On the Spectrum of L^∞ (G)

Abstract

We investigate the algebraic structure of the spectrum Ω of L∞(G) for a locally compact group G. In contrast to the compact and discrete cases, when G has neither of these properties, Ω is never a semigroup. For σcompact G we determine exactly when the product of two elements of Ω. is in Ω, but we present an example which suggests that for general groups the underlying set theory may have an effect. Our principal tool, which has independent interest, is a topological structure theorem for the LB‐compactification of an arbitrary locally compact group.

Slowly oscillating functions in semigroup compactifications and convolution algebras

“…Unlike G LUC , the spectrum Ω of L ∞ (G) is a semigroup with respect to the Arens multiplication if and only if G is discrete or compact [29]. If G is discrete, then L ∞ (G) = LUC(G) = ∞ (G), and so Ω, G LUC and the Stone-Čech compactification βG of G all agree.…”

Second duals of measure algebras

“…Indeed, we have noted in Corollary 6.3 that (Φ {e} , 2 ) is a left-zero semigroup. The above proposition does not extend to all locally compact groups G. Indeed, it is shown in [81,Corollary 4.4] that (Φ, 2 ) is a semigroup if and only if G is either compact or discrete; we state this result in the following form. Proposition 8.3.…”

“…Of course, Φ is a clopen subset of G, and π(Φ) = G. Thus we may suppose that the family {Ω i : i ∈ I} of subsets of G described in Proposition 4.8 contains the singletons {x} for x ∈ G and the compact space Φ. The space Φ is the topic of the paper [81], where it is called the spectrum of L ∞ (G). For x ∈ G, we set…”

“…Proof. The topological space (G, τ ) is neither compact nor discrete, and so, by [81], (Φ(τ ), 2 ) is not a semigroup. It follows from Proposition 8.23 that ( G, 2 ) is not a semigroup.…”

Banach algebras on semigroups and on their compactifications

“…(The character space Φ L ∞ (G) is discussed in [101]; it is a very complicated object, and it is not a semigroup unless G is discrete or compact. See also [24] for a long discussion of Φ L ∞ (G) .…”