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In this paper, we consider the question of contractivity vs. completecontractivity for domains in $\mathbb{C}^2$, which are unit balls with respectto some norm. We show that for a large class of Reinhardt domains, thecorresponding Banach spaces do not have Property P, which implies that thereexists contractive homomorphisms on these domains which are not completelycontractive. At the end, we present a simple proof of the fact that the complexBanach spaces $(\mathbb{C}^2,\|\cdot\|_{\infty})$ and$(\mathbb{C}^3,\|\cdot\|_{\infty})$ have Property P.

Contractivity vs. Complete Contractivity via property P