Bounded degree of algebraic topological algebras

Relatively Weakly Open Sets in Closed Balls of $C^*$-Algebras

“…It would be interesting to find a proof of Theorem 2.5 that did not involve the theory of JB * -triples and the nonassociative techniques in [7] or [8]. For some of the consequences of Theorem 2.5 suggested above, we are in fact provided with such an autonomous proof.…”

“…Then U ∩ A x is a nonempty relatively w-open subset of B A x with diam(U ∩ A x ) < 2, so that, by Lemma 2.2, Ω x is finite, and hence A x is finite-dimensional. Since x is arbitrary in U, and U has nonempty n-interior in A, it follows from either [7,Theorem 1] or [8,Theorem D] that in fact A x is finite-dimensional for every x in A. Now, for every x in A, the set σ(x) := {λ ∈ R : x − λ1 is not invertible in the unital hull of A x } is finite.…”

“…Because of the conjugate-linear behaviour of triple products of JB * -triples in the middle variable, and the essentially linear nature of the arguments in [7,8], the application of such arguments in our setting needs a formal consideration of 'real algebraic' elements, and certainty that 'real algebraic' elements and (complex) algebraic elements coincide, thanks to the fact that C is finite-dimensional over R. …”

Linearly compact algebraic Lie algebras and coalgebraic Lie coalgebras

“…From the point of view that some recent results provide this fact can be explained as follows. Endowed with the weak linear topology the dual algebra A becomes a topological linearly compact algebra [9,10] and since linearly compact algebras are Baire algebras the result of the type "algebraic implies algebraic of bounded degree" is true for them even in alternative or Jordan case [4].…”