Formally Real Involutions on Central Simple Algebras

Abstract

An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r ^# r where r ∊ 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∗ a, where ∗ is some transpose involution on R and a is an invertible matrix such that a∗ = ±a. In Section 1 we characterize formal reality of # in terms of a and ∗| _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, Φ) that extend to a formally real involution on the split algebra D ⊗ _F K ≅ 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 ↦ tr(x ^# x) is not positive semidefinite.

Key Words:

• Central simple algebras
• Crossed products
• Involutions
• Sums of hermitian squares

2000 Mathematics Subject Classification:

• 16K20
• 16W10
• 12D15

ACKNOWLEDGMENTS

I would like to thank Prof. Tom Craven for sharing with me his experience. He proved an early version of Theorem 1 (with ∊ = 1 and ∗ = id) and read later versions of the manuscript. I would also like to thank to Dr. Thomas Unger for his comments on Section 5.

Notes

Communicated by A. Prestel.