y):=a(y)\xi(y)\,.\] For any \(h\in\textsf{G}\,,\,y\in Y\) and \(\xi\in\ell^2(Y)\) one defines \[\label{partilix} \big[u_h(\xi)\big](y):= \begin{cases} \xi\big(\Theta_{h^{-1}}(y)\big) &\textrm{if}\ \;y\in X_h\,,\\ 0\, &\textrm{if}\ \;y\notin X_h\,. \end{cases}\] Then \(\big(\pi,u,\ell^2(Y)\big)\) is a covariant representation of \(\big(C(X),\theta,\textsf{G}\big)\) and the integrated form representation \[\Pi\equiv\pi\!\rightthreetimes u\!:C(X)\!\rightthreetimes_\theta\!\textsf{G}\to\mathbb B\big[\ell^2(Y)\big]\,,\] which acts on \(\ell^1_\theta\big(\textsf{G};C(X)\big)\) as \(\Pi(\Phi)=\sum_{h\in\textsf{G}}\pi\big[\Phi(h)\big]u_h\) and which satisfies \[\begin{equation} \Pi\big(a\otimes\delta_h\big)=\pi(a)u_h\,,\quad\forall\,h\in\textsf{G}\,,\,a\in C(X_h)\,.\tag{3.1} \end{equation}\] To express it explicitly, we need some notations. For \(y,z\in Y\) we set \(g_{yz}\) for the unique element of the group such that \(\Theta_{g_{yz}}(y)=z\) ; we use the fact that \(Y\) is an orbit with trivial isotropy. Note the relations \[\begin{equation} g_{yz}^{-1}=g_{zy}\,,\quad g_{zy}=g_{xy}g_{zx}\,,\quad\forall\,x,y,z\in Y. \tag{3.2} \end{equation}\] A direct computation, relying on ((3.2)), shows that on \(\ell^1_\theta\big(\textsf{G};C(X)\big)\) the representation \(\Pi\) reads \[\begin{equation} \big[\Pi(\Phi)\xi\big](y)=\sum_{z\in Y}\Phi\big(g_{zy},y\big)\xi(z)\,, \tag{3.3} \end{equation}\] where we set \(\big[\Phi(k)\big](y)=:\Phi(k,y)\) .
Theorem 3.1 Suppose that \({\textsf{G}}\) is countable and rigidly symmetric and that \(Y\) is a open dense orbit of the space \(X\), with trivial isotropy. Let \(\Phi\in\ell^1_\theta\big({\textsf{G}};C(X)\big)\) .
(i). If the operator \(\,\Pi(\Phi)\) is invertible, there exists \(\Psi\in\ell^1_\theta\big({\textsf{G}};C(X)\big)\) such that \(\Pi(\Phi)^{-1}\!=\Pi(\Psi)\) .
(ii). If \(\,\Pi(\Phi)\) is Fredholm, at least one of its inverses \(T\) modulo \(\mathbb K\big[\ell^2(Y)\big]\) belongs to \(\Pi[\ell^1_{\theta}\big({\textsf{G}};C(X)\big)]\) .
Proof. (i). Having in view Proposition (3.1), we only need to check that the representation \(\Pi\) is faithful; see also Theorem (2.1)(iv). The universal and the reduced partial crossed products here are the same, cf. Remark (1.2). Note that the restriction of \(\Pi\) to \(C(X)\) coincides with \(\pi\) (set \(h=\textsf{e}\) in ((3.1)). By (Exel, Laca, and Quigg 2002, Th. 2.6), injectivity of \(\Pi\) would follow if we prove that \(\pi\) is injective and the action is topologically free. The injectivity of \(\pi\) is obvious: \(\pi(a)=0\) means that the restriction of \(a\) to \(Y\) is null, which implies that \(a\) is null, since the orbit \(Y\) was supposed dense. Topological freeness (cf. (Exel, Laca, and Quigg 2002, Def. 2.1.)) is clear, since \(Y\) is dense: having trivial isotropy, it is disjoint from any set of the form \[F_g:=\big\{x\in X_{g^{-1}}\,\big\vert\,\Theta_g(x)=x\big\}\,,\quad g\ne\textsf{e}\,,\] so this one must have empty interior.
