Lattice-like subsets of Euclidean Jordan algebras

TL;DR

This paper investigates the properties of lattice-like subsets in Euclidean Jordan algebras, showing that Jordan subalgebras are lattice-like but not all such sets are subalgebras, especially in higher-rank cases.

Contribution

It establishes that Jordan subalgebras are lattice-like sets and explores the restrictions on lattice-like convex sets in simple Euclidean Jordan algebras of rank at least three.

Findings

Jordan subalgebras are lattice-like sets.

Not all lattice-like sets are Jordan subalgebras.

In higher-rank simple Euclidean Jordan algebras, lattice-like proper convex sets with interior points do not exist.

Abstract

While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder and Tao have introduced generalized lattice operations based on the projection onto the cone of squares. In two recent papers of the authors of the present paper it has been shown that these lattice-like operators and their generalizations are important tools in establishing the isotonicity of the metric projection onto some closed convex sets. The results of this kind are motivated by metods for proving the existence of solutions of variational inequalities and methods for finding these solutions in a recursive way. It turns out, that the closed convex sets admitting isotone projections are exactly the sets which are invariant with respect to these lattice-like operations, called lattice-like sets. In this paper it is shown that the Jordan subalgebras are lattice-like sets, but the converse in general is not true. In the case of simple Euclidean Jordan algebras of rank at least three the lattice-like property is rather restrictive, e.g., there are no lattice-like proper closed convex sets with interior points.