Signatures, sums of hermitian squares and positive cones on algebras with involution

We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and"ordered"algebras with involution.

(and also [23]) we extended these results in the noncommutative direction, more precisely to central simple F -algebras with involution and hermitian forms over such algebras.
The study of central simple algebras with involution was initiated by Albert in the 1930s [1] and is still a topic of current research as testified by The Book of Involutions [19]; see also [10] and the copious references therein for a list of open problems in this area. A large part of present day research in algebras with involution is driven by the deep connections with linear algebraic groups, first observed by Weil [35]; see also Tignol's 2 ECM exposition [34]. Some work has been done on algebras with involution over formally real fields, for example [22], [30], but this part of the theory is relatively underdeveloped. This observation, together with the fact that algebras with involution are a natural generalization of quadratic forms, are motivating factors for our research.
This article is an expanded version of the prepublication [8], from the Séminaire de Structures Algébriques Ordonnées, Universities Paris 6 and 7.

Signatures
Let (A, σ) be an F -algebra with involution, by which we mean that A is a finite dimensional simple F -algebra with centre a field K ⊇ F and σ is an F -linear anti-automorphism of A of order 2 (which implies that [K : F ] 2). Let W (A, σ) denote the Witt group of (A, σ), i.e. the W (F )-module of Witt equivalence classes of nondegenerate hermitian forms h : M × M → A, where M is a finitely generated right A-module (cf. [18,Chap. I] or [32,Chap. 7]). We identify hermitian forms with their Witt class in W (A, σ), unless indicated otherwise. Given an ordering P ∈ X F we wish to define a signature at P , i.e. a morphism of groups W (A, σ) → Z.
Following the approach of [11] we do this by extending scalars to a real closure F P of F at P and realizing that, by Morita equivalence, the Witt group of any F P -algebra with involution is isomorphic to either Z, 0 or Z/2Z. In the last two cases, the only sensible definition is to take the signature at P to be identically zero. In this case we call P a nil-ordering and we write Nil[A, σ] for the set of all nil-orderings, noting that it only depends on the Brauer class of A and the type of σ. Furthermore, Nil[A, σ] is clopen in X F , cf. [4,Corollary 6.5].
In the first case, the Witt group where − denotes (quaternion) conjugation, each one in turn being isomorphic to Z via the usual Sylvester signature of quadratic or hermitian forms. The composite map s P , given by At first sight, one way to fix a sign would be to demand that s P ( 1 σ ) is positive, as is the case for quadratic forms. This is the approach taken in [11], but it may not always work, since it may happen that s P ( 1 σ ) is in fact 0, as illustrated in [4,Rem. 3.11 and Ex. 3.12]. Our solution to this dilemma is to show that there exists a hermitian form η over (A, σ), called a reference form, such that s P (η) is always nonzero whenever P ∈ X F := X F \ Nil[A, σ], cf. [5,Prop. 3.2]. Using this, given P ∈ X F , we define the signature at P with respect to the reference form η, sign η P : W (A, σ) → Z, to be the map s P , multiplied by −1 in case s P (η) < 0, so that sign η P (η) > 0. The map sign η P does not depend on the Morita equivalence used in its computation and so we may use the explicit Morita equivalence presented in [24] in all practical situations.
Remark 2.1. In case (A, σ) = (F, id F ), we may take η = 1 and sign η P is then the usual Sylvester signature sign P of quadratic forms.
Remark 2.2. The signature map is defined for all hermitian forms over (A, σ), not just the nondegenerate ones as the notation above (which makes use of W (A, σ)) might suggest. It suffices to replace a form by its nondegenerate part (cf.   [5, §3]. In fact, this is the approach used in [4]. We collect some immediate properties of the signature map: (1) Let h be a hyperbolic form over (A, σ), then sign η P h = 0. (2) Let h 1 , h 2 ∈ W (A, σ), then sign η P (h 1 ⊥ h 2 ) = sign η P h 1 + sign η P h 2 . (3) Let h ∈ W (A, σ) and q ∈ W (F ), then sign η P (q · h) = sign P q · sign η P h. (4) (Going-up) Let h ∈ W (A, σ) and let L/F be an algebraic extension of ordered fields. Then Property (4) is complemented by the following going-down result: [4,Thm 8.1]). Let L/F be a finite extension of ordered fields and assume P ∈ X F extends to L. Let h ∈ W (A⊗ F L, σ ⊗ id). Then where Tr * A⊗ F L h denotes the Scharlau transfer induced by the A-linear homo- for all h ∈ W (A, σ) and all P ∈ X F . . For every f ∈ C(X F , Z) [A,σ] there exists n ∈ N such that 2 n f ∈ Im sign η . In other words, the cokernel of sign η is a 2-primary torsion group.
The stability index of (A, σ) is the smallest k ∈ N such that 2 k C(X F , Z) [A,σ] ⊆ Im sign η if such a k exists and ∞ otherwise. It is independent of the choice of η. The group coker sign η is up to isomorphism independent of the choice of η. We denote it by S η (A, σ) and call it the stability group of (A, σ).

