On Division Algebras of Degree 3 with Involution

Generic Algebras with Involution of Degree 8m

The centers of the generic central simple algebras with involution are interesting objects in the theory of central simple algebras. These fields also arise as invariant fields for linear actions of projective orthogonal or symplectic groups. In this paper, we prove that when the characteristic is not 2, these fields are retract rational, in the case the degree is 8m and m is odd. We achieve th...

Division Algebras with an Anti-automorphism but with No Involution

In this note we give examples of division rings which posses an anti-automorphism but no involution. The motivation for such examples comes from geometry. If D is a division ring and V a finite-dimensional right D-vector space of dimension ≥ 3, then the projective geometry P(V ) has a duality (resp. polarity) if and only if D has an anti-automorphism (resp. involution) [2, p. 97, p. 111]. Thus,...

GENERIC ALGEBRAS WITH INVOLUTION OF DEGREE 8 mDavid

Algebras of Degree Eight 265 on Primary Normal Division Algebras

Normed algebras with involution

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...