Invariant means and invariant ideals in L∞( G) for a locally compact group G

Abstract

For a nondiscrete σ-compact locally compact Hausdorff group G, L ∞( G) is a commutative Banach algebra under pointwise multiplication which has many nonzero proper closed invariant ideals; there is at least a continuum of maximal invariant ideals { N α } such that N α 1 + N α 2 = L ∞( G) whenever α 1 ≠ α 2. It follows from the construction of these ideals that when G is also amenable as a discrete group, then LIM⧹TLIM contains at least a continuum of mutually singular elements each of which is singular to any element of TLIM. The supports of left-invariant means are in the maximal ideal space of L ∞( G); the structure of these supports leads to the notion of stationary and transitive maximal ideals. To prove that both these types of maximal ideals are dense among all maximal ideals, one shows that the intersection of all nonzero closed invariant ideals is zero. This is the case even though the intersection of any sequence of closed invariant ideals is not zero and the intersection of all the maximal invariant ideals is not zero.