(ii). Since \(Y\) is discrete, the elements of \(C_0(Y)\) are transformed by the representation \(\pi\) into compact multiplication operators in \(\ell^2(Y)\) . The \(\theta\)-invariant ideal \(C_0(Y)\) gives rise to the ideal \(C_0(Y)\!\rightthreetimes_\theta\!\textsf{G}\) of the partial crossed product \(C(X)\!\rightthreetimes_\theta\!\textsf{G}\) and one has \(\Pi\big[C_0(Y)\!\rightthreetimes_\theta\!\textsf{G}\big]=\mathbb K\big[\ell^2(Y)\big]\) . Therefore one can apply Corollary (2.2) and conclude that \(\Pi[\ell^1_{\theta}\big(\textsf{G};C(X)\big)]\subset\mathbb B\big[\ell^2(Y)\big]\) is Fredholm inverse closed. Then the assertion follows easily from Atkinson’s Theorem; see also Example (2.1).
3.2 Toeplitz-Bunce-Deddens operators
In the Hilbert space \(\ell^2(\mathbb{N})\) , with canonical orthonormal base \(\{\delta_n\}_{n\in\mathbb{N}}\) , each function \(a\in\ell^\infty(\mathbb{N})\) defines a bounded multiplication operator \(\textsf{M}_a\) . Let us denote by \(\textsf{S}\) the right shift uniquely determined by \(\textsf{S}\,\delta_n\!:=\delta_{n+1}\) for every \(n\in\mathbb{N}\) . For \(q\in\mathbb{N}\) , we say that \(a\in\ell^\infty(\mathbb{N})\) is \(q\)-periodic if \(a(n+q)=a(n)\,,\,\forall\,n\in\mathbb{N}\) . We denote by \(\ell^\infty(\mathbb{N};q)\) the \(C^*\)-subalgebra of all \(q\)-periodic functions. The starting point in constructing a Toeplitz-Bunce-Deddens algebra is an infinite family \(\mathbf q:=\{q_i\!\mid\!i\in\mathbb{N}\}\) of strictly increasing positive integers such that each \(q_i\) divides \(q_{i+1}\) , i.e. \(q_{i+1}=r_i q_i\) with \(r_i\ge 2\) .
Definition 3.1 The Toeplitz-Bunce-Deddens algebra \(\mathfrak D(\mathbf q)\) associated to the multi-integer \(\mathbf q\) is the \(C^*\)-algebras of operators in \(\ell^2(\mathbb{N})\) generated by the family \[\begin{equation} \Big\{\textsf{ S}_a:=\textsf{ S}\textsf{ M}_a\,\Big\vert\,a\in\bigcup_{i\in\mathbb{N}}\ell^\infty(\mathbb{N};q_i)\Big\}\,. \tag{3.4} \end{equation}\]
Proposition 3.2 The Toeplitz-Bunce-Deddens algebra is topologically graded over \(\mathbb{Z}\) . Its \(\ell^1\)-Banach \(^*\)-algebra \(\ell^1\big[\mathfrak D(\mathbf q)\big]\) is reduced, symmetric and inverse closed in \(\mathfrak D(\mathbf q)\) and in \(\mathbb B\big[\ell^2(\mathbb{N})\big]\) (an explicit description is included in the proof).
Proof. By (Exel 1994b), a canonical action \(\alpha\) of the torus \(\mathbb{T}=\widehat{\mathbb{Z}}\) on \(\mathfrak D(\mathbf q)\) is defined as follows: For every \(z\in\mathbb T\) the unitary multiplication operator determined by \(U_z\delta_n:=z^n\delta_n\) (\(n\in\mathbb{N}\)) defines on \(\mathbb B\big[\ell^2(\mathbb{N})\big]\) the inner automorphism \(T\to U_z TU_z^*\). Because of \[\begin{equation} \alpha_z(S_a)=U_z S_aU_z^*=zS_a\,,\quad\forall\,a\in\ell^\infty(\mathbb{N})\,, (eq:costarica) \end{equation}\] \(\big(\mathfrak D(\mathbf q),\alpha,\mathbb{T}\big)\) is indeed a (full) continuous action. We also write \(\ell^\infty(\mathbb{N};\mathbf q)\) for the (unital, Abelian) \(C^*\)-subalgebra of \(\ell^\infty(\mathbb{N})\) generated by \(\bigcup_{i\in\mathbb{N}}\ell^\infty(\mathbb{N};q_i)\) . The spectral subspace \(\mathfrak D(\mathbf q)_k\) is generated by operators of the ordered form \[\begin{equation} S_{a_1}\dots S_{a_n}S^*_{b_1}\dots S^*_{b_m}\,,\quad a_1,\dots,a_n,b_1,\dots,b_m\in\ell^\infty(\mathbb{N};\mathbf q)\,,\ n-m=k\,. \tag{3.5} \end{equation}\] This follows easily from the relations \(S^*_a S_b=M_{\overline ab}\) and \(M_a S_b=S_{\tilde ab}\) , where \(\tilde a_n:=a_{n-1}\) if \(n\ge 1\) . Since \(\tilde a_0\) does not appear in the definition of the weighted shift operator, it can be fixed such that \(\tilde a\) is periodic. Other forms of the elements in \(\mathfrak D(\mathbf q)_k\) could involve the final projection \(Q:=SS^*\). These having been settled, the assertions concerning the \(\ell^1\)-Banach algebra follow from Theorem (2.2).
