Gauge-invariant uniqueness theorems for \(P\)-graphs (Q6561306)

``Uniqueness theorem'' means in this paper an answer to the following question: given a \(C^*\)-algebra \(A\), when is a homomorphism \(\pi:A\to B\) faithful? The ``gauge-invariant'' phrase comes from an equivariance condition: if \(A\) carries an action \(\Gamma \curvearrowright A\) of a group \(\Gamma\), we can ask for the existence of an action \(\Gamma \curvearrowright B\). In this paper, a version of the gauge-invariant uniqueness theorem is proved using maximal co-actions in the context of discrete groups, since that maximal co-actions provide a natural starting point to state and prove such uniqueness theorems.\N\NAn \textit{ordered group} is a pair \((G,P)\), where \(G\) is a group and \(P\) a submonoid such that \(P\cap P^{-1}=\{e\}\). There is a partial order on \(G\) defined by \(a \leq b\) if \(a^{-1}b\in P\). A \textit{weakly quasi-lattice ordered} (WQLO) group is an ordered group \((G,P)\) such that for all \(p,\,q\in P\), if \(\{p,q\}\) is bounded, then it has a least upper bound in \(P\), denoted by \(p\vee q\). In this paper, \(P\) will always refer to a WQLO group \((G,P)\). A \(P\)-graph is a countable category \(\Lambda\) together a functor \(d:\Lambda\to P\) such that for all \(\alpha\in\Lambda\) and \(p,\,q\in P\) such that \(d(\alpha)=pq\), there exist unique \(\beta,\,\gamma\in\Lambda\) such that \(\alpha=\beta\gamma\) and \(d(\beta)=p\) (and hence \(d(\gamma)=q\)). The elements of \(\Lambda\) are called \textit{paths} and the objects are identified with the identity morphisms, called \textit{vertices}. The set of all vertices is denoted by \(\Lambda^0\). We say that \(\Lambda\) is \textit{finitely aligned} if \(\bigvee S=\{\mu\in\bigcap\limits_{\gamma\in S}\gamma \Lambda:d(\mu)=\vee d(S)\}\) is finite for all finite \(S\subset\Lambda\).\N\NA \textit{representation} of a finitely aligned \(P\)-graph \(\Lambda\) in a \(C^*\)-algebra \(B\) is a map \(t:\Lambda\to B\) such that for all \(\alpha,\,\beta\in \Lambda\) we have that \(t_\alpha t_\beta=t_{\alpha\beta}\) whenever \(s(\alpha)=r(\beta)\); \(t^*_\alpha t_\alpha=t_{s(\alpha)}\); and \(t_\alpha t^*_\alpha t_\beta t^*_\beta=\sum\limits_{\gamma\in\alpha\vee\beta}t_\gamma t^*_\gamma\). If in addition for every \(\nu\in \Lambda^0\) and every finite exhaustive set \(E\subset \nu \Lambda\), \(t_\nu=\bigvee\limits_{\alpha\in E}t_\alpha t^*_\alpha\), the representation \(t\) is called \textit{tight}. A representation \(t\) of \(\Lambda\) is \(\Lambda\)-faithful if \(t_\alpha\neq 0\) for all \(\alpha\in\Lambda\). A \textit{gauge co-action} is a co-action \(\delta:C^*(t)\to C^*(t)\otimes C^*(G)\) satisfaying \(\delta(t_\alpha)=t_\alpha\otimes d(\alpha)\) for \(\alpha\in \Lambda\) and if such a \(\delta\) exists we say that \(t\) is gauge-compatible. If futhermore the co-action \(\delta\) is maximal, we say that \(t\) has \textit{maximal gauge co-action}.\N\NThe gauge-invariant uniqueness theorem that applies to all finitely aligned \(P\)-graphs is the following.\N\NTheorem. Let \(\Lambda \) be a finitely aligned \(P\)-graph and let \(t:\Lambda\to B\) a \(\Lambda\)-faithful tight representation with a maximal gauge co-action \(\varepsilon\). Then the associated \(^*\)-homomorphism \(\pi:\mathcal{O}(\Lambda)\to B\) is faithful, where \(\mathcal{O}(\Lambda)\) denotes the Cuntz-Krieger algebra.\N\N\NA representation \(t\) of \(\Lambda\) is co-universal if it is terminal in the category of \(\Lambda\)-faithful gauge-compatible representations, in which case \(C^*(t)\) is a co-universal \(C^*\)-algebra of \(\Lambda\). Finally, the authors prove the following result.\N\NLet \(\Lambda\) be a finitely aligned \(P\)-graph and let \(\delta\) be the canonical co-action on the Cuntz-Krieger algebra \(\mathcal{O}(\Lambda)\). A \(\Lambda\)-faithful gauge-compatible representation \(t\) of \(\Lambda\) is co-universal if and only if \(\delta_t\) is normal. In particular, the regular representation of \(\Lambda\) is co-universal and \(\mathcal{O}_r(\Lambda)\) is a co-universal \(C^*\)-algebra of \(\Lambda\).