Group representation theory is one of the most powerful tools to study groups. The unitary group is an important research branch of group theories. We study a class of algebraic structures generated by unitary groups，and we prove that Alg( H) is a von Neumann algebra for any subgroup H of the unitary group U( d) . Then we show that group algebra ［G］，generated by a finite group G，is isomorphic to a direct sum of some von Neumann algebras of Type Ⅰn for distinct values of n. We further discuss what kinds of von Neumann algebra of Type Ⅰ can be generated by a finite unitary subgroup. It is of great significance to study the algebraic structure of the generation of subgroups of unitary groups for the establishment of Schur-Weyl duality of unitary groups and permutation groups.