METRIC SEMIGROUPS THAT DETERMINE LOCALLY COMPACT GROUPS ANTHONY TO-MING LAU, CHI-KEUNG NG, AND NGAI-CHING WONG Abstract. Let G be a locally compact group. Let A be any one of the (complex) Banach algebras: L 1 (G), M(G), WAP(G) and LUC(G), consisting of integrable functions, regular Borel complex measu- res, weakly almost periodic functions, and bounded left uniformly continuous functions, respectively, on G. We show that the metric semigroup A 1 + := {f ∈ A : f ≥ 0 and ∥f ∥ =1} (the convex structure is not considered) is a complete invariant for G. 1. Introduction In this paper, we fnd several new and simple complete invariants for locally compact groups. Let G and H be locally compact groups. Wendel showed in [24] (respectively, Johnson showed in [10]) that G and H are isomorphic if and only if there exists an isometric algebra isomorphism Φ: L 1 (G) → L 1 (H) (respectively, Φ: M (G) → M (H)). Optimistically, as Φ(sf )= sΦ(f ), information in the one dimensional subspace {sf : s ∈ C} is somehow encoded in the element {f }. This leads to a quest of a “smaller invariant”. As a candidate, however, the unit sphere of L 1 (G) is not closed under the convolution product and hence cannot be served as an invariant for G. On the other hand, Kawada showed in [11] that G and H are isomorphic whenever there is an algebra isomorphism Ψ: L 1 (G) → L 1 (H) satisfying: Ψ(f ) ≥ 0 if and only if f ≥ 0. Observe that L 1 (G) 1 + , the positive part of the unit sphere of L 1 (G), is closed under the convolution product. This suggests us to consider L 1 (G) 1 + as a candidate of a complete invariant of G. In this article, we will show that the metric and the semigroup structures of L 1 (G) 1 + , or those of M (G) 1 + , (note that the convexity is not needed) determines G. This result supplements the above mentioned results of Wendel [24], Johnson [10] and Kawada [11]. Furthermore, Ghahramani, Lau and Losert ([8]), as well as Lau and McKennon ([13]), showed that either one of the dual Banach algebras LUC(G) * and WAP(G) * determines G, too. We will also show that the positive parts of the unit spheres of LUC(G) * and WAP(G) * are complete invariants for G. For a subset S ⊆ E of an ordered Banach space E, we set S 1 + := { f ∈ S : ∥f ∥ = 1; f ≥ 0 } . Our main results (namely, Theorems 5 and 6) can be subsumed and simplifed in the following statement. Theorem 1. Two locally compact groups G and H are isomorphic as topological groups if and only if any one of the following holds (1) L 1 (G) 1 + ∼ = L 1 (H) 1 + as metric semigroups; (2) M (G) 1 + ∼ = M (H) 1 + as metric semigroups. (3) (WAP(G) * ) 1 + ∼ = (WAP(H) * ) 1 + as metric semigroups; Date: QJM accepted version as of October 9, 2017. 2010 Mathematics Subject Classifcation. 43A10, 43A20, 46H20, 46L30. Key words and phrases. locally compact groups; group Banach algebras, normal state spaces; W * -algebras. Corresponding author: Ngai-Ching Wong, wong@math.nsysu.edu.tw. The authors are partially supported by NSERC Grant ZC912, National Natural Science Foundation of China (11471168) and Taiwan MOST grant (106-2115-M-110-006-MY2). 1