The second dual L 1 (G) * * of the group algebra has a well-known Arens multiplication (μ, ν) → μν which is weak * continuous in the μ variable. In contrast, if ν → μν is weak * continuous at every point, then μ ∈ L 1 (G). A significant question is whether continuity at all points ν is necessary for this conclusion, and there has been a long-standing conjecture that continuity at just two specified points might be enough. The main conclusion of the present paper is that the minimal number is in fact just one. However, a second theorem considers a closely related question for which two points are required. The methods yield similar answers to corresponding problems about the quotient algebra LUC(G)

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

No abstract

Abstract. In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal compact subgroup. In particular, if G = KAN is an Iwasawa decomposition of the group G, then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer M of AN in K. Complete descriptions are given for the enveloping semigroups in the cases G = SL(2, C) and G = SL(2, R).

No abstract