The primitive spectrum of a semigroup of Markov operators (Q2174930)

Let \(K\) be a compact space and \(C(K)\) the Banach lattice of continuous complex-valued functions on \(K\). The space of linear operator on \(C(K)\) is denoted by \(\mathcal{L}(C(K))\). A positive operator \(T \in \mathcal{L}(C(K))\) is a Markov operator if \(T\mathbb{I}=\mathbb{I}\), where \(\mathbb{I}(x)=1\) for all \(x \in K\). Let \(\mathcal{S} \subset \mathcal{L}(C(K))\) be a semigroup of Markov operator and \(I \subset C(K)\) closed proper ideal. \(I\) is said to be an \(\mathcal{S}\)-ideal if it is \(\mathcal{S}\)-invariant, i.e., \(SI \subset I\) for each \(S\in \mathcal{S}\). It is called maximal if it is maximal among all \(\mathcal{S}\)-ideals with respect to inclusion. It is well known that, for each \(I\) \(\mathcal{S}\)-ideal, there is a subset \(L \subset K\) such that \(I=\left\{f \in C(K)\mid f_{|L}=0\right\}.\) The set of of invariant probability measures on \(K\) is denoted by \(P_{\mathcal{S}}(K)\). It turns out that, for each \(\mu \in P_{\mathcal{S}}(K)\), the space \(L^1(K,\mu)\) is the completion of the quotient \(C(K)/I_{\mu}\), where \(I_{\mu}=\big\{f \in C(K)\mid\int|f(x)|\, d\mu(x)=0\big\}.\) Therefore, every \(S \in \mathcal{S}\) induces an operator \(S_{\mu}\) on \(L^1(K,\mu)\). We thus obtain a semigroup \(\mathcal{S}_{\mu}=\big\{S_{\mu}\mid S \in \mathcal{S}\}\) of Markov operators induced by \( \mathcal{S}\). We denote by \(\operatorname{fix}( \mathcal{S}_{\mu})\) the space of invariant functions under the action of each \(S_{\mu} \in \mathcal{S}_{\mu}\). \(\mu\) is said to be ergodic if \(\operatorname{fix}( \mathcal{S}_{\mu})\) is one-dimensional. An \(\mathcal{S}\)-ideal \(p\) is called primitive if there is an ergodic measure \(\mu \in P_{\mathcal{S}}(K)\) such that \(p = I_{\mu}\). The set of all primitive \(\mathcal{S}\)-ideals is called the primitive spectrum of \(\mathcal{S}\), denoted by \(\operatorname{Prim}(\mathcal{S}).\) For subsets \(A \subset \operatorname{Prim}(\mathcal{S})\) and \(I \subset C(K )\), we set \[ \operatorname{ker}(A) :=\bigcap_{p \in A}p,\qquad \operatorname{hull}(I ):= \big\{ p \in \operatorname{Prim}(\mathcal{S}) \mid I\subset p\big\}. \] For a subset \(I \subset C(K )\), the \(\mathcal{S}\)-radical of \(I\) is given by \[ \operatorname{rad}_{\mathcal{S}}=\operatorname{ker}(\operatorname{hull}(I)). \] The semigroup \(\mathcal{S}\) is radical free if the zero ideal is a radical \(\mathcal{S}\)-ideal. In this setting, the author proves a Dauns-Hofmann-type theorem showing that, if \(\mathcal{S}\) is radical free, then the space of continuous functions on the primitive spectrum \(C(\operatorname{Prim}(\mathcal{S}))\) is canonically isomorphic to the fixed space \(\operatorname{fix}(\mathcal{S})\) of the semigroup \(\mathcal{S}\). He further states some application to ergodic theory.