We prove that, if A and B are complete normed non-associative algebras, and if B is strongly semisimple and algebraic, then dense range homomorphisms from A to B are continuous. As a consequence, we obtain that homomorphisms from complete normed algebras into a complete normed power-associative algebraic algebra B are continuous if and only if B has no nonzero elements with zero square. We also...

We show that most of the theory of Hermitian Banach algebras can be proved for normed ∗-algebras without the assumption of completeness. The condition r(x) ≤ p(x) for all x (where p(x) = r(x∗x)1/2 is the Pták function), which is essential in the theory of Hermitian Banach algebras, is replaced for normed ∗-algebras by the condition r(x + y) ≤ p(x) + p(y) for all x, y. In case of Banach ∗-algebr...

A celebrated theorem of Hopf, Bott, Milnor, and Kervaire [11],[1],[12] states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified [6],[3],[9], the problem of classifying all 8-dimensional real quadratic division algebras is still open. We...

A celebrated theorem of Hopf, Bott, Milnor, and Kervaire [11],[1],[12] states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified [6],[3],[9], the problem of classifying all 8-dimensional real quadratic division algebras is still open. We...

In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to define Clifford algebras with scalars in arbitrary rings and provides new suggestions for an infinite-dimensional approach. Furthermore, I give a quick review...

Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of 'noncommutative topology', agreeing with some results of noncommutative geometry but lacking the metric aspects of C*-algebras. Here, we make such similarity stri...

Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of ‘noncommutative topology’, agreeing with some results of noncommutative geometry but lacking the metric aspects of C∗-algebras. Here, we make such similarity stri...

We show that the complexification (Ã, τ̃) of a real locally pseudoconvex (locally absorbingly pseudoconvex, locally multiplicatively pseudoconvex, and exponentially galbed) algebra (A,τ) is a complex locally pseudoconvex (resp., locally absorbingly pseudoconvex, locally multiplicatively pseudoconvex, and exponentially galbed) algebra and all elements in the complexification (Ã, τ̃) of a commutati...