  The sandwiched R\'enyi $\alpha$-divergences of two finite-dimensional quantum
states play a distinguished role among the many quantum versions of R\'enyi
divergences as the tight quantifiers of the trade-off between the two error
probabilities in the strong converse domain of state discrimination. In this
paper we show the same for the sandwiched R\'enyi divergences of two normal
states on an injective von Neumann algebra, thereby establishing the
operational significance of these quantities. Moreover, we show that in this
setting, again similarly to the finite-dimensional case, the sandwiched R\'enyi
divergences coincide with the regularized measured R\'enyi divergences, another
distinctive feature of the former quantities. Our main tool is an approximation
theorem (martingale convergence) for the sandwiched R\'enyi divergences, which
may be used for the extension of various further results from the
finite-dimensional to the von Neumann algebra setting.
  We also initiate the study of the sandwiched R\'enyi divergences of pairs of
states on a $C^*$-algebra, and show that the above operational interpretation,
as well as the equality to the regularized measured R\'enyi divergence, holds
more generally for pairs of states on a nuclear $C^*$-algebra.
