On invariant convex subsets in algebras defined on a locally compact group \(G\) (Q2859343)

Let \(G\) be a locally compact group with left invariant Haar measure and let \(f\) be a function on \(G\). If \(x \in G\), then \(L_x f\) denotes the left translate of \(f\), that is \(f_x(y) = f(x^{-1}y)\) for \(y \in G\). In this paper translation invariant closed subspaces of various algebras are investigated. We will now describe some of these algebras and corresponding results.NEWLINENEWLINESet \(P^1(G) = \{ \phi \in L^1(G): \phi \geq 0, \| \phi \| _1 =1 \}\). A subset \(\mathcal{X}\) of \(L^{\infty}(G)\) is said to be topologically invariant if \(\phi \ast f \in \mathcal{X}\) for all \(\phi \in P^1(G)\) and \(f \in \mathcal{X}\), where \(\ast\) denotes the usual convolution of functions. Also, \(\mathcal{X}\) is said to be left translation invariant if \(L_x f \in \mathcal{X}\) whenever \(f \in \mathcal{X}\) and \(x \in G\). The space \(L^{\infty}(G)\) can be embedded in \(\mathcal{B}(L^1(G), L^{\infty}(G))\) by the linear map \(T\) given by \(T(f)(\phi) = \phi \ast f\) where \(f \in L^{\infty}(G)\) and \(\phi \in L^1(G)\). Denote by \(\tau_c\) the topology on \(L^{\infty}(G)\) induced from the strong operator topology on \(\mathcal{B}(L^1(G), L^{\infty}(G))\). The \(\tau_c\)-topology is not weaker than the \(\text{weak}^{\ast}\) topology and not stronger than the norm topology on \(L^{\infty}(G)\). The author proves that if \(G\) is compact and if \(\mathcal{X}\) is a \(\tau_c\)-closed convex subset of \(L^{\infty}(G)\), then \(\mathcal{X}\) is left translation invariant if and only if it is topologically invariant. Using this result, it can be shown that if \(G\) is abelian, in addition to being compact, then a \(\tau_c\)-closed subspace of \(L^{\infty}(G)\) is left translation invariant if and only if it is introverted.NEWLINENEWLINELet us return to the case where \(G\) is only assumed to be locally compact. Let \(C_b(G)\) be the Banach algebra of continuous, bounded, complex-valued functions on \(G\) with the sup norm. Let \(LUC(G)\) denote the closed subspace of \(C_b(G)\) that consists of bounded left uniformly continuous functions on \(G\). A result is given that shows a \(\text{weak}^{\ast}\) closed subspace \(\mathcal{I}\) of \(LUC(G)^{\ast}\) is a left ideal precisely when it is invariant under \(L_x^{\ast\ast}\) for each \(x \in G\).NEWLINENEWLINEA function \(f \in C_b(G)\) is said to be weakly almost periodic if \(\{ L_x f : x \in G\}\) is relatively weakly compact in \(C_b(G)\). The set of all weakly almost periodic functions on \(G\) is denoted by \(\mathrm{Wap }(G)\). The corresponding result for this space is that a closed convex subset \(\mathcal{I}\) of Wap \((G)\) is left translation invariant if and only if it is topologically invariant.NEWLINENEWLINEThe paper concludes with some results for the case when \(G\) is amenable. In particular, 1-dimensional left ideals exist in \(LUC(G)^{\ast} (L^1(G)^{\ast \ast})\) precisely when \(G\) is amenable.