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.)

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