Doubly $\Lambda$-commuting row isometries, universal models, and classification

The goal of the paper is to study the structure of the k-tuples of doubly $\Lambda$-commuting row isometries and the $C^*$-algebras they generate from the point of view of noncommutative multivariable operator theory. We obtain Wold decompositions, in this setting, and use them to classify the $k$-tuples of doubly $\Lambda$-commuting row isometries up to a unitary equivalence. We introduce a universal model in this setting, describe its invariant subspaces, and develop a dilation theory on $\Lambda$-polyballs.