Igor Klep (Universities of Maribor and Ljubljana)

will speak on

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

Time: 4:00PM
Date: Mon 14th June 2010
Location: Mathematical Sciences Seminar Room [map]

Abstract: Consider a central simple algebra $A$ with involution $*$.
The involution is called emph{positive} if the involution trace
form $xmapsto r(x^*x)$ 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 $xmapsto r(x^*bx)$ 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 $d imes d$ 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(A_1,ldots,A_n))geq0$ for all $A_iin M_2(R)$;
item
there exist a nonvanishing central polynomial $c$,
and a polynomial identity $h$ of
$2 imes 2$ 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 and Number Theory series.)

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