Remark 3.1 In (Exel 1994b) one gets the isomorphism \[\begin{equation} \mathfrak D(\mathbf q)\overset{\sim}{\longrightarrow}\big[c_0(\mathbb{N})\oplus\ell^\infty(\mathbb{N};\mathbf q)\big]\!\rightthreetimes_\theta\mathbb{Z}\cong C\big[\mathbb{N}(\mathbf q)\big]\!\rightthreetimes_\theta\mathbb{Z}\,. \tag{3.6} \end{equation}\] We refer to (Exel 1994b) for the Gelfand spectrum \(\mathbb{N}(\mathbf q)=\mathbb{N}\sqcup\Sigma(\mathbf q)\) of the Abelian \(\alpha\)-fixed point \(C^*\)-algebra \[\mathfrak D(\mathbf q)_0\!:=c_0(\mathbb{N})\oplus\ell^\infty(\mathbb{N};\mathbf q)\] (a compactification of the discrete set \(\mathbb{N}\) by a Cantor set \(\Sigma(\mathbf q)\)) and for the interesting explicit form of the partial action \(\theta\) in the Gelfand realization (induced by a partial homeomorphism of \(\mathbb{N}(\mathbf q)\)).
We may also state that \(\ell^1\big[\mathfrak D(\mathbf q)\big]=\bigoplus_{k\in\mathbb{Z}}^{1,\alpha}\mathfrak D(\mathbf q)_k\) is Fredholm inverse closed in \(\mathfrak D(\mathbf q)\subset\mathbb B\big[\ell^2(\mathbb{N})\big]\) . This follows from Corollary (2.2), where \(\mathbb{K}=c_0(\mathbb{N})\) and \(\Pi\) is the inverse of the isomorphism appearing in ((3.6)), since \(\Pi\big[c_0(\mathbb{N})\!\rightthreetimes_\theta\!\mathbb{Z}\big]=\mathbb K\big[\ell^2(\mathbb{N})\big]\) . One calls the quotient \[\begin{equation} \mathfrak D(\mathbf q)/\mathbb K\big[\ell^2(\mathbb{N})\big]\cong\ell^\infty(\mathbb{N};\mathbf q)\!\rtimes_\theta\mathbb{Z}\cong C\big[\Sigma(\mathbf q)\big]\!\rtimes_\theta\mathbb{Z} \tag{3.7} \end{equation}\] the Bunce-Deddens algebra. The action "at infinity" of \(\mathbb{Z}\) on the Cantor set \(\Sigma(\mathbf q)\) is induced by an odometer map. It is a global action, so this quotient can be dealt with by usual crossed products.
3.3 UHF algebras
An UHF-algebra is the inductive limit \[\label{limindiz} \mathfrak C:=\underset{m\in\mathbb{N}}{\textrm{lim} \,\textrm{ind}\,\mathbf M^{p_m}} \] of a sequence of full matricial \(C^*\)-algebras, with unital and injective connecting morphisms, where each \(p_m\) divides \(p_{m+1}\). In (Exel 1995) Exel treated general approximately finite algebras. He defined on \(\mathfrak C\) a regular and semi-saturated \(\mathbb T\)-action and then he applied his theory from (Exel 1994a) to get an isomorphism between \(\mathfrak C\) and a partial crossed product \(\mathcal{A}\rightthreetimes_\theta\mathbb{Z}\) , where \(\mathcal{A}\) is an Abelian almost finite \(AF\)-algebra. To simplify, we decided to treat only UHF algebras; the general case is similar, but it would involve more complicated notations. To state a result on symmetric subalgebras, we make use of Theorem (2.2), only invoking the circle action and its spectral subspaces; the partial crossed product will not be mentioned.
