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# On Valuations, Places and Graded Rings Associated to $*$-Orderings

Published:2007-03-01
Printed: Mar 2007
• Igor Klep
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## Abstract

We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.
 Keywords: $*$--orderings, valuations, rings with involution
 MSC Classifications: 14P10 - Semialgebraic sets and related spaces 16S30 - Universal enveloping algebras of Lie algebras [See mainly 17B35] 16W10 - Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx]