Topologically nilpotent normed algebras

We introduce in a nonassociative setting the notions of spectral radius of a bounded subset of a normed algebra, as well as that of topologically nilpotent normed algebra. We generalize and refine most known results on topologically nilpotent associative algebras to the nonassociative context, and prove some new results both in the associative and nonassociative setting. Among them, we emphasize the one asserting that an associative normed algebra A is topologically nilpotent if and only if so is the normed Jordan algebra obtained by symmetrizing the product of A, as well as the one asserting that, if A is a topologically nilpotent complete normed algebra, then its full multiplication algebra FM(A) is a radical algebra (equivalently, A coincides with its weak radical).