A formally real field \(F\) satisfies the Strong Approximation Property (SAP) if every closed and open subset of its space of orderings \(X_F\) has the form \(\{P\mid a>_P0\}\) for some \(a\in F\). In [R. Elman, T.-Y. Lam, A. Prestel, Math. Z. 134, 291-301 (1973; Zbl 0277.15013)], [A. Prestel, Math. Z. 133, 319-342 (1973; Zbl 0275.12013)], this property is shown to be equivalent to the following weak Hasse principle: every totally indefinite quadratic form \(q\) over \(F\) is weakly isotropic, i.e. there is an integer \(m\) such that \(m\times q\) is isotropic.
The paper under review investigates corresponding notions for central simple \(F\)-algebras \(A\) with an involution \(\sigma\) of the first kind (i.e. \(\sigma\) is an \(F\)-linear anti-automorphism of \(A\) such that \(\sigma^2=I\)). The involution \(\sigma\) is called totally indefinite if for all \(P\in X_F\) the signature \(\text{sig}_P\sigma\) satisfies \(\text{sig}_P\sigma<\deg A\). It is called weakly isotropic if there exist nonzero \(x_1,\dots,x_m\in A\) (for some integer \(m\geq 1\)) such that \(\sigma(x_1)x_1+\cdots+\sigma(x_m)x_m=0\). Thus, a quadratic form is totally indefinite (resp. weakly isotropic) if and only if its adjoint involution has the same property, and the weak Hasse principle for quadratic forms is subsumed in the following weak Hasse principle for central simple algebras with involution: every totally indefinite involution of the first kind on a central simple \(F\)-algebra is weakly isotropic. The authors show that the latter holds if and only if the field \(F\) satisfies the Effective Diagonalization (ED) Property, which is equivalent to requiring SAP for \(F\) and all its quadratic extensions. As an application, they show that an involution of the first kind on a central simple algebra over an ED-field is weakly isotropic if and only if the involution trace form \(T_\sigma(x)=\text{Trd}_A(\sigma(x)x)\) is weakly isotropic.