Existence of optimal transport maps in very strict $CD(K,\mathrm{\infty })$ -spaces

We introduce a more restrictive version of the strict $CD(K,\mathrm{\infty })$ -condition, the so-called very strict $CD(K,\mathrm{\infty })$ -condition, and show the existence of optimal maps in very strict $CD(K,\mathrm{\infty })$ -spaces despite the possible lack of uniqueness of optimal plans.