Almost contractive maps between C⁎-algebras with applications to Fourier algebras

We prove that unital almost contractive maps between C⁎-algebras enjoy approximately certain properties of unital positive maps (such as selfadjointness or the Kadison-Schwarz inequality). Our main application is a description of almost contractive homomorphisms between Fourier algebras and Fourier-Stieltjes algebras: they are actually contractive if their norm is smaller than 1.00018. For surjective isomorphisms of Fourier algebras, the bound 1.0005 is sufficient in order to obtain an isometry.