Ideals and morphisms
Let R be a commutative ring and let M be an R-module. We introduce ideals of R-modules as follows: An ideal of M is a pair (I, N ) where I is an ideal of R and N is a submodule of M such that I · M ⊆ N . An ideal (I, N ) of M is prime if I is a prime ideal of R (we assume that all prime ideals are proper), N is a proper submodule of M , and for every r ∈ R and m ∈ M , r · m ∈ N implies that r ∈ I or m ∈ N .
These definitions are in part motivated by the following natural example: The pair (ker sign P , ker sign η P ) is a prime ideal of the W (F )-module W (A, σ) whenever P ∈ X F . We obtain a classificationà la Harrison and Lorenz-Leicht [25]: (1) If 2 ∈ I, then one of the following holds: (i) There exists P ∈ X F such that (I, N ) = (ker sign P , ker sign η P ).
(ii) There exist P ∈ X F and a prime p > 2 such that (I, N ) = ker(π p • sign P ), ker(π • sign η P ) , where π p : Z → Z/pZ and π : Im sign η P → Im sign η P /(p · Im sign η P ) are the canonical projections.
Remark 3.2. When 2 ∈ I, N is completely determined by I. This is however not the case when 2 ∈ I, cf. [5, Ex. 6.8].
The pair (sign P , sign η P ) is again a natural example of a (W (F ), Z)-morphism from W (A, σ) to Z and is trivial if and only if P ∈ Nil[A, σ].
The classification of prime ideals of W (A, σ) yields the following description of signatures as morphisms:

Sums of hermitian squares
In the field case, Pfister's local-global principle can be used to give a short proof of the fact that sums of squares are exactly the elements that are nonnegative at every ordering. In [7] we showed that the same approach directly yields a similar result for F -division algebras with involution and, with some extra effort, for all F -algebras with involution.
Let A × denote the set of invertible elements of A, Sym(A, σ) the set of σsymmetric elements of A and Sym(A, σ) × := Sym(A, σ) ∩ A × . We say that an element a ∈ Sym(A, σ) is η-maximal at an ordering P ∈ X F if sign η P a σ is maximal among all sign η P b σ for b ∈ Sym(A, σ). In the field case, this means sign P a = 1, in other words a ∈ P \ {0}. For elements b 1 , . . . , b t ∈ F × we denote the Harrison set {P ∈ X F | b 1 , . . . , b t ∈ P } by H(b 1 , . . . , b t ).  H(b 1 , . . . , b t ). Assume that a ∈ Sym(A, σ) × is η-maximal at all P ∈ Y . Let u ∈ Sym(A, σ). The following statements are equivalent: The presence of the element a as well as the hypothesis on η-maximality correspond in the field case to the fact that 1 belongs to every ordering. Here 1 does not play a particular role since it may not have maximal signature at some orderings. We replace it by the element a and only consider a set of orderings Y on which a has maximal signature.  The general answer to this question is negative as shown in [17], but we can now describe cases where the answer is positive, and also propose a natural reformulation (inspired by signatures of hermitian forms) of the question that has a positive answer.

