[Paper Review] A non-commutative, analytic version of Hilbert's 17-th problem in type II$_1$ von Neumann algebras

Florin Rădulescu|ArXiv.org|Apr 26, 2004

Advanced Operator Algebra Research6 references10 citations

TL;DR

This paper establishes a non-commutative, analytic analogue of Hilbert's 17th problem in type II₁ von Neumann algebras, proving that a symmetric analytic non-commutative series with non-negative trace under all representations in type II₁ factors is cyclically equivalent to a weak limit of sums of squares. The key result links this to the Connes embedding conjecture, showing it is equivalent to a matrix trace inequality condition on such series.

ABSTRACT

We prove a non-commutative version of the Hilbert's 17th problem, giving a characterization of the class of non-commutative polynomials in n-undeterminates that have positive trace when evaluated in n-selfadjoint elements in arbitrary II1 von Neumann algebra. As a corollary we prove that Connes's embedding conjecture is equivalent to a statement that can be formulated entirely in the context of finite matrices.

Motivation & Objective

• To establish a non-commutative, analytic version of Hilbert’s 17th problem in the context of type II₁ von Neumann algebras.
• To characterize symmetric analytic non-commutative series that yield non-negative traces under all representations in type II₁ factors.
• To show that such series are cyclically equivalent to weak limits of sums of squares in the non-commutative analytic algebra.
• To prove the Connes embedding conjecture is equivalent to a matrix trace inequality condition on symmetric analytic series.

Proposed method

• Define the algebra of analytic non-commutative series $ \mathbb{C}_{\rm an}[Y_1,\ldots,Y_n] $ as formal series with coefficients decaying rapidly enough to ensure convergence for all radii.
• Introduce a canonical involution extending $ (a_I Y_I)^* = \overline{a_I} Y_{I^{\rm op}} $, and define the real subspace $ \mathbb{C}_{\rm an}^{\rm sym}[Y_1,\ldots,Y_n] $ of auto-adjoint series.
• Define cyclic equivalence as difference being a weak limit of scalar multiples of $ Y_I - Y_{\tilde{I}} $, where $ \tilde{I} $ is a cyclic permutation of $ I $.
• Define a series to be a sum of squares if it is in the weak closure of finite sums $ \sum_s b_s^* b_s $ with $ b_s \in \mathbb{C}_{\rm an}[Y_1,\ldots,Y_n] $.
• Use the GNS construction to associate a cyclic vector $ Z_\Phi $ to a moment sequence $ (\theta_I)_{I \in \mathcal{I}_n} $, and show that cyclic symmetry of the moments implies unitary invariance of the state.
• Prove that if a symmetric analytic series $ p $ satisfies $ \tau(p(X_1,\ldots,X_n)) \geq 0 $ for all self-adjoint $ X_i $ in any type II₁ factor $ M $, then $ p $ is cyclically equivalent to a weak limit of sums of squares.

Experimental results

Research questions

• RQ1 Under what conditions is a symmetric analytic non-commutative series $ p \in \mathbb{C}_{\rm an}^{\rm sym}[Y_1,\ldots,Y_n] $ guaranteed to be cyclically equivalent to a weak limit of sums of squares?
• RQ2 How does the trace positivity of $ p(X_1,\ldots,X_n) $ in all type II₁ von Neumann algebras relate to its algebraic structure?
• RQ3 Can the Connes embedding conjecture be reformulated purely in terms of matrix trace inequalities for analytic non-commutative polynomials?
• RQ4 What is the precise relationship between the moment sequences of type II₁ factors and those of finite matrix algebras?

Key findings

• A symmetric analytic non-commutative series $ p $ is cyclically equivalent to a weak limit of sums of squares if $ \tau(p(X_1,\ldots,X_n)) \geq 0 $ for all self-adjoint $ X_i $ in any type II₁ von Neumann algebra $ M $ with faithful trace $ \tau $.
• The Connes embedding conjecture holds if and only if every symmetric analytic series $ p \in \mathbb{C}_{\rm an}^{\rm sym}[Y_1,\ldots,Y_n] $ with $ \mathop{\rm tr}(p(X_1,\ldots,X_n)) \geq 0 $ for all self-adjoint matrices $ X_i \in M_N(\mathbb{C}) $, for all $ N $, is cyclically equivalent to a weak limit of sums of squares.
• The moment sequence $ (\theta_I)_{I \in \mathcal{I}_n} $ of a type II₁ factor arises as the weak limit of moment sequences from finite matrix algebras if and only if the sequence is cyclically symmetric and satisfies a uniform growth condition.
• The GNS construction applied to a cyclically symmetric, rapidly decaying moment sequence yields a type II₁ von Neumann algebra with a faithful trace, and the associated vector is cyclic and separating.
• The set of moment sequences from finite matrix algebras is weakly dense in the set of moment sequences from type II₁ factors if and only if the Connes embedding conjecture holds.