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Noncommutative polynomial
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In mathematics, a noncommutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. Every noncommutative polynomial can be written as a sum of monomials in which no two monomials contain the same two products of variables. Noncommuting polynomials on matrix algebras is of interest in mathematics mainly due to showing that Connes' embedding conjecture is equivalent to an algebraic assertion involving the trace of polynomial values on matrices. This has motivated the consideration of linear span values of a noncommutative polynomial on the matrix algebra. For example, a class of algebras Q can be found such that it satisfies the following property: for every nontrivial noncommutative polynomial f(Y<sub>1</sub>, ... ,Y<sub>n</sub>), the linear span of all its values f(c<sub>1</sub>, ... ,c<sub>n</sub>), c<sub>i</sub> ∈ Q, equals Q. This class includes the algebras of all bounded and compact operators on an infinite dimensional Hilbert space. See Also
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