Let \(A\) be an associative complex algebra. By \(A^*\) we denote its algebraic dual and \(A'\subseteq A^*\) will denote a total subspace (i.e., separates the points of \(A\)). Whenever \(A\) is endowed with a Hausdorff locally convex topology, \(A'\) will be the topological dual of \(A\). The weak topology defined on \(A\) by \(A'\) will be denoted by \(\sigma(A,A')\). An element \(a\in A\) is said to be left quasi-invertible if there exists some \(b\in A\) such that \(ba-a-b=0\). The Jacobson radical of \(A\) is \(\mathrm{Rad}(A)=\{ a\in A:\lambda a+ax \text{ is left quasi-invertible for all } \lambda\in {\mathbb C},\, x\in A\}\). the algebra \(A\) is radical if \(\mathrm{Rad}(A)=A\).
Theorems 4 and 5 are the main results of the paper. The former says that a Banach algebra \(A\) is either finite-dimensional or has uncountably many closed ideals of finite codimension if the multiplication is continuous with respect to the \(\sigma(A, A')\) topology. Theorem 5 reads as follows. Let \(A\) be a radical algebra (without any topology) and let \(A'\) be a total subspace of \(A^*\). Then \((A, \sigma(A, A'))\) has continuous multiplication if, and only if, up to a topological isomorphism, it is a subalgebra of a product of finite-dimensional radical algebras.