Abstract
Let F = R or C. For i = 1; 2, let K-i be a locally compact (Hausdorff) topological space and let Hi be a closed subspace of C-0(K-i, F) such that each point of the Choquet boundary Ch(Hi) Ki of Hi is a weak peak point.
We show that if there exists an isomorphism T : H-1 -> H-2 with parallel to T parallel to.parallel to T-1 parallel to < 2, then Ch(H1) K-1 is homeomorphic to Ch(H2) K-2. We then provide a one-sided version of this result.
Finally we prove that under the assumption on weak peak points the Choquet boundaries have the same cardinality provided H-1 is isomorphic to H-2.