On every full matrix algebra \(\mathbf M^p\) one defines \[\label{inerix} \alpha^p:\mathbb T\to\textrm{ Aut}\big(\mathbf M^p\big)\,,\quad\alpha^p_z\big[(c_{ij})_{i,j}\big]=\big(z^{i-j}c_{ij}\big)_{i,j}\,.\] The connecting morphisms \(\big\{\mu^{m}\!:\!\mathbf M^{p_m}\to\mathbf M^{p_{m+1}}\big\}\) are covariant with respect to the actions \(\big\{\alpha^{p_m}\big\}\). The inductive limit comes with the canonical monomorphisms \(\big\{\nu^{m}\!:\mathbf M^{p_m}\!\to\mathfrak C\big\}\) and \(\bigcup_m\nu^{(m)}\!\big(\mathbf M^{p_m}\big)\) is dense in \(\mathfrak C\) . For every \(z\in\mathbb T\,,m\in\mathbb{N}\) and \(\Phi\in\mathbf M^{p_m}\) we set \[\label{tetaniz} \alpha_z\big[\nu^{m}(\Phi)\big]:=\nu^{m}\big[\alpha^{p_m}_z(\Phi)\big]\,.\] It is shown in (Exel 1995, Sect. 2) that \(\alpha\) extends to an action on \(\mathfrak C\) (which is semi-saturated and regular).
To make Theorem (2.2) concrete, one needs the spectral subspaces. For each \(|k|\le p\) , let us denote by \(\mathbf M^p_k\) the vector subspace of \(p\!\times\!p\)-matrices with non-empty entries only on the \(k'\)th diagonal (\(c_{ij}=0\) if \(i-j\ne k\)). If \(|k|>p\) , we set simply \(\mathbf M^p_k\!:=\{0\}\) . Then \(\mathbf M^p_k\) is the \(k'\)th spectral subspace of the action \(\alpha^p\). It is follows from the definitions and from the fact that each canonical morphism \(\nu^{m}\) is injective that the spectral subspaces of the action \(\alpha\) are the closures in \(\mathfrak C\) of the unions over \(m\) of the images through \(\nu^{m}\) of the subspaces \(\mathbf M^{p_m}_k\).
Theorem 3.2 The Banach \(^*\)-algebra \[\label{perru} \ell^1(\mathfrak C)=\bigoplus^{1,\alpha}_{k\in\mathbb{N}}\overline{\bigcup_{m\in\mathbb{Z}}\nu^{m}\big(\mathbf M^{p_m}_k\big)}\] is reduced, symmetric and inverse closed in \(\mathfrak C\) .
Remark 3.2 In (Ta?) Takesaki (see also (Fell and Doran 1988, 1:VIII.17)) provided an approach to UHF algebras which would lead to a different grading, and consequently also to a symmetric \(\ell^1\)-type Banach \(^*\)-algebra, this time over the infinite restricted product \(\textsf{G}:=\prod^\prime_{m\in\mathbb{N}}\textsf{G}_m\) with the discrete topology, where each \(\textsf{G}_m\) is a cyclic group of a certain order \(q_m\) . Actually, the UHF algebra \(\mathfrak C\) is isomorphic to an infinite tensor product \(\otimes_m\mathbf M^{q_m}\), where \(q_m:=p_{m+1}/p_m\) . In its turn, this one is shown to be isomorphic to the full crossed product \(C(\widehat{\textsf{G}}\,)\!\rtimes\textsf{G}\) . The dual \(\widehat{\textsf{G}}\) can be identified with the product group \(\prod_{m\in\mathbb{N}}\textsf{G}_m\) .
3.4 CAR algebras
Let \(\big(\mathcal R,(\cdot|\cdot)\big)\) be an infinitely dimensional separable Hilbert space and \(a:\mathcal R\to\mathbb B(\mathcal H)\) a representation of the canonical anticommutation relations in the Hilbert space \(\mathcal{H}\) , generating the unital \(C^*\)-algebra \(\textrm{ CCR}(\mathcal R)\subset\mathbb B(\mathcal H)\) . Thus \(a\) is lineal and for every \(r,s\in\mathcal R\) one has \[\label{anticom} a(r)a(s)+a(s)a(r)=0\,,\quad a^*(r)a(s)+a^*(s)a(r)=(r|s)\,.\] It is known that \(\textrm{ CCR}(\mathcal R)\) is isomorphic to the UHF algebra \(\mathbf M(2^\infty):=\underset{m\in\mathbb{N}}{\textrm{ lim\,ind}\,\mathbf M^{2^m}}\) as well as with the infinite tensor product \(\bigotimes_{m\in\mathbb{N}}\mathbf M^2\).
