In the series of papers [1-4], L. Barker developed a general notion of convergence for sequences of Hilbert spaces and related objects (vectors, operators. . . ). In this paper, we remark that Barker’s convergence for sequences of operators provides a notion of contraction of Lie group (unitary) representations and we compare it to the usual one introduced by J. Mickelsson and J. Niederle. This allows us to illustrate Barker’s convergence of operators by various examples taken from contraction theory.