4 - Unicity Subspaces and Property A

Summary

Introduction

Both Theorems 3.1 and 3.3 (see also Corollary 3.4) provide us with criteria for exactly determining when a given finite-dimensional subspace U of C(K) is a unicity space for C₁(K, μ). Nevertheless there are at least two drawbacks to these criteria. In the first place the criteria are generally very difficult, if not impossible, to verify. Except for certain specific subspaces it is almost always impossible to check whether there exist h and u* satisfying the conditions of Theorem 3.1, or whether the inequalities of Corollary 3.4 are always valid. Secondly, the above-mentioned criteria are measure dependent. That is, a specific subspace U may be a unicity subspace for C₁(K, μ₁), but not for C₁(K, μ₂), where μ₁ and μ₂ are both ‘admissible’ measures on K. See Examples 3.1 and 3.3, and Exercise 8 of Chapter 3 where this phenomenon occurs.

It is therefore natural to ask for a characterization of finite-dimensional subspaces U of C(K) with the property that U is a unicity subspace for C₁(K, μ) for all ‘nice’ measures μ. Perhaps such a characterization, while it would be interesting in and of itself, might also be easier to verify. Perhaps such a characterization will be an intrinsic property of the subspace U. Perhaps it might be possible to list all such subspaces. This is the problem which we investigate in this chapter.

In Section 2 we delineate a condition called Property A.