Standard dilations of q-commuting tuples

Here we study dilations of q-commuting tuples. In [BBD] the authors gave the correspondence between the two standard dilations of commuting tuples and here these results have been extended to q-commuting tuples. We are able to do this when q-coefficients `$q_{ij}$' are of modulus one. We introduce `maximal q-commuting subspace ' of a $n$-tuple of operators and `standard q-commuting dilation'. Our main result is that the maximal q-commuting subspace of the standard noncommuting dilation of q-commuting tuple is the `standard q-commuting dilation'. We also introduce q-commuting Fock space as the maximal q-commuting subspace of full Fock space and give a formula for projection operator onto this space. This formula for projection helps us in working with the completely positive maps arising in our study. The first version of the Main Theorem (Theorem 19) of the paper for normal tuples using some tricky norm estimates and then use it to prove the general version of this theorem.