THE PROCESI–SCHACHER CONJECTURE AND HILBERT’S 17TH PROBLEM FOR ALGEBRAS WITH INVOLUTION

IGOR KLEP AND THOMAS UNGER

Abstract. In 1976 Procesi and Schacher developed an Artin–Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.

Dedicated to David W. Lewis on the occasion of his 65th birthday.

1. Introduction

Artin’s 1927 affirmative solution of Hilbert’s 17th problem (Is every nonnegative real polynomial a sum of squares of rational functions?) arguably sparked the beginning of the field of real algebra and consequently real algebraic geometry (cf. [BCR, PD]).

Starting with Helton’s seminal paper [Hel], in which he proved that every positive semidefinite real or complex noncommutative polynomial is a sum of hermitian squares of polynomials, variants of Hilbert’s 17th problem in a noncommutative setting have become a topic of current interest with wide-ranging applications (e.g. in control theory, optimization, engineering, mathematical physics, etc.); see [dOHMP] for a nice survey. Most of these results have a functional analytic flavour and are what Helton et al. call dimensionfree, that is, they deal with evaluations of noncommutative polynomials in matrix algebras of arbitrarily large size.

Procesi and Schacher in their 1976 Annals of Mathematics paper [PS] introduce a notion of orderings on central simple algebras with involution, prove a real Nullstellensatz, and a weak noncommutative version of Hilbert’s 17th problem. A strengthening of the latter is proposed as a conjecture [PS, p. 404]: In a central simple algebra with involution, a totally positive element is always a sum of hermitian squares.

We explain in Section 5 how these results can be applied to study non-dimensionfree positivity of noncommutative polynomials. Roughly speaking, a noncommutative polynomial all of whose evaluations in n × n matrices (for fixed n) are positive semidefinite, is a sum of hermitian squares with denominators and weights.

Date: 18 October 2008.

2000 Mathematics Subject Classification. Primary 11E25, 13J30; Secondary 16W10, 16R50.

Key words and phrases. central simple algebra, involution, quadratic form, ordering, trace, noncommutative polynomial.

A brief outline of the rest of the paper is as follows: in Section 2 we fix terminology and summarize some of the Procesi–Schacher results in a modern language. Then in Section 3 we present counterexamples to the Procesi–Schacher conjecture, while Section 4 contains a study of examples (mainly in the split case) where the conjecture is true.

For general background on central simple algebras with involution we refer the reader to [KMRT] and for the theory of quadratic forms over fields we refer to [Lam].

2. The Procesi–Schacher Conjecture

Let F be a formally real field and let A be a central simple algebra with involution σ and centre K. Assume that F is the fixed field of σ (i.e., σ|_{F} = id_{F}). The involution σ is of the first kind if K = F, and of the second kind (also called unitary) otherwise. In this case [K : F] = 2 and σ|_{K} is the non-trivial element in Gal(K/F).

Let D be a division algebra over K with involution τ and fixed field F. Let h be an n-dimensional hermitian or skew-hermitian form over (D, τ). Then h gives rise to an involution on M_{n}(D), the adjoint involution ad_{h}, defined by

ad_{h}(X) = H · τ(X)^{t} · H^{−1},

for all X ∈ M_{n}(D), where H is the Gram matrix of h, t denotes the transpose map on M_{n}(D) and τ(X) signifies applying τ to the entries of X. It is well-known that every central simple algebra with involution (A, σ) is of the form (M_{n}(D), ad_{h}), where n is unique, D is unique up to isomorphism and h is unique up to multiplicative equivalence (see [KMRT, 4.A]).

If σ is of the first kind, then σ is called orthogonal or symplectic if σ becomes adjoint to a quadratic or alternating form, respectively, after scalar extension to a splitting field of A (i.e., an extension field L of K such that A ⊗_{K} L M_{n}(L)). We denote the subspace of σ-symmetric elements of A by Sym(A, σ).

Let ≤ be an ordering on F. We identify ≤ with its positive cone P = {x ∈ F | 0 ≤ x} via

x ≤ y ⇔ y − x ∈ P

for all x, y ∈ F. In this case we also write ≤_{P} instead of ≤.

Procesi and Schacher [PS, §1] consider central simple algebras A, equipped with a positive involution σ, i.e., an involution whose involution trace form T_{σ} is positive semidefinite with respect to the ordering ≤_{P} on F,

T_{σ}(x) := Trd(σ(x)x) ≥_{P} 0 for all x ∈ A.

Here Trd : A → F (the trace) denotes the reduced trace Trd_{A/F} if σ is of the first kind and the composition Trd_{K/F}∘ Trd_{A/K}if σ is of the second kind. The form T_{σ} is a nonsingular quadratic form over F, cf. [KMRT, §11]. If dim_{K}A = n, then dimT_{σ} = n if σ is of the first kind and dimT_{σ} = 2n if σ is of the second kind.

Remark 2.1. The notion of positive involution seems to have been considered first by Weil in his groundbreaking paper [Wei].

Procesi and Schacher also define a notion of positive elements in (A, σ), cf. [PS, §V]. For greater clarity we have adapted their definitions as follows:

Definition 2.2.

(1) An ordering ≤_{P} of F is called a σ-ordering if it makes the involution σ positive,

i.e., if

0 ≤_{P} Trd(σ(x)x) for all x ∈ A.

