Just-infinite Jordan Banach algebras

By analogy with the well-established notions of just-infinite groups and just-infinite algebras, in particular [Formula: see text]-algebras, we initiate a study of just-infinite [Formula: see text]-algebras, i.e. infinite-dimensional [Formula: see text]-algebras for which all proper quotients are finite-dimensional. We investigate the connections between a just-infinite [Formula: see text]-algebra [Formula: see text] and its Jordan algebra [Formula: see text] of self-adjoint elements. We also show that any just-infinite [Formula: see text]-algebra [Formula: see text] either is an infinite-dimensional spin factor or there exist a [Formula: see text]-algebra [Formula: see text] and just-infinite, norm-closed real ∗-subalgebras [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text]