Standard noncommuting and commuting dilations of commuting tuples

We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction, there are two commonly used dilations in multivariable operator theory. First there is the minimal isometric dilation consisting of isometries with orthogonal ranges, and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of the Cuntz algebra coming from dilations of commuting tuples.