(2) Suppose ≤_{P} is a σ-ordering on F. An element a ∈ Sym(A, σ) is called σ-positive with respect to ≤_{P} if the quadratic form Trd(σ(x)ax) is positive semidefinite with respect to ≤_{P}. That is, if

0 ≤_{P} Trd(σ(x)ax) for all x ∈ A.

(3) An element a ∈ Sym(A, σ) is called totally σ-positive if it is positive with respect to all σ-orderings on F.

Elements of the form σ(x)x with x ∈ A are called hermitian squares. The set of hermitian squares of A is clearly a subset of Sym(A, σ). It is also clear that the hermitian squares of K are all in F.

Example 2.3. Sums of hermitian squares and sums of traces of hermitian squares are examples of totally σ-positive elements, as easy verifications will show.

One of the main results in [PS] explains that these are essentially the only examples. It can be considered as a noncommutative analogue of Artin’s solution to Hilbert’s 17th problem:

Theorem 2.4. [PS, Theorem 5.4] Let A be a central simple algebra with involution σ, centre K and fixed field F. Let α_{1}, …, α_{m} ∈ F be elements appearing in a diagonalisation of the quadratic form Trd(σ(x)x). Then for a ∈ Sym(A, σ) the following statements are equivalent:

(i) a is totally σ-positive; (ii) there exist x_{i,ε} ∈ A with

a = ∑_{ε∈{0,1}^{m}}α^{ε} ∑_{i} σ(x_{i,ε})x_{i,ε}.

(As usual, α^{ε} denotes α_{1}^{ε_{1}} ⋯ α_{m}^{ε_{m}}.)

In the case n = degA = 2, the weights α_{j} are superfluous (we will come back to this later). Procesi and Schacher [PS, p. 404] conjecture that this is also the case for n > 2:

The PS Conjecture. In a central simple algebra A with involution σ, every totally σpositive element is a sum of hermitian squares. (Equivalently: the trace of a hermitian square is always a sum of hermitian squares.)

Remark 2.5. The two statements in the PS Conjecture are indeed equivalent: the necessary direction follows from the fact that traces of hermitian squares are totally σ-positive, as observed in Example 2.3.

For the sufficient direction, assume that the trace of a hermitian square is always a sum of hermitian squares. Let a ∈ Sym(A, σ) be totally σ-positive. Then a can be expressed in terms of the entries in a diagonalization of the form Trd(σ(x)x) as in Theorem 2.4(ii). Let β be such an entry. Thus, β = Trd(σ(y)y) for some y ∈ A. By the assumption there are x_{1}, …, x_{ℓ} ∈ A such that β = ∑ _{i} σ(x_{i})x_{i}. Since β ∈ F, the expression in Theorem 2.4(ii) can now be rewritten as a sum of hermitian squares.

As mentioned a few lines earlier, Procesi and Schacher provide supporting evidence for their conjecture for the case degA = 2. Another case where the PS Conjecture is true has been well-known since the 1970s:

Example 2.6. Let A be the full matrix ring M_{n}(F) over a formally real field F endowed with the transpose involution σ = t. Since Trd = tr, every ordering of F is a σordering. We claim that a ∈ Sym(A, σ) is totally σ-positive if and only if a is a positive semidefinite matrix in A ⊗_{F} R = M_{n}(R) for any real closed field R containing F (equivalently: for any real closure of F).

Indeed, if a is totally σ-positive, then for all x ∈ A, tr(x^{t}ax) is positive with respect to every (σ-)ordering of F, i.e., tr(x^{t}ax) ∈ ∑ F^{2}. A diagonalisation of the quadratic form tr(x^{t}ax) will contain only sums of squares in F (as it would otherwise violate the total σ-positivity). Hence this quadratic form remains positive semidefinite under every ordered field extension of F.

The converse implication is also easy: if a is positive semidefinite over M_{n}(R) for every real closed field R ⊇ F, then the trace of x^{t}ax for x ∈ A is nonnegative under the ordering of R and hence under all orderings of F. By definition, this means that a is totally σ-positive.

Moreover, every totally σ-positive element of (A, σ) is a sum of hermitian squares. Essentially, this goes back to Gondard and Ribenboim [GR] and has been reproved several times [Djo, FRS, HN, KS]. It also follows easily from Theorem 2.4 for it suffices to show that the trace of a hermitian square is a sum of hermitian squares. But this is clear: if a = [a_{ij}]_{1≤i,j≤n} ∈ A, then

Trd(σ(a)a) = ∑_{i,j=1}^{n} a_{ij}^{2}

is obviously a sum of (hermitian) squares in F.

The reader will have no problems extending this example to the case K = F( \sqrt{−1}) and A = M_{n}(K) endowed with the conjugate transpose involution t.

3. The Counterexamples

When the transpose involution in the previous example is replaced by an arbitrary orthogonal involution σ on M_{n}(F) (i.e., an involution which is adjoint to a quadratic form over F), the equivalence between totally σ-positive elements and sums of hermitian squares is in general no longer true, as we proceed to show in this section. We assume throughout that F_{0} is a formally real field.

Lemma 3.1. Let F = F_{0}((X))((Y)), the iterated Laurent series field in two commuting variables X and Y. The quadratic form

q = ⟨X, Y, XY⟩

does not weakly represent 1 over F. In fact this is already true over the rational function field F_{0}(X, Y).

