The uniqueness of the complete norm topology in complete normed nonassociative algebras (Q1081812)

A subalgebra C of an associative algebra B is said to be a full subalgebra of B if C contains the quasi-inverses of the elements of C that are quasi-invertible in B. If A is a nonassociative algebra, the full subalgebra of L(A) (the associative algebra of all linear operators on A) generated by the operators of left and right multiplication by elements of A is called the full multiplication algebra of A and denoted by FM(A). The maximal FM(A)-invariant subspace of A contained in \(\{a\in A:L_ a,R_ a\in Rad(FM(A))\}\) (where Rad stands for the Jacobson radical and \(L_ a(x)=ax\), \(R_ a(x)=xa\) for x in A) is called the weak radical of A. The main result of this paper states that every complete normed nonassociative algebra with zero weak radical has a unique complete algebra norm topology. This theorem contains the well-known one by Johnson for associative Banach algebras and the recent analogous result by Aupetit for Banach-Jordan algebras. Also it has nontrivial applications to some Banach-Lie algebras.