Let also \(V:\widehat{\textsf{G}}\to\mathbb B(\mathcal R)\) be a strongly continuous unitary representation of the compact Abelian group \(\widehat{\textsf{G}}\) . Associated to \(V\) there is a continuous action \(\upsilon:\widehat{\textsf{G}}\to\textrm{ Aut}\big[\textrm{ CCR}(\mathcal R)\big]\) , uniquely defined on generators by \[\label{creatrix} \upsilon_\chi\big[a(r)\big]=a\big[V_\chi(r)\big]\,,\quad\forall\,r\in\mathcal R\,,\,\chi\in\widehat{\textsf{G}}\,.\]
We denote by \(\textsf{G}\) the dual of \(\widehat{\textsf{G}}\) ; it is a discrete Abelian group. We already have the grading \[\label{carnatix} \textrm{ CCR}(\mathcal R)=\widetilde\bigoplus_{g\in\textsf{G}}\textrm{ CCR}(\mathcal R)^{\upsilon}_g\] in terms of spectral subspaces of the action \(\upsilon\) . The next corollary follows directly from Theorem (2.1). Note the generality: any representation \(\big(\widehat{\textsf{G}},V\big)\) provides a result.
Corollary 3.1 The corresponding \(\ell^1\)-algebra \[\begin{equation} \ell^1\big(\textrm{CCR}(\mathcal R)\big)=\bigoplus_{g\in{\textsf{G}}}^{1,\upsilon}\textrm{ CCR}(\mathcal R)^{\upsilon}_g \tag{3.8} \end{equation}\] is symmetric and inverse closed in \(\textrm{ CCR}(\mathcal R)\) and in \(\mathbb B(\mathcal H)\).
It is clear that \(a(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_g\) if and only if \(r\in\mathcal R^V_g\), meaning by definition that \(V_\chi(r)=\chi(g)r\) for all \(\chi\in\widehat{\textsf{G}}\) . Similarly, \(a^*(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_g\) if and only if \(r\in\mathcal R^V_{g^{-1}}\) .
Example 3.1 If \(V:\mathbb{T}\to\mathbb B(\mathcal R)\) is given by \(V_\tau(r):=e^{2\pi i\tau}r\) and the group duality is implemented by \(\tau(k):=e^{2\pi ik\tau}\) (\(k\in\mathbb{Z}\)) , then \(\mathcal R^V_1=\mathcal R\) , while \(\mathcal R^V_k=\{0\}\) for any \(k\ne 1\) . Thus \(a(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_1\) and \(a^*(r)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_{-1}\) , from which it follows that \[\label{followix} a^*(r_1)\dots a^*(r_n)a(s_1)\dots a(s_m)\in\textrm{ CCR}(\mathcal R)^{\upsilon}_{m-n}\,,\quad\forall\,r_1,\dots,r_n,s_1,\dots s_m\in\mathcal R\,.\]
3.5 Wiener-Hopf algebras associated to quasi-lattice ordered groups
One can treat this subject by using partial crossed products, as in Subsection 2.2, due to results from (Quigg and Raeburn 1997; Exel, Laca, and Quigg 2002; Exel 2017). We found it easier for our purposes to use the initial article (Nica 1992), combined with Theorem (2.1).
We fix a sub-monoid \(\textsf{P}\) of a discrete amenable group \(\textsf{G}\) such that \(\textsf{P}\cap\textsf{P}^{-1}=\{\textsf{e}\}\) . One defines a left-invariant order relation in \(\textsf{G}\) by \(g\leq h\) iff \(g^{-1}h\in \textsf{P}\). The ordered group \((\textsf{G},\textsf{P})\) is quasi-lattice ordered if any \(g\in\textsf{P}\textsf{P}^{-1}\) has a least upper bound in \(\textsf{P}\). Many examples may be found at (Nica 1992, pag.23).
Consider the Hilbert space \(\ell^2(\textsf{P})\) with its usual orthonormal basis \(\{e_q\}_{q\in\textsf{P}}\). For \(p\in\textsf{P}\), consider the bounded linear operator \(W_p\in \mathbb{B}(\ell^2(\textsf{P}))\) defined by \[W_p(e_q)=e_{pq}\,,\quad\forall\,q\in\textsf{P}.\] We refer to \(W\) as the regular semigroup of isometries of \(\textsf{P}\). The \(C^*\)-algebra of operators on \(\ell^2(\textsf{P})\) generated by the range of \(W\) is called the Wiener-Hopf algebra of \(\,\textsf{P}\) is denoted by \(\mathfrak{W}(\textsf{P})\) .