Proof. Assume for the sake of contradiction that m × q represents 1 for some positive integer m. Then the form

φ := ⟨1⟩ ⊥ m × ⟨−X, −Y, −XY⟩

is isotropic over F. This leads to a contradiction by repeated application of Springer’s theorem on fields which are complete with respect to a discrete valuation, cf. [Lam, Chapter VI, §1]. Since F_{0}(X, Y) embeds into F the proof is finished.

Theorem 3.2. Let F = F_{0}(X, Y). Let A = M_{3}(F) and σ = ad_{q}, where

q = ⟨X, Y, XY⟩.

The (σ-symmetric) element XY is totally σ-positive, but is not a sum of hermitian squares in (A, σ).

Proof. It is clear that XY ∈ Sym(A, σ) since XY ∈ F.

We first show that XY is totally σ-positive. Since T_{σ} ≃ q ⊗ q (see [Lew, p. 227] or [KMRT, 11.4]) we have

sign_{P}T_{σ} = (sign_{P}q)^{2} ∈ {1, 9}

for any ordering P ∈ X_{F}. (Here sign_{P}T_{σ} denotes the signature of the quadratic form T_{σ} with respect to the ordering P.) Hence, the set of σ-orderings on F is not empty. It is exactly the set of P ∈ X_{F} with sign_{P}T_{σ} = 9. (Note that F has orderings for which both X and Y, and thus XY, are positive so that the value sign_{P}T_{σ} = 9 can indeed be attained.)

Let P be any σ-ordering on F. Then we have for any a ∈ A,

Trd(σ(a)a) ≥_{P} 0

(by definition) and so for any a ∈ A,

Trd(σ(a)XYa) = XY Trd(σ(a)a) ≥_{P} 0,

since XY ≥_{P} 0 (for otherwise sign_{P}T_{σ} = 1 and P would not be a σ-ordering on F). Hence, XY is totally σ-positive. An alternative argument showing that XY is totally σ-positive can be given by observing that XY = Trd(σ(b)b) for

b = [ \begin{matrix}0 & X & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}].

Next we show that XY is not a sum of hermitian squares in (A, σ) = (M_{3}(F), ad_{q}). We identify XY with XYI_{3} in M_{3}(F), where I_{3} denotes the 3×3 identity matrix. Assume for the sake of contradiction that XYI_{3} is a sum of elements of the form σ(a)a with a = [a_{ij}]_{1≤i,j≤3} ∈ M_{3}(F). Recall that

σ(a)a = ad_{q}(a)a = [\begin{matrix}X \\ & Y \\ & & XY \end{matrix}]· a^{t} · [\begin{matrix}X \\ & Y \\ & & XY \end{matrix}]^{−1} · a.

The (3, 3)-entry of σ(a)a is equal to

Ya_{13}^{2} + Xa_{23}^{2} + a_{33}^{2}.

By our assumption there are s_{1}, s_{2}, s_{3} ∈ ∑ F^{×2} such that

XY = Ys_{1} + Xs_{2} + s_{3},

which is equivalent with

1 = X^{−1}s_{1} + Y^{−1}s_{2} + X^{−1}Y^{−1}s_{3}.

Thus, 1 is weakly represented by the quadratic form

⟨X^{−1}, Y^{−1}, X^{−1}Y^{−1}⟩ ≃ ⟨X, Y, XY⟩ = q,

which is impossible by Lemma 3.1. This finishes the proof.

The previous theorem gives us a counterexample to the PS Conjecture. It shows that the conjecture is in general not true for full matrix algebras equipped with an orthogonal involution. In contrast, when we equip a full matrix algebra with a symplectic involution, we will show in Theorem 4.7 below that the conjecture does hold.

Thus, we could ask if the PS Conjecture also holds for non-split central simple algebras with symplectic involution. The answer is “no”:

Theorem 3.3. Let F = F_{0}(X, Y). Let A = M_{3}(F) ⊗_{F} H M_{3}(H), where H = (−1, −1)_{F} is Hamilton’s quaternion division algebra over F. Equip A with the involution σ = ad_{q}⊗γ, where γ is quaternion conjugation and σ = ad_{q} for

q = ⟨X, Y, XY⟩.

The algebra A is central simple over F of degree 6 and the involution σ is symplectic. The (σ-symmetric) element XY is totally σ-positive, but is not a sum of hermitian squares in (A, σ).

Proof. The assertion about (A, σ) is clear, as is the fact that XY ∈ Sym(A, σ) since XY ∈ F.

It is easy to verify that the involution trace form of γ, T_{γ}, is isometric to ⟨2⟩ ⊗ N_{H}, where N_{H} = ⟨1, 1, 1, 1⟩ is the norm form of H. Here N_{H}(x) := Nrd_{H}(x) for all x ∈ H, where Nrd_{H} denotes the reduced norm on H. Since T_{σ} = T_{ad_{q} ⊗γ} ≃ T_{ad_{q}} ⊗ T_{γ}, we have

sign_{P}T_{σ} = (sign_{P}T_{ad_{q}})(sign_{P}T_{γ}) = 4 sign_{P}T_{ad_{q}} ∈ {4, 36}

for any ordering P ∈ X_{F}. Hence, the set of σ-orderings on F is not empty. It is exactly the set of P ∈ X_{F} with sign_{P}T_{σ} = 36. (Note again that this value can indeed be attained since there are orderings on F for which both X and Y, and thus XY, are positive.) Arguing similarly as in the proof of Theorem 3.2 we can verify that XY is totally σ-positive.

