Formally Real Involutions on Central Simple Algebras |
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Authors: | J Cimprič |
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Institution: | 1. Department of Mathematics , University of Ljubljana , Ljubljana, Slovenija cimpric@fmf.uni-lj.si |
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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. |
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Keywords: | Central simple algebras Crossed products Involutions Sums of hermitian squares |
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