Given a finite-dimensional CCR C*-algebra $\mathcal{A}$ (you can find the details of how CCR algebras are introduced in many books, for example see "Operator algebras and quantum statistical mechanics" vol. II, by Bratteli and Robinson). Finite-dimensional here means that the algebra is generated by a finite number of couples of operators $a_i, a_{i}^{*}$, $i = 1, ..., N$, with the CCR relations that are $[a_i,a_{j}^{*} ] = \delta_{i,j} \mathcal{I}$ ($\mathcal{I}$ is the identity operator).
Given two representations $\pi_1, \pi_2$, let's assume that there is a unitary operator $U$ such that $\pi_2 (a_i) = U \pi_1 (a_i) U^{-1}, \forall i = 1, ..., N.$
However, let's assume that the limit $N \to \infty$ of the unitary operator $U$ does not exist. My doubts here is what can we say on the two representations, given that the operator $U$ exists only in the finite dimensional case and does not in the infinite case.
More generally speaking, is enough to prove that one sequence of unitary operators $U_N$ (let me write down explicitly the dependence on the degrees of freedom here) does not converge despite each $U_N$ links the two representations when the degrees of freedom are $N$ finite, to say that there are no unitary operators linking the two representations at all and then the two representations are unitarily inequivalent?