Before proceeding, note that the involution γ is adjoint to the hermitian form ⟨1⟩_{γ} over (H, γ). Hence, σ is adjoint to the hermitian form h = q ⊗ ⟨1⟩_{γ} = ⟨X, Y, XY⟩_{γ} over (H, γ). Thus

h(x, y) = γ(x_{1})Xy_{1} + γ(x_{2})Yy_{2} + γ(x_{3})XYy_{3}

for vectors x = (x_{1}, x_{2}, x_{3}) and y = (y_{1}, y_{2}, y_{3}) in the right H-vector space H^{3}.

Next we show that XY is not a sum of hermitian squares in (A, σ) = (M_{3}(H), ad_{h}). We identify XY with XYI_{3} in M_{3}(H), where I_{3} denotes the 3×3 identity matrix. Assume for the sake of contradiction that XYI_{3} is a sum of elements of the form σ(a)a with a = [a_{ij}]_{1≤i,j≤3} ∈ M_{3}(H). Recall that

σ(a)a = ad_{h}(a)a = [\begin{matrix}X \\ & Y \\ & & XY \end{matrix}]· γ(a)^{t} · [\begin{matrix}X \\ & Y \\ & & XY \end{matrix}]^{−1} · a,

where γ(a) = [γ(a_{ij})]_{1≤i,j≤3} . The (3, 3)-entry of σ(a)a is equal to

γ(a_{13})Ya_{13} + γ(a_{23})Xa_{23} + γ(a_{33})a_{33} = YN_{H}(a_{13}) + XN_{H}(a_{23}) + N_{H}(a_{33}).

Since N_{H} = ⟨1, 1, 1, 1⟩, each of N_{H}(a_{13}), N_{H}(a_{23}), N_{H}(a_{33}) is a sum of four squares in F. Thus, by our assumption there are s_{1}, s_{2}, s_{3} ∈ ∑ F^{×2} such that

XY = Ys_{1} + Xs_{2} + s_{3}.

We can now finish the proof with an appeal to Lemma 3.1, as in the proof of Theorem 3.2.

Remark 3.4. By tensoring (M_{3}(F), ad_{q}) with Hamilton’s quaternion division algebra, equipped with a unitary involution one obtains a counterexample in the non-split unitary case. We leave the details, which are similar to those in the proof of Theorem 3.3, to the diligent reader.

Remark 3.5. From a real algebra perspective it is clear that these counterexamples to the PS Conjecture can easily be seen to work over any formally real field F that admits a proper semiordering (see [PD, §5] for details and unexplained terminology). Given such a field F, endowed with a proper semiordering, take negative a, b ∈ F such that ab is negative as well. Then q = ⟨a, b, ab⟩ does not weakly represent 1 (the quadratic module generated by {−a, −b, −ab} is proper) and thus in M_{3}(F), endowed with the involution σ = ad_{q}, the element ab is totally σ-positive, but not a sum of hermitian squares (as the proof of Theorem 3.2 shows).

4. Positive Results

Procesi and Schacher [PS, p. 404 and 405] prove their conjecture for central simple algebras A of degree two, i.e., quaternion algebras, with arbitrary involution σ by appealing to matrices and the Cayley–Hamilton theorem. We start this section by giving an alternative argument motivating some of the generalizations that follow.

Throughout this section we assume that the base field F is formally real.

Proposition 4.1. Let A be a quaternion algebra (not necessarily division) with centre K, equipped with an arbitrary involution σ. Let F be the fixed field of (A, σ). Each entry occurring in a diagonalisation of T_{σ} is a sum of hermitian squares.

Proof. (i) We first consider involutions of the first kind on A. Let A be the quaternion algebra (a, b)_{F} with F-basis {1, i, j, k} where i, j and k anti-commute, ij = k, i^{2} = a and j^{2} = b.

If σ is symplectic, then σ is the unique quaternion conjugation involution γ on A. An easy computation gives T_{σ} = T_{γ} ≃ ⟨2⟩ ⊗ ⟨1, −a, −b, ab⟩. We have

1 = γ(1)1, −a = γ(i)i, −b = γ(j)j, ab = γ(k)k.

If σ is orthogonal, then σ = Int(u) ∘ γ, where u ∈ A satisfies γ(u) = −u. From [KMRT, 11.6] we know that

T_{σ} ≃ ⟨2⟩ ⊗ ⟨1, Nrd_{A}(u), − Nrd_{A}(s), − Nrd_{A}(su)⟩

for some s ∈ A with σ(s) = s = −γ(s). Now,

Nrd_{A}(u) = uγ(u) = uγ(u)u^{−1}u = σ(u)u;
− Nrd_{A}(s) = −γ(s)s = σ(s)s;
− Nrd_{A}(su) = − Nrd_{A}(s) Nrd_{A}(u) = −γ(s)s Nrd_{A}(u) = σ(s)σ(u)us = σ(us)us.

