Transformation groups of automorphisms of \(C(X)\) (Q2528103)

Let \(X\) be a completely regular \(T_1\)-space, and let \(C(X)\) be the ring of continuous, real-valued functions on \(X\). Give \(C(X)\) the compact-open topology; then it becomes a topological ring. If \((X,T,\pi)\) is a transformation group, where \(T\) is a locally compact topological group and \(X\) satisfies the conditions described above, define the induced transformation group \(\pi^*\colon C(X)\times T\to C(X)\) by \(\pi^*(f,t)(x)=f(xt^{-1})\). Then \((C(X),T,\pi^*\)) is a transformation group of ring isomorphisms of \(C(X)\). The main theorem in this article asserts that when \(X\) is locally compact, the above can be reversed. More precisely, Theorem: If \((C(X),T,\varphi)\) is a transformation group of ring isomorphisms on \(C(X)\), and if \(X\) is locally compact and satisfies the above conditions, then there exists a transformation group \((K,T,\pi)\) such that the induced transformation group \((C(X),T,\pi^*)\) is \((C(X),T,\varphi)\). Once this has been established, it is routine to carry over many classical dynamical properties of \((X,T,\pi)\) to \((C(X),T,\pi^*)\). These properties take the form of purely algebraic statements about maximal ideals in \(C(X)\). Some indications are given below. Associated with any ideal in \(C(X)\) there is a unique ideal \(m(J)\) defined by \(m(J) = \{f\in C(X): \) there exists a \(g\in \ni fg=f\}\). Also, let \(M_p = \{f\in C(X): f(p) =0\}\); every maximal ideal in \(C(X)\) is an \(M_p\) when the conditions described above are imposed on \(X\). \([f]\) denotes the principal ideal generated by \(f\). Finally, if \(Q\) is an ideal in \(C(X)\) and \(S\) is any subset of \(T\), \(J(Q;S) = \displaystyle\cap_{t\in S} \pi_t^*(Q)\). As a sample, consider the concept of periodicity. A maximal ideal \(M_p\) is periodic under \(T\) if and only if there exists a compact set \(K\subset T\) such that \(J(M_p;K)\subset J(M_p;T)\). Theorem. \(M_p\) is periodic under \(T\) if and only if \(p\) is periodic under \(T\). More generally, Definition: Let \(\mathcal A\) be a class of subsets of \(T\), called admissible sets. The maximal ideal \(M_p\) is \(\mathcal A\)-recursive under \(T\) provided that for each \(f\in m(M_p)\) there is an admissible set \(A\in\mathcal A\) such that \([f]\subset J(M_p;A)\). Theorem. \(M_p\) is recursive under \(T\) if and only if \(p\) is recursive under \(T\). It is then clear how to proceed in general. In order to define all of the classical recursive properties for maximal ideals \(M_p\) in \(C(X)\), simply replace the term admissible set by an appropriate phrase. In a slightly different direction, the following theorem is established. Theorem: A necessary and sufficient condition that a transformation group \((X,T,\pi)\) be minimal is that the only ideals in \(C(X)\) which are invariant under \(\mathcal G = \{\pi_t^*: t\in T\}\) are the ideals \(\{0\}\) and \(C(X)\).