Gel'fand transform of locally convex algebras (Q1061354)

In what follows A is a compact LCA (locally convex algebra - understood here as a locally convex space together with an associative separately continuous multiplication) with unit element. Let SpA be the set of all continuous non-zero multiplicative-linear functionals on A equipped with the \(\sigma\) (A',A) topology, and the algebra C(SpA) is equipped with the topology of uniform convergence on compact subsets of SpA. The author proves that the Gelfand map \(A\to C(SpA)\) is continuous provided A is a Mackey space (Proposition 1); that every closed maximal ideal in A is a kernel of a functional in SpA, provided A is commutative and the operation \(x\to x^{-1}\) is \(\sigma\) (A,A')-continuous (what does not happen too often, even in the case of a Banach algebra - referee's remark) (Proposition 2); that \(A={\mathbb{C}}\), provided A is a division algebra and the operation \(x\to x^{-1}\) is weakly sequentially continuous (Proposition 3); that for the projective tensor product A\({\hat \otimes}B\) of two complete commutative LCA (here the multiplication is assumed to be jointly continuous) it is Sp(A\({\hat \otimes}B)=SpA\times SpB\) (Proposition 4).