On topologizable algebras
Let A be a linear associative algebra. By [3] it is always possible to define a topology on A which makes of A a locally convex algebra with separately continuous multiplication (i.e. xα, x, y ∈ A, xα → x implies xαy → xy, yxα → yx). On the other hand (cf. [3]) in general it is not possible to introduce a topology on A which makes of A a locally convex algebra with jointly continuous multiplication (i.e. xα → x, yβ → y ⇒ xαyβ → xy). The aim of this note is to exhibit two examples which continue these investigations. In the first example we construct a commutative algebra which admits no topology. This gives a negative answer to the question raised in [2]. In the second example we construct a topological algebra which admits no locally convex topology. All algebras in this paper will be complex (this condition, however, is not essential). We say that an algebra A is topologizable (topologizable as a locally convex algebra) if there exists a topology on A which makes of A a topological (locally convex) algebra with jointly continuous multiplication. It is easy to see that an algebra A is topologizable if and only if there exists a system V of subset of A (zero-neighbourhoods in A) satisfying (1) ⋂