For every \(g\in\textsf{P}\textsf{P}^{-1}\) we define the closed subspace \[\mathfrak{W}(\textsf{P})_g:=\overline{\textrm{span}}\,\big\{W_p W_q^*\,\big\vert\,pq^{-1}\!=g\big\}\] and set \(\mathfrak{W}(\textsf{P})_g=\{0\}\) if \(g\notin \textsf{P}\textsf{P}^{-1}\). A characterization of the Abelian \(C^*\)-subalgebra \(\mathfrak{W}(\textsf{P})_\textsf{e}\) is \[\mathfrak{W}(\textsf{P})_\textsf{e} =\left\{T\in\mathfrak{W}(\textsf{P})\!\mid\! T \textrm{ has a diagonal matrix relatively to the canonical basis of }\ell^2(\textsf{P})\right\}.\] In Section 3 of Nica’s paper (Nica 1992) it is shown that that the collection \(\big\{\mathfrak{W}(\textsf{P})_g\,\big\vert\,g\in\textsf{G}\big\}\) provides a topologically graded structure of the Wiener-Hopf algebra. We can apply now Theorem (2.1) and get
Corollary 3.2 The Banach \(^*\)-algebra \(\ell^1\Big( \widehat{\bigoplus}_{g\in{\textsf{G}}}\,\mathfrak{W}({\textsf{P}})_g \Big)\) is symmetric and inverse closed in \(\mathbb B\big[\ell^2({\textsf{P}})\big]\) .
Remark 3.3 Suppose that \({\textsf{P}}\) is finitely generated (a slightly more general assumption is possible). Then \(\ell^1\Big( \widehat{\bigoplus}_{g\in{\textsf{G}}}\,\mathfrak{W}({\textsf{P}})_g \Big)\) is also Fredholm inverse closed. We only sketch a proof. As in (Quigg and Raeburn 1997; Exel, Laca, and Quigg 2002; Exel 2017), one has an isomorphism \(\mathfrak W({\textsf{P}})\cong\mathfrak W({\textsf{P}})_\textsf{e}\rightthreetimes_\theta{\textsf{G}}\) for a certain partial action \(\theta\) of \({\textsf{G}}\) on the diagonal Abelian algebra. This one is induced from a (Bernoulli-type) partial topological action \(\Theta\) on its Gelfand spectrum \(\Omega_\textsf{e}\) , which is a compactification of the monoid \({\textsf{P}}\). This compactification is regular (\({\textsf{P}}\) is an open dense orbit) under the stated hypothesis on \({\textsf{P}}\). It is shown in (Nica 1992, 6.3) that this regularity is equivalent with the fact that the ideal of all compact operators, identified with \(C({\textsf{P}})\rightthreetimes_\theta{\textsf{G}}\) , is contained in \(\mathfrak W({\textsf{P}})\,.\) Then the result follows easily from our Corollary (2.2).
3.6 Higher rank graph algebras
In this subsection we are going to use constructions from (Kumjian and Pask 2000) and (Raeburn 2016).
Definition 3.2 A \(\textrm{ k}\)-graph ( higher rank graph) is a countable category \(\Lambda\) endowed with a functor \(\textrm{l}:\Lambda\to\mathbb{N}^\textrm{ k}\) ( the degree functor) satisfying the factorization property: For every \(\lambda\in\Lambda\) such that \(\textrm{ l}(\lambda)=\mathfrak m+\mathfrak n\) , there exist unique elements \(\mu,\nu\in\Lambda\) such that \(\textrm{l}(\mu)=\mathfrak m\) , \(\textrm{l}(\nu)=\mathfrak n\) and \(\lambda=\mu\nu\) .
If \(\textrm{ k}=1\) , one identifies a \(1\)-graph with the path category \(E^*\!:=\bigsqcup_{n\in\mathbb{N}}E^n\) of a directed graph \(\big(s,r:E^1\to E^0\big)\) and \(\textrm{ l}(\lambda)\) is the usual length of a path \(\lambda:=e_1\dots e_{\textrm{ l}(\lambda)}\) . Accordingly, for the \(\textrm{ k}\)-graph \(\big(\textrm{ l}:\Lambda\to\mathbb{N}^\textrm{ k}\big)\) , one denotes by \(\textsf{s},\textsf{r}:\Lambda\to\Lambda^0\equiv\textrm{ Obj}(\Lambda)\) the source and the range map in the category. One has the decomposition \[\Lambda=\bigsqcup_{\mathfrak n\in\mathbb{N}^\textrm{ k}}\Lambda^{\mathfrak n}\equiv\bigsqcup_{\mathfrak n\in\mathbb{N}^\textrm{ k}}\textrm{ l}^{-1}(\mathfrak n)\,.\] For elements from the object set \(\Lambda^0\) (called vertices) one prefers notations as \(v,w\) . We also set set \(\Lambda^{\mathfrak n}(v):=\Lambda^{\mathfrak n}\cap\textrm{ r}^{-1}(v)\) ("paths" of "length" \(\mathfrak n\in\mathbb{N}^\textrm{ k}\) ending in \(v\in\Lambda^0\) ).