(ii) Finally, let K = F( \sqrt{δ}) and let A be a quaternion algebra over K with unitary involution σ whose restriction to K is τ, where τ is determined by τ( \sqrt{δ}) = − \sqrt{δ}. By a well-known result of Albert [KMRT, 2.22] there exists a unique quaternion Fsubalgebra A_{0} ⊆ A such that

A = A_{0} ⊗_{F} K and σ = γ_{0} ⊗ τ,

where γ_{0} is quaternion conjugation on A_{0}. Then T_{σ} ≃ T_{γ_{0}} ⊗ T_{τ} ≃ T_{γ_{0}} ⊗ ⟨1, −δ⟩. Since τ( \sqrt{δ}) \sqrt{δ} = −δ, we are finished by the symplectic part of the proof.

This shows in particular that the PS Conjecture is true for full matrix algebras of degree two over a formally real field F since these are just split quaternion algebras.

Part (ii) of the proof of Proposition 4.1 motivates the following more general result:

Theorem 4.2. Let A and B be central simple algebras with centre K, equipped with arbitrary involutions σ and τ, respectively. Assume that (A, σ) and (B, τ) have the same fixed field F. If the PS Conjecture holds for (A, σ) and (B, τ), it also holds for the tensor product (A ⊗_{K} B, σ ⊗ τ).

Proof. This is a simple computation, using the fact that T_{σ⊗τ} ≃ T_{σ} ⊗ T_{τ} and that elements of A commute with elements of B in the tensor product A ⊗_{K} B.

Corollary 4.3. Let (Q_{1}, σ_{1}), …, (Q_{ℓ}, σ_{ℓ}) be quaternion algebras with arbitrary involution over K and with common fixed field F. The PS Conjecture holds for the tensor product ⨂ _{i=1}^{ℓ}(Q_{i}, σ_{i}).

Proof. This is an immediate consequence of Proposition 4.1 and Theorem 4.2.

Corollary 4.4. Let A = M_{n}(F) be a split algebra of 2-power degree n = 2^{ℓ}, equipped with an orthogonal involution σ which is adjoint to an n-fold Pfister form over F. The PS Conjecture holds for (A, σ).

Proof. By Becher’s proof of the Pfister Factor Conjecture [Bec], (A, σ) decomposes as

(A, σ) ⨂_{i=1}^{ℓ} (Q_{i}, σ_{i}),

where (Q_{1}, σ_{1}), …, (Q_{ℓ}, σ_{ℓ}) are quaternion algebras with involution. An appeal to Corollary 4.3 finishes the proof.

Corollary 4.5. Let A = M_{n}(K) be a split algebra of 2-power degree n = 2^{ℓ}, equipped with a hyperbolic involution σ of any kind. Let F be the fixed field of (A, σ). The PS Conjecture holds for (A, σ).

Proof. Recall from [BST, Theorem 2.1] that the involution σ is hyperbolic if there exists an idempotent e ∈ A such that σ(e) = 1 − e or, equivalently, if the adjoint (quadratic, alternating or hermitian) form of σ is hyperbolic.

If ℓ = 1 this is just the split version of Proposition 4.1. Assume now that ℓ ≥ 2. By [BST, Theorem 2.2], (A, σ) decomposes as

(A, σ) ⨂_{i=1}^{ℓ} (Q, σ_{i}),

where Q = M_{2}(K) and σ_{1}, …, σ_{ℓ} are involutions on Q. An appeal to Corollary 4.3 finishes the proof.

Corollary 4.6. Let A = M_{n}(F) be a split algebra of 2-power degree n = 2^{ℓ}, equipped with a symplectic involution σ. The PS Conjecture holds for (A, σ).

Proof. If σ is a symplectic involution, it is hyperbolic (since it is adjoint to an alternating form over F which is automatically hyperbolic) and we are finished by Corollary 4.5.

In fact, the PS Conjecture is true for any split algebra with symplectic involution. Such an algebra is always of even degree.

Theorem 4.7. Let A = M_{n}(F) be a split algebra of even degree n = 2m, equipped with a symplectic involution σ. The PS Conjecture holds for (A, σ).

Proof. Since σ is symplectic, the quadratic form T_{σ} is hyperbolic (see [Lew, p. 227] or [KMRT, Proof of 11.7]). Thus T_{σ} ≃ m × ⟨1, −1⟩ and it suffices to show that −1 is a sum of hermitian squares in A. We identify −1 with −I_{n}, where I_{n} denotes the n × n identity matrix in A = M_{n}(F).

Since σ is symplectic, we have σ = Int(S) ∘ t, where t denotes transposition and S ∈ GL_{n}(F) satisfies S^{t} = −S. Since S is skew-symmetric, there exists a matrix P ∈ GL_{n}(F) such that P^{t}SP = B, where B is the block diagonal matrix with m blocks [\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}]on the diagonal.

Let X be the block diagonal matrix with m blocks [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] on the diagonal. Then X^{t}BX = B^{−1}. Hence with Y = PXP^{t}, we have Y^{t}SY = S^{−1}. Thus

σ(SY)SY = S(SY)^{t}S^{−1}SY = SY^{t}S^{t}Y = SY^{t}(−S)Y = −SS^{−1} = −I_{n}.

5. Positive noncommutative polynomials

5.1. Algebras of generic matrices with involution. After studying the PS Conjecture in the setting of central simple algebras with involution, we proceed to interpret these results as well as Theorem 2.4 for non-dimensionfree positivity of noncommutative (NC) polynomials.

