# Contractivity vs. Complete Contractivity via property P

## TL;DR

This paper investigates the distinction between contractivity and complete contractivity for certain domains in complex space, showing that many do not have Property P and providing proofs for specific Banach spaces.

## Contribution

It demonstrates that a broad class of Reinhardt domains lack Property P, leading to contractive but not completely contractive homomorphisms, and offers a simple proof for specific Banach spaces.

## Key findings

- Many Reinhardt domains lack Property P.
- Existence of contractive but not completely contractive homomorphisms.
- Proof that certain $oldsymbol{ ext{C}^n}$ spaces have Property P.

## Abstract

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