Definition 3.3 One says that the \(\textrm{ k}\)-graph is admissible if every \(\Lambda^{\mathfrak n}(v)\) is finite (raw-finite \(\textrm{ k}\)-graph) and non-void (no sources) and \(\Lambda^0\) is finite (insuring that the \(C^*\)-algebra below is unital, with unit \(\sum_{v\in\Lambda^0}\textsf{ S}_v\)).
Definition 3.4 Let \(\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}\) be an admissible \(\textrm{ k}\)-graph. A Cuntz-Krieger \(\Lambda\)-family is a set \(\{\textsf{ T}_\lambda\!\mid\!\lambda\in\Lambda\}\) of partial isometries in a \(C^*\)-algebra \(\mathfrak D\) such that
the elements \(\big\{\textsf{ T}_v\!\mid\!v\in\Lambda^0\big\}\) are mutually orthogonal projections,
if \(\textrm{ s}(\lambda)=\textrm{ r}(\mu)\) , then \(\textsf{ T}_\lambda\textsf{ T}_\mu=\textsf{ T}_{\lambda\mu}\) ,
\(\textsf{ T}^*_\lambda\textsf{ T}_\lambda=\textsf{ T}_{\textrm{ s}(\lambda)}\) for every \(\lambda\in\Lambda\) ,
\(\textsf{ T}_v=\sum_{\lambda\in\Lambda^{\mathfrak n}(v)}\textsf{ T}_\lambda\textsf{ T}^*_\lambda\) for every \(v\in\Lambda^0\) and \(\mathfrak n\in\mathbb{N}^\textrm{ k}\).
We define \(C^*(\Lambda,\textrm{ l})\equiv\mathfrak C(\Lambda)\) to be the universal \(C^*\)-algebra generated by a Cuntz-Krieger \(\Lambda\)-family \(\{\textsf{ S}_\lambda\!\mid\!\lambda\in\Lambda\}\) . Universality means that for every Cuntz-Krieger \(\Lambda\)-family \(\{\textsf{ T}_\lambda\!\mid\!\lambda\in\Lambda\}\subset\mathfrak D\) , there exists a unique \(C^*\)-morphism \(\pi_\textsf{ T}:\mathfrak C(\Lambda)\to\mathfrak D\) such that \(\pi_\textsf{ T}(\textsf{ S}_\lambda)=\textsf{ T}_\lambda\) for every \(\lambda\) .
We identify now families of symmetric Banach \(^*\)-algebras, starting with functors \(\textsf{ F}:\Lambda\to\textsf{G}\) , where \(\textsf{G}\) is supposed to be a discrete Abelian group. One particular case is \(\textsf{G}=\mathbb{N}^\textrm{ k}\) and \(\textsf{ F}=\textrm{ l}\) , but many others are possible. For every \(g\in\textsf{G}\) one sets \[\label{bebex} \mathfrak{C}(\Lambda)_g:=\overline{\textrm{ span}}\big\{\textsf{ S}_\lambda\textsf{ S}^*_{\mu}\,{\big\vert\,\textsf{ F}}(\lambda)\textsf{ F}(\mu)^{-1}=g\big\}\,.\] It turns out that these are spectral subspaces of a dual action \(\alpha^\textsf{ F}:\widehat{\textsf{G}}\to\textrm{ Aut}\big[\mathfrak C(\Lambda)\big]\) , uniquely determined (use the universal property of \(\mathfrak C(\Lambda)\)) by \[\alpha^\textsf{ F}_\chi(\textsf{ S}_\lambda)=\chi\big[\textsf{ F}(\lambda)\big]\textsf{ S}_\lambda\,,\quad\forall\,\lambda\in\Lambda\,,\,\chi\in\widehat{\textsf{G}}\,.\]
Applying Theorem (2.2) one gets
Corollary 3.3 Let \(\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}\) be an admissible \(\textrm{ k}\)-graph and \(\textsf{ F}:\Lambda\to\textsf{G}\) a functor with values in an Abelian discrete group. Then \[\ell^1\big(\mathfrak C(\Lambda)\big):=\bigoplus_{g\in\textsf{G}}^{1,\alpha^{\textsf{F}}}\,\overline{\textrm{ span}}\big\{\textsf{ S}_\lambda\textsf{ S}^*_\mu\,{\big\vert\,\textsf{F}}(\lambda)\textsf{ F}(\mu)^{-1}=g\big\}\] is a reduced and symmetric Banach \(^*\)-algebra, which is inverse closed in \(\mathfrak C(\Lambda)\) .
