Algebraic isomorphisms and Jordan derivations of J-subspace lattice algebras

It is shown that every algebraic isomorphism between standard subalgebras of J-subspace lattice algebras is quasi-spatial and every Jordan derivation of standard subalgebras of J-subspace lattice algebras is an additive derivation. Also, it is proved that every finite rank operator in a J-subspace lattice algebra can be written as a finite sum of rank one operators each belonging to that algebra. As an additional result, a multiplicative bijection of a J-subspace lattice algebra onto an arbitrary ring is proved to be automatically additive. Those results can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.