# Noncommutative Positivstellensätze for pairs representation-vector

@article{Cimpric2010NoncommutativePF, title={Noncommutative Positivstellens{\"a}tze for pairs representation-vector}, author={J. Cimpric}, journal={Positivity}, year={2010}, volume={15}, pages={481-495} }

We study non-commutative real algebraic geometry for a unital associative *-algebra $${\mathcal {A}}$$ viewing the points as pairs (π, v) where π is an unbounded *-representation of $${\mathcal A}$$ on an inner product space which contains the vector v. We first consider the *-algebras of matrices of usual and free real multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general $${\mathcal {A}}$$. Finally, we compare our… Expand

#### 4 Citations

Quasi-convex free polynomials

- Mathematics
- 2012

Let $\Rx$ denote the ring of polynomials in $g$ freely non-commuting variables $x=(x_1,...,x_g)$. There is a natural involution * on $\Rx$ determined by $x_j^*=x_j$ and $(pq)^*=q^* p^*$ and a free… Expand

Finsler's Lemma for matrix polynomials

- Mathematics
- 2015

Abstract Finsler's Lemma characterizes all pairs of symmetric n × n real matrices A and B which satisfy the property that v T A v > 0 for every nonzero v ∈ R n such that v T B v = 0 . We extend this… Expand

Non-commutative polynomials with convex level slices

- Mathematics
- 2015

Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the… Expand

A Real Nullstellensatz for free modules

- Mathematics
- 2013

Abstract Let A be the algebra of all n × n matrices with entries from R [ x 1 , … , x d ] and let G 1 , … , G m , F ∈ A . We will show that F ( a ) v = 0 for every a ∈ R d and v ∈ R n such that G i (… Expand

#### References

SHOWING 1-10 OF 21 REFERENCES

A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

- Mathematics
- Canadian Mathematical Bulletin
- 2009

Abstract We present a new approach to noncommutative real algebraic geometry based on the representation theory of ${{C}^{*}}$ -algebras. An important result in commutative real algebraic geometry is… Expand

A non-commutative real Nullstellensatz and Hilbert's 17th problem

- Mathematics
- 1976

1. Generalities Our object of study will be algebras with involution, or *-algebras as we will often call them. All our rings will be algebras over a field F of characteristic # 2; F is endowed with… Expand

A non-commutative Positivstellensatz on isometries

- Mathematics
- 2004

A symmetric non-commutative polynomial p when evalu- ated at a tuple of operators on a flnite dimensional, real Hilbert space H has a value which is a symmetric operator. We show that any such… Expand

A positivstellensatz for non-commutative polynomials

- Mathematics
- 2004

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the… Expand

Maximal Quadratic Modules on ∗-rings

- Mathematics
- 2007

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to ∗-rings and discuss the relation of this generalization to recent developments in… Expand

AN ELEMENTARY AND CONSTRUCTIVE SOLUTION TO HILBERT'S 17TH PROBLEM FOR MATRICES

- Mathematics
- 2006

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let A be an n x n symmetric matrix with… Expand

Pure states, positive matrix polynomials and sums of hermitian squares

- Mathematics
- 2010

Let M be an archimedean quadratic module of real t × t matrix polynomials in n variables, and let S ⊆ ℝ n be the set of all points where each element of M is positive semidefinite. Our key finding is… Expand

“Positive” noncommutative polynomials are sums of squares

- Mathematics
- 2002

Hilbert's 17th problem concerns expression of polynomials on R n as a sum of squares. It is well known that many positive polynomials are not sums of squares; see [Re], [D'A] for excellent surveys.… Expand

Rings with involution as partially ordered abelian groups

- Mathematics
- 1981

Let (S, *) be a ring with involution *. The involution is positive definite if, for all finite subsets {r,-} of S, £ w * = 0 implies all the r{ are zero. Then the set of self-adjoint elements of S,… Expand

Unbounded Operator Algebras and Representation Theory

- Mathematics
- 1990

1. Preliminaries.- I. O*-Algebras and Topologies.- 2. O-Families and Their Graph Topologies.- 3. Spaces of Linear Mappings Associated with O-Families and Their Topologization.- 4. Topologies for… Expand