Transformations of the unitary group on a Hilbert space

Let H be an infinite dimensional separable complex Hilbert space and U be the group of all unitary operators on H. Motivated by the algebraic properties of surjective isometries of U that have recently been revealed, and also by some classical results related to automorphisms of the unitary groups of operator algebras, we determine the structures of bijective transformations of U that respect certain algebraic operations. These are, among others, the usual product of operators, the Jordan triple product, the inverted Jordan triple product, and the multiplicative commutator. Our basic approach to obtain these results is the use of commutativity preserving transformations on the unitary group.