Strict contractions of symmetric cones (Q1574467)

Let \(\Omega\) be the open cone of squares in the Euclidean Jordan algebra \(V\). Then \(\Omega\) is in particular a Riemannian symmetric space, hence has a natural Riemannian metric which is invariant under the group \(G(\Omega)\) of all linear automorphisms of the cone. Let \(T_\Omega := \Omega + i V\) denote the tube domain corresponding to \(\Omega\) and \(\Gamma_\Omega\) the semigroup of all those biholomorphic automorphisms of \(T_\Omega\) mapping \(\Omega\) into itself. In the paper under review it is shown that each element in the interior of \(\Gamma_\Omega\) acts as a strict contraction for the Riemannian metric on \(\Omega\), hence in particular that every such element has a unique fixed point. Related results have been obtained in the context of infinite-dimensional bounded symmetric domains by the reviewer in the article ``Compressions of infinite-dimensional bounded symmetric domains'' which is to appear in Semigroup Forum.