  In this note, we consider the smallest submaximal space structure {\mu}(X) on
a Banach space X. We derive a characterization of {\mu}(X) up to complete
isometric isomorphism in terms of a universal property. Also, we show that an
injective Banach space has a unique submaximal space structure and we explore
some duality relations of {\mu}-spaces.
  Key Words: operator spaces, maximal operator spaces, submaximal spaces,
{\mu}- spaces.
