Parallel addition and fixed points of compressions on symmetric cones

Let V be a Euclidean Jordan algebra with identity e, and let Ω be the corresponding symmetric cone. In this paper, we introduce a partially ordered commutative semigroup structure on the closed convex cone Ω̄ extending the binary operation a : b = (a^-1 + b^-1)^-1 on Ω and consider compressions of the symmetric cone Ω of the form φ_a,b,kP(w)(x) = a + kP(w)(x : b), a ∈ Ω, b ∈ Ω̄, kP(w) ∈ Aut(V) P(V) where P is the quadratic representation of the Jordan algebra V and Aut(V) is the Jordan automorphism group of V. The aim of this paper is to show that φ_a,b,kP(w) has a unique fixed point p(a,b,kP(w)) on Ω and the fixed point map p : Ω × Ω̄ × V × Aut(V) → (a,b,w,k) → p(a,b,kP(w)) is continuous.