John Ringrose has made major contributions to the theory of algebras of operators on Hilbert spaces, especially in long-standing collaboration with Richard V. Kadison. They showed that certain von Neumann algebras have only inner derivations — a result extended by Shoichiro Sakai to all von Neumann algebras and to some other C*-algebras. John and Richard also developed the fundamental properties of the automorphism group of a C*-algebra, as a topological group with the uniform topology.

Of particular note is their theorem that every automorphism at distance less than 2 from the identity is the exponential of a derivation and so lies on a norm-continuous, one-parameter subgroup, and is implemented by a unitary operator in the von Neumann closure in every representation. In addition, they laid the foundations of a cohomology theory for C*-algebras.

John has worked also on the triangular representation of compact linear operators (especially those in the Schmidt class), and has initiated a study of nests of subspaces and the corresponding triangular operator algebras, or ‘nest algebras’.