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. Firstly
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 Cuntz algebra O_n coming from dilations of commuting tuples.Comment: 18 pages, Latex, 1 commuting diagra