Motivated by problems in optimization and control theory, Helton [Hel] proved that a symmetric real or complex NC polynomial, all of whose images under algebra ∗- homomorphisms into M_{n}( ), n ∈ , are positive semidefinite (i.e., a dimensionfree positive NC polynomial), is a sum of hermitian squares. What we are interested in, is positivity under evaluations in M_{n}( ) for a fixed n.

To tackle this problem we introduce the language of generic matrices [Row]. Verifying a condition on evaluations of an NC polynomial in the algebra of n×n matrices is often conveniently done in the algebra of generic matrices. In this subsection we recall the definition of generic matrices with involution, while our main result on positive NC polynomials (i.e., a Positivstellensatz) is presented in the next subsection.

As in the classical construction of the algebra of generic matrices (see e.g. [Row, §1.3]), it is possible to construct the algebra of generic matrices with involution [PS, §II]. To each type of involution (orthogonal, symplectic and unitary) an algebra of generic matrices with involution can be associated, as we now explain. We assume from now on that K is a field of characteristic 0 with involution ∗ and fixed field F.

Let K⟨\overline{X}, \overline{X}^{∗}⟩ be the free algebra with involution over (K, ∗), i.e., the algebra with involution, freely generated by the noncommuting variables \overline{X} := (X_{1}, X_{2}, …). Its elements (called NC polynomials) are (finite) linear combinations of words in (the infinitely many) letters \overline{X}, \overline{X}^{∗}.

Fix a type J ∈ \{orthogonal, symplectic, unitary\}. Let _{J_{n}} ⊆ K⟨\overline{X}, \overline{X}^{∗}⟩ denote the ideal of all identities satisfied by degree n central simple K-algebras with type J involution. That is, f = f (X_{1}, …, X_{k}, X_{1}^{∗}, …, X_{k}^{∗}) ∈ K⟨\overline{X}, \overline{X}^{∗}⟩ is an element of _{J_{n}} if and only if for every central simple algebra A of degree n with type J involution σ and every a_{1}, …, a_{k} ∈ A,

f (a_{1}, …, a_{k}, σ(a_{1}), …, σ(a_{k})) = 0.

Then GM_{n}(K, J) := K⟨\overline{X}, \overline{X}^{∗}⟩/ _{J_{n}} is the algebra of generic n × n matrices with type J involution.

Remark 5.1. An alternative description of the algebra of generic matrices with involution can be obtained as follows. Let ζ := (ζ_{ij}^{(ℓ)} | 1 ≤ i, j ≤ n, ℓ ∈ ) denote commuting variables and form the polynomial algebra K[ζ] endowed with the involution extending ∗ and fixing ζ_{ij}^{(ℓ)} pointwise. Consider the n × n matrices Y_{ℓ} := [ζ_{ij}^{(ℓ)}]_{1≤i,j≤n} ∈ M_{n}(K[ζ]), ℓ ∈ . Each Y_{ℓ} is called a generic matrix.

(a) If J ∈ {orthogonal, unitary}, then the (unital) K-subalgebra of M_{n}(K[ζ]) generated by the Y_{ℓ} and their transposes is (canonically) isomorphic to GM_{n}(K, J).

(b) If J = symplectic, then n is even, say n = 2m. Consider the usual symplectic involution

[\begin{matrix}x & y \\ z & w \end{matrix}] ↦ [\begin{matrix}w^{t} & −y^{t} \\ −z^{t} & x^{t} \end{matrix}]

on M_{2m}(K[ζ]). Then the (unital) K-subalgebra of M_{n}(K[ζ]) generated by the Y_{ℓ} and their images under this involution is (canonically) isomorphic to GM_{n}(K, J).

If n = 1, then J ∈ {orthogonal, unitary} and GM_{1}(K, J) is isomorphic to K[ζ] endowed with the involution introduced above. Hence in the sequel we will always assume n ≥ 2.

Let J ∈ {orthogonal, symplectic, unitary}. For n ≥ 2, GM_{n}(K, J) is a PI algebra and a domain (cf. [PS, §II]). Hence its central localization is a division algebra UD_{n}(K, J) with involution, which we call the generic division algebra with type J involution of degree n. As we will only consider the canonical involution on GM_{n}(K, J) and UD_{n}(K, J) we use ∗ to denote it.

5.2. A Positivstellensatz. Let ∈ { , } be endowed with the complex conjugation involution . Our aim in this subsection is to deduce a non-dimensionfree version of Helton’s sum of hermitian squares theorem. We will describe symmetric NC polynomials f all of whose evaluations in M_{n}( ) are positive semidefinite, see Theorem 5.3.

We start with a lemma characterizing total ∗-positivity in the algebra of generic matrices GM_{n}( , J). The proof of the following proposition uses some elementary model theory, e.g. Tarski’s transfer principle for real closed fields. All the necessary background can be found in [PD, §1 and §2] or, alternatively, [BCR, §1].

Lemma 5.2. Let n ∈ . If = , let J = orthogonal and if = , let J = unitary. If a = a^{∗} ∈ GM_{n}( , J) is totally σ-positive under each ∗-homomorphism from GM_{n}( , J) to M_{n}( ) endowed with a positive type J involution σ, then a is totally ∗-positive (in UD_{n}( , J)).

