Acta Math. Hungar. 63 (3) (1994), Z91-303. COMPLETELY CONTRACTIVE HILBERT MODULES AND PARROTT'S EXAMPLE G. MISRA 1 (Bangalore) 1. Introduction In two earlier papers [9, 10] the present author together with Sastry studied certain finite dimensional Hilbert modules C~ +1 over the function algebra A(~) for fl a domain in C m. This paper is a continuation of that work and provides partial answers to some of the questions raised in [10] for the poly disk algebra. While most of the terminology and notations are from the two papers [9, 10] and will be used without any further apology, we point out in Remark 3.8 that the contractive module C2N n (respectively completely contractive) gives rise to a matricially normed 2m-dimensional vector space and a contractive (respectively completely contractive) linear map on it and conversely. In the two papers cited above the main result showed that a contractive module C n+l over the ball algebra A(B m) is completely bounded by v/m and examples were given to show that the bound is attained. This, in particular shows that for m _>__ 2, contractive modules are not necessarily completely contractive over the ball algebra. However, for the poly disk algebra .A (Dm), we know via Ando's theorem [2] that every contractive module over .A (D 2) is completely contractive while Parrott [11] provides an example of a contractive module over A (D 3) which is not completely contractive. As Paulsen [12] points out, it would be good to know the difference in the internal structure of JI (D 2) and A (D 3) that leads to this situation, see 4.4 for a partial answer. Our approach is to actually work out Parrot's example using the notion of complete contractivity rather than dilation, these notions are of course equivalent [cf. 12]. The methods of [9, 10], seem to work well in the context of the ball algebra but the actual computations over the poly disk algebra seem to be very messy. In fact, we are not able to produce an example of a contractive module over .A (D 3) which is not completely contractive within the class of the very simple modules considering in [9, 10], however see Remark 4.8. Therefore, we are forced to consider slightly more general module action than those of This work was supported by a grant from the NSF.