Formally real involutions on central simple algebras

TL;DR

This paper characterizes when involutions on central simple algebras are formally real, linking this property to algebraic conditions on matrices and involutions, and explores implications for crossed product division algebras.

Contribution

It provides a characterization of formal reality of involutions on central simple algebras and applies this to crossed product division algebras, revealing new examples.

Findings

Characterization of formal reality in terms of matrix and involution properties

Existence of formally real involutions not extending from subalgebras

Counterexamples where hermitian trace form is not positive semidefinite

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.