Proof. Suppose a ∈ GM_{n}( , J) is not totally ∗-positive. Then there is a ∗-ordering ≤ of the fixed field Z of the centre of UD_{n}( , J)) under which Trd(x^{∗}ax) is not positive semidefinite. Let ⟨α_{1}, …, α_{m}⟩ be the diagonalisation of Trd(x^{∗}x) with α_{i} = α_{i}^{∗} ∈ Z. (Here m = n^{2} if the involution is of the first kind and m = 2n^{2} otherwise.) Given that Z is the field of fractions of the symmetric centre Z_{0} of GM_{n}( , J), we may even assume α_{i} ∈ Z_{0}. We also diagonalise Trd(x^{∗}ax) as ⟨β_{1}, …, β_{m}⟩ with β_{i} ∈ Z_{0}. Clearly, α_{i} > 0 and one of the β_{i}, say β_{1}, is negative with respect to the given ∗-ordering ≤. Let \overline{Z}^{rc} denote the real closure of Z with respect to this ordering and form A := UD_{n}( , J)⊗_{Z}\overline{Z}^{rc} endowed with the involution σ = ∗ ⊗ id. Then A is a central simple algebra over a real closed (if J = orthogonal) or algebraically closed field (if J = unitary). Moreover, its involution σ is positive. Hence by the classification result [PS, Theorem 1.2] of Procesi and Schacher, A is either M_{n}(\overline{Z}^{rc}) endowed with the transpose (if J = orthogonal) or M_{n}(\overline{Z}) endowed with the complex conjugate transpose involution (if J = unitary). Here \overline{Z}is the algebraic closure \overline{Z}^{rc}( \sqrt{−1}) of \overline{Z}^{rc} and the complex conjugate maps r+t \sqrt{−1} ↦ r − t \sqrt{−1} for r, t ∈ \overline{Z}^{rc}.

For b ∈ GM_{n}( , J) let b̂ ∈ ⟨\overline{X}, \overline{X}^{∗}⟩ denote a preimage of b under the canonical map ⟨\overline{X}, \overline{X}^{∗}⟩ → GM_{n}( , J). Every ∗-homomorphism GM_{n}( , J) → M_{n}(L) for a ∗-field extension L of , where M_{n}(L) is given a type J involution, yields a ∗-homomorphism ⟨\overline{X}, \overline{X}^{∗}⟩ → M_{n}(L), so is essentially given by a point s ∈ M_{n}(L)^{}describing the images of the X_{i} under this induces map.

By construction, the image β_{1} ⊗ 1 of β_{1} under the embedding of algebras with involution GM_{n}( , J) → A is not σ-positive. Let s denote the corresponding evaluation point. By Example 2.6, this means that β̂_{1}(s, \overline{s}^{t}) = β_{1} ⊗ 1 is not positive semidefinite. Consider the following elementary statement:

∃ n × n matrices x = (x_{1}, …, x_{N}) : α̂_{i}(x, \overline{x}^{t}) is positive semidefinite∧
β̂_{1}(x, \overline{x}^{t}) is not positive semidefinite.

(1)

(N is the maximal number of variables appearing in one of the α̂_{i}, β̂_{1}.)

Obviously such n × n matrices x_{i} can be found over \overline{Z}^{rc} or \overline{Z}; just take x_{i} = s_{i}. By Tarski’s transfer principle, the above elementary statement (1) can be satisfied in . This yields a ∗-homomorphism ⟨\overline{X}, \overline{X}^{∗}⟩ → M_{n}( ) endowed with the (positive) involution t and in turn (by universality) a ∗-homomorphism GM_{n}( , J) → (M_{n}( ), t). By the construction, the image of a under this mapping will not be positive semidefinite. This finishes the proof.

In order to state the Positivstellensatz, we need to recall the notion of central polynomials for n × n matrices. These are f ∈ K⟨\overline{X}, \overline{X}^{∗}⟩ whose image in GM_{n}(K, J) is central. Equivalently, the image of f under a ∗-homomorphism from K⟨\overline{X}, \overline{X}^{∗}⟩ to M_{n}(K) endowed with a type J involution, is always a scalar matrix. If it is nonzero, we call f nonvanishing. The existence of nonvanishing central polynomials is nontrivial; we refer to [Row, §1; Appendix A] for details.

Theorem 5.3 (Positivstellensatz). Suppose ∈ { , } is endowed with the complex conjugate involution . Let g = g^{∗} ∈ ⟨\overline{X}, \overline{X}^{∗}⟩, n ∈ and fix a type J ∈ \{orthogonal, unitary\} according to the type of involution on . Choose α_{1}, …, α_{m} ∈ ⟨\overline{X}, \overline{X}^{∗}⟩ whose images in GM_{n}( , J) form a diagonalisation of the quadratic form Trd(x^{∗}x) on UD_{n}( , J). Then the following are equivalent:

(i) for any s ∈ M_{n}( )^{}, g(s, \overline{s}^{t}) is positive semidefinite; (ii) there exists a nonvanishing central polynomial h ∈ ⟨\overline{X}, \overline{X}^{∗}⟩ for n × n matrices

and p_{i,ε} ∈ ⟨\overline{X}, \overline{X}^{∗}⟩ with

h^{∗}gh ≡ ∑_{ε∈{0,1}^{m}}α^{ε} ∑_{i} p_{i,ε}^{∗}p_{i,ε} (mod _{J_{n}}).

