# UCD School of Mathematics and Statistics Scoil na Matamaitice agus na Staitisticí UCD

Central Simple Algebras, the Procesi-Schacher Conjecture, and Positive Polynomials

Speaker: Igor Klep (Universities of Maribor and Ljubljana)

Time: 4:00PM
Date: Mon 14th June 2010

Location: Mathematical Sciences Seminar Room

Abstract
Consider a central simple algebra A with involution ∗. The involution is called emph{positive} if the involution trace form xmapstor(x∗x) is positive semidefinite (w.r.t.~a fixed ordering of the center F of A). A symmetric element b is defined to be emph{positive} if the scaled involution trace form xmapstor(x∗bx) is positive semidefinite, giving rise to an emph{ordering} of the central simple algebra A. We discuss how these can be used to give a Positivstellensatz characterizing polynomials in noncommuting variables that are positive semidefinite or trace-positive on dimesd matrices. Along the way we give a counterexample to a conjecture of Procesi and Schacher.
Here is a sample result:

egin{theorem}
For a real polynomial f in n free noncommuting variables, the following are equivalent:
egin{enumerate}[ m (i)]
item
r(f(A1,ldots,An))geq0 for all AiinM2(R);
item
there exist a nonvanishing central polynomial c, and a polynomial identity h of 2imes2 matrices, such that
[
c f c^* in h + os.]
end{enumerate}
Here os denotes the set of all polynomials that can be written as sums of hermitian squares g∗g and commutators pq−qp.
end{theorem}

We shall also explain how this statement fails for d>2, and how this fact pertains to the Procesi-Schacher conjecture.

igskip
The talk is partially based on joint work with Thomas Unger.

(This talk is part of the Algebra/Claude Shannon Institute series.)