Just-infinite Jordan Banach algebras

By analogy with the well-established notions of just-infinite groups and just-infinite algebras, in particular $C^{\ast }$-algebras, we initiate a study of just-infinite $JB$-algebras, i.e. infinite dimensional $JB$-algebras for which all proper quotients are finite dimensional. We investigate the connections between a just-infinite $C^{\ast }$-algebra $A$ and its Jordan algebra $H(A,\ast )$ of self-adjoint elements. We also show that any just-infinite $JB$-algebra $J$ either is a infinite-dimensional spin factor or there exists a $C^{\ast }$-algebra $A$ and just-infinite norm-closed real $\ast$-subalgebras $A_1$ and $A_2$ of $A$ such that $H(A_1,\ast )⊴J\subseteq H(A_2,\ast )\mathrm{.}$