Proof. Given a congruence as in (ii), it is clear that (i) holds whenever h(s, \overline{s}^{t}) 0. As the set of all such s is Zariski dense, (i) holds for all s ∈ M_{n}( )^{}.

For the converse implication note that by Lemma 5.2, g + _{J_{n}} is totally ∗-positive in UD_{n}( , J). Hence by Theorem 2.4 we obtain a positivity certificate

g + _{J_{n}} = ∑_{ε∈{0,1}^{m}}(α + _{J_{n}})^{ε} ∑_{i} (x′_{i,ε})^{∗}x′_{i,ε}

for some x′_{i,ε} ∈ UD_{n}( , J). Clearing denominators, there are x_{i,ε} ∈ GM_{n}( , J) and a nonzero central r ∈ GM_{n}( , J) with

r^{∗}(g + _{J_{n}})r = ∑_{ε∈{0,1}^{m}}(α + _{J_{n}})^{ε} ∑_{i} x_{i,ε}^{∗}x_{i,ε}.

Lifting this equality to the free algebra yields the desired conclusion.

When n = 2, the weights α are redundant (cf. §4 or [PS, p. 405]) and we obtain the following strengthening:

Corollary 5.4. Suppose ∈ { , } is endowed with the complex conjugate involution . Let g = g^{∗} ∈ ⟨\overline{X}, \overline{X}^{∗}⟩, n ∈ and fix a type J ∈ \{orthogonal, unitary\} according to the type of involution on . Then the following are equivalent:

(i) for any s ∈ M_{2}( )^{}, g(s, \overline{s}^{t}) is positive semidefinite; (ii) there exists a nonvanishing central polynomial h ∈ ⟨\overline{X}, \overline{X}^{∗}⟩ for 2 × 2 matrices

and p_{i} ∈ ⟨\overline{X}, \overline{X}^{∗}⟩ with

h^{∗}gh ≡ ∑_{i} p_{i}^{∗}p_{i} (mod _{J_{2}}).

Remark 5.5. By Tarski’s transfer principle, Theorem 5.3 and Corollary 5.4 hold with replaced by any real closed or algebraically closed field of characteristic 0.

We conclude the paper with an open problem: can Theorem 5.3 be used to give a proof of Helton’s sum of hermitian squares theorem?

Acknowledgments

Both authors wish to thank David Lewis for some very useful observations. The second author wishes to thank Jaka Cimprič for arranging a very nice stay at the University of Ljubljana in May 2008. The first author was supported by the Slovenian Research Agency (project no. Z1-9570-0101-06). The second author was supported by the Science Foundation Ireland Research Frontiers Programme (project no. 07/RFP/MATF191).

References

[BST] E. Bayer-Fluckiger, D.B. Shapiro, J.-P. Tignol, Hyperbolic involutions, Math. Z. 214 (1993), no. 3, 461–476.

[Bec] K.J. Becher, A proof of the Pfister factor conjecture, Invent. Math. 173 (2008), no. 1, 1–6.

[BCR] J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry, Springer-Verlag, Berlin, 1998.

[dOHMP] M.C. de Olivera, J.W. Helton, S.A. McCullough, M. Putinar, Engineering Systems and Free Semi-Algebraic Geometry, Emerging Applications of Algebraic Geometry, 17–62, IMA Vol. Math. Appl. 149, Springer, 2008.

[Djo] D.Ž. Djoković, Positive semi-definite matrices as sums of squares, Linear Algebra and Appl. 14 (1976), no. 1, 37–40.

[FRS] J.F. Fernando, J.M. Ruiz, C. Scheiderer, Sums of squares of linear forms, Math. Res. Lett. 13 (2006), no. 5-6, 947–956.

[GR] D. Gondard, P. Ribenboim, Le 17e problème de Hilbert pour les matrices, Bull. Sci. Math. (2) 98 (1974), no. 1, 49–56.

[Hel] J.W. Helton, “Positive” noncommutative polynomials are sums of squares, Ann. of Math. (2) 156 (2002), no. 2, 675–694.

[HN] C.J. Hillar, J. Nie, An elementary and constructive solution to Hilbert’s 17th problem for matrices, Proc. Amer. Math. Soc. 136 (2008), no. 1, 73–76.

[KS] I. Klep, M. Schweighofer, Positivity certificates for matrix polynomials, in preparation.

[KMRT] M.-A. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Coll. Pub. 44, Amer. Math. Soc., Providence, RI (1998).

[Lam] T.Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67, American Mathematical Society, Providence, RI (2005).

[Lew] D.W. Lewis, Trace forms, Kronecker sums, and the shuffle matrix, Linear and Multilinear Algebra 40 (1996), no. 3, 221–227.

[PD] A. Prestel, C.N. Delzell, Positive polynomials. From Hilbert’s 17th problem to real algebra, Springer-Verlag, Berlin, 2001.

[PS] C. Procesi, M. Schacher, A non-commutative real Nullstellensatz and Hilbert’s 17th problem, Ann. of Math. (2) 104 (1976), no. 3, 395–406.

[Row] L.H. Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, 84, Academic Press, Inc., New York-London (1980).

[Wei] A. Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 1960 589–623 (1961).

University of California San Diego, Department of Mathematics, 9500 Gilman Drive, La Jolla, California 92093-0112, USA

  E-mail address: iklep@math.ucsd.edu

School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

  E-mail address: thomas.unger@ucd.ie