  We associate to any Riemannian symmetric space (of finite or infinite
dimension) a L$^*$-algebra, under the assumption that the curvature operator
has a fixed sign. L$^*$-algebras are Lie algebras with a pleasant Hilbert space
structure. The L$^*$-algebra that we construct is a complete local isomorphism
invariant and allows us to classify Riemannian symmetric spaces with fixed-sign
curvature operator. The case of nonpositive curvature is emphasized.
