Formally real involutions on central simple algebras

Abstract

An involution $#$ on an associative ring $R$ is \textit{formally real} if a sum of nonzero elements of the form $r^# r$ where $r \in R$ is nonzero. Suppose that $R$ is a central simple algebra (i.e. $R=M_n(D)$ for some integer $n$ and central division algebra $D$) and $#$ is an involution on $R$ of the form $r^# = a^{-1} r^\ast a$, where $\ast$ is some transpose involution on $R$ and $a$ is an invertible matrix such that $a^\ast=\pm a$. In section 1 we characterize formal reality of $#$ in terms of $a$ and $\ast|_D$. In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on $D=(K/F,\Phi)$ that extend to a formally real involution on the split algebra $D \otimes_F K \cong M_n(K)$. Every such involution is formally real but we show that there exist formally real involutions on $D$ which are not of this form. In particular, there exists a formally real involution $#$ for which the hermitian trace form $x \mapsto \tr(x^#x)$ is not positive semidefinite.