Definition 3.5 On \(\mathbb{N}^\textsf{ k}\) one considers the order \(\mathfrak m\le\mathfrak n\,\Leftrightarrow\,\mathfrak m_i\le\mathfrak n_i\,,\forall\,i\) . Let \(\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}\) be an admissible \(\textrm{ k}\)-graph.
We say that \(\rho\) is a minimal common extension of \(\mu,\nu\in\Lambda\) , and we write \(\rho\in\textrm{ MCE}(\mu,\nu)\) , if \(\textrm{ l}(\rho)=\max\{\textrm{ l}(\mu),\textrm{ l}(\nu)\}\) and \(\rho=\mu\mu'=\nu\nu'\) for some \(\mu',\nu'\in\Lambda\) .
The \(\textsf{ k}\)-graph is called aperiodic if for every \(\mu,\nu\in\Lambda\) with \(\textrm{ s}(\mu)=\textrm{ s}(\nu)\) there exists \(\lambda\in\Lambda\) with \(\textrm{ r}(\lambda)=\textrm{ s}(\mu)\) and \(\textrm{ MCE}(\mu\lambda,\nu\lambda)=\emptyset\) .
The two notions are easier to visualize for \(\textsf{ k}=1\) . In this case, for instance, \(\textrm{ MCE}(\mu,\nu)\) is either void, or a singleton (one of the two elements) and aperiodicity boils down to condition (L).
Proposition 3.3 Let \(\Lambda\overset{\textrm{ l}}{\longrightarrow}\mathbb{N}^\textrm{ k}\) be an admissible and aperiodic \(\textrm{ k}\)-graph and \(\textsf{ F}:\Lambda\to{\textsf{G}}\) a functor with values in an Abelian discrete group. Let \(\{\textsf{ T}_\lambda\!\mid\!\lambda\in\Lambda\}\) be a Cuntz-Krieger \(\Lambda\)-family of partial isometries in the Hilbert space \(\mathcal{H}\) such that \(\textsf{ T}_v\ne 0\) for every \(v\in\Lambda^0\). Then (the quite obvious action \(\beta^\textsf{ F}\) is explained in the proof) \[\begin{equation} \bigoplus_{g\in{\textsf{G}}}^{1,\beta^{\textsf{ F}}}\,\overline{\textrm{ span}}\big\{\textsf{ T}_\lambda\textsf{ T}^*_{\mu}\,{\big\vert\,\textsf{ F}}(\lambda)\textsf{ F}(\mu)^{-1}\!=g\big\} \tag{3.9} \end{equation}\] is inverse closed in \(\mathbb B(\mathcal{H})\) .
Proof. Under the stated assumptions, "the Cuntz-Krieger unicity theorem" (see (Kumjian and Pask 2000; Si?) for example) states that the unique representation \(\pi_\textsf{ T}:\mathfrak C(\Lambda)\to\mathbb B(\mathcal{H})\) satisfying \[\pi_{\textsf{T}}(\textsf{ S}_{\lambda})=\textsf{ T}_{\lambda}\,,\quad\forall\lambda\in\Lambda\] (provided by the universal property) is injective. We denote by \(\mathfrak C(\textsf{ T})\) its range, so that \(\mathfrak C(\Lambda)\) and \(\mathfrak C(\textsf{ T})\) are isomorphic. Theorem (2.1) states then that \(\pi_\textsf{ T}\Big[\ell^1\big(\mathfrak C(\Lambda)\big)\Big]\) is inverse closed in \(\mathfrak C(\textsf{ T})\) and in \(\mathbb B(\mathcal{H})\) . There is an action \(\beta^\textsf{ F}\!:\widehat{\textsf{G}}\to\textrm{ Aut}\big[\mathfrak C(\textsf{ T})\big]\) such that \[\beta^\textsf{ F}_\chi(\textsf{ T}_\lambda)=\chi[\textsf{ F}(\lambda)]\textsf{ T}_\lambda\,,\quad\forall\,\chi\in\widehat{\textsf{G}}\,,\,\lambda\in\Lambda\] and \(\pi_T\) intertwines the two actions. It follows that \(\pi_\textsf{ T}\Big[\ell^1\big(\mathfrak C(\Lambda)\big)\Big]\) coincides with ((3.9)).