Positive cones
The results presented thus far suggest that there could be a notion of "ordering" on central simple algebras with involution, whose behaviour would be similar to that of orderings on fields. The purpose of this final section is to present such a notion. (P2) P + P ⊆ P; (P3) σ(a) · P · a ⊆ P for every a ∈ A; (P4) P F := {u ∈ F | uP ⊆ P} is an ordering on F . (P5) P ∩ −P = {0} (we say that P is proper ). We say that a prepositive cone P is over P ∈ X F if P F = P .
Remark 5.2. Axiom (P4) is necessary if we want our prepositive cones to consist of either positive semidefinite (PSD) matrices with respect to P , or of negative semidefinite (NSD) matrices with respect to P , in the case of (M n (F ), t), see [6,Rem. 3.13].
If P is a prepositive cone, then −P is also a prepositive cone. This is due to the fact that prepositive cones are meant to contain elements of maximal signature, and the sign of the signature can vary with a change of the reference form.
It can be shown that there is a prepositive cone over P ∈ X F on (A, σ) if and only if P ∈ X F , cf. [6, Prop. 6.6].
(2) The set of PSD matrices, and the set of NSD matrices with respect to some P ∈ X F are both prepositive cones over P on (M n (F ), t).
Remark 5.4. Other notions of orderings have been introduced for division rings with involution, most notably Baer orderings, * -orderings and their variants and an extensive theory has been developed around them. Craven's surveys [13] and [14] provide more information on these topics. Without going into the details, the main difference in the definitions is that positive cones were developed to correspond to a pre-existing algebraic notion, namely signatures of hermitian forms (e.g. axiom (P4) reflects the fact that the signature is a morphism of modules, cf. Proposition 2.4(3); see also the sentence after Theorem 5.8) and as a consequence are not required to induce total orderings on the set of symmetric elements.
We obtain the desired results linking prepositive cones and W (A, σ): where P is a prepositive cone on (M n (D), ϑ t ).
We use prepositive cones to consider the question of the existence of positive involutions: Theorem 5.7 ([6, Thm. 6.8]). Let P ∈ X F . The following statements are equivalent: (i) There is an involution τ on A which is positive at P and of the same type as σ; The notion of prepositive cone can be seen as somewhat equivalent to that of preordering or Prestel's pre-semiordering [27], [28], so it is natural to consider in more detail the maximal prepositive cones, which we simply call positive cones. They can be completely described and match the examples provided above. To see this we define, for P ∈ X F and S ⊆ Sym(A, σ), (the smallest, possibly nonproper, prepositive cone over P containing S), and we denote by X (A,σ) the set of all positive cones on (A, σ).
In particular, the only positive cones over P on (D, ϑ) are M η P (D, ϑ) and −M η P (D, ϑ) and therefore the examples above are essentially the only positive cones on (A, σ), cf. [6,Props. 4.3 and 4.9]. It follows that the PSD matrices over P and the NSD matrices over P are the only positive cones over P on (M n (F ), t). (See also Proposition 5.6.) Using this description, it is possible to make the link with the results presented in Section 4, and to obtain results similar to the Artin-Schreier and Artin theorems.
The second statement is a trivial consequence of the first one, but it is still included here to point out that while the element a in it obviously belongs to a prepositive cone (namely C P (a)), the element b in the third statement may not belong to any prepositive cone on (A, σ), contrary to what could be expected from the field case (see [6,Rem. 7.10]).
and let a ∈ Sym(A, σ) × be such that, for every P ∈ X (A,σ) with P F ∈ Y , a ∈ P ∪ −P. Then As a consequence of our study of positive involutions, given Q ∈ X F , there always exist a and Y that satisfy the hypothesis of Theorem 5.10 with Q ∈ Y , cf. Remark 4.2.
The element a in this theorem plays the same role as the element a in Theorem 4.1, and chooses a prepositive cone from {P, −P} in a uniform way. This is not necessary in the field case, because 1 belongs to every ordering. In the special case where a = 1 can be used for this purpose, we obtain a result more similar to the usual one: Corollary 5.11 ([6,Cor. 7.15]). Assume that for every P ∈ X (A,σ) , 1 ∈ P ∪ −P. Then The hypothesis of Corollary 5.11 is exactly X σ = X F in the terminology of Section 4. More precisely, as seen therein, this property characterizes the algebras with involution for which there is a positive answer to (PS'), cf. [7,Section 4.2].
A natural question is to ask if signatures of hermitian forms over (A, σ) can now also be defined with respect to positive cones on (A, σ). As shown in [6, §8.2], this can indeed be done using decompositions of hermitian forms, reminiscent of Sylvester's decomposition for quadratic forms: Theorem 5.12 ([6, Cor. 8.14, Lemma 8.15]). There exists an integer t, depending on (A, σ), such that for every P ∈ X (A,σ) and for every h ∈ W (A, σ) there exist u 1 , . . . , u t ∈ P := P F , a 1 , . . . , a r ∈ P ∩ A × and b 1 , . . . , b s ∈ −P ∩ A × such that n 2 P × u 1 , . . . , u t ⊗ h ≃ a 1 , . . . , a r σ ⊥ b 1 , . . . , b s σ , where n P is the matrix size of A ⊗ F F P , and r and s are positive integers, uniquely determined by P and the rank of h.
Proposition 5.13 ([6, Prop. 9.2]). The topologies T σ and T × σ are equal. Recall that spectral topologies, defined in [16], are precisely the topologies of the spectra of commutative rings, and that a map between spectral spaces (i.e. spaces equipped with spectral topologies) is called spectral if it is continuous and the preimage of a quasicompact open set is quasicompact.