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Abstract

We study dilations of $q$-commuting tuples. Bhat, Bhattacharyya and Dey gave the correspondence between the two standard dilations of commuting tuples and here these results are extended to $q$-commuting tuples. We are able to do this when the $q$-coefficients $q_{ij}$ are of modulus one. We introduce a “maximal $q$-commuting subspace” of an $n$-tuple of operators and a “standard $q$-commuting dilation”. Our main result is that the maximal $q$-commuting subspace of the standard noncommuting dilation of a $q$-commuting tuple is the standard $q$-commuting dilation. We also introduce the $q$-commuting Fock space as the maximal $q$-commuting subspace of the full Fock space and give a formula for a projection operator onto this space. This formula helps us in working with the completely positive maps arising in our study. We prove the first version of the Main Theorem (Theorem 21) of the paper for normal tuples by applying some tricky norm estimates and then use it to prove the general version of this theorem. We also study the distribution of a standard tuple associated with the $q$-commuting Fock space and related operator spaces.