Invariant means on double coset spaces
- PDF / 377,602 Bytes
- 8 Pages / 439.37 x 666.142 pts Page_size
- 45 Downloads / 193 Views
Invariant means on double coset spaces László Székelyhidi1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract In this paper we study invariant means on and amenability of double coset spaces. We prove the amenability of Gelfand pairs. As an application we prove a stability theorem for a functional equation related to spherical functions. Keywords Invariant mean · Amenability · Gelfand pair · Stability Mathematics Subject Classification 43A07 · 43A90 · 39A30
1 Introduction Given a Hausdorff topological space X we denote by C (X ), resp. BC (X ) the set of all continuous complex valued functions, resp. the set of all bounded continuous complex valued functions on X . We equip C (X ) with the topology of uniform convergence on compact sets, and BC (X ) with the topology arising from the supremum norm. Then C (X ) is a locally convex topological vector space and BC (X ) is a Banach space. The identity mapping is a continuous embedding of BC (X ) into C (X ). Let G be a locally compact group, and K a compact subgroup with normed Haar measure ω. For each x in G the double coset K x K of x with respect to K is defined as the set K x K = {kxl : k, l ∈ K }. Obviously, every double coset is compact. Proposition 1.1 All double cosets form a partition of G.
The research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. K111651.
B 1
László Székelyhidi [email protected] Institute of Mathematics, University of Debrecen, Debrecen, Hungary
123
L. Székelyhidi
Proof We have to show only that given x, y in G the two double cosets K x K and K y K are either disjoint, or they coincide. Suppose that z belongs to the intersection of K x K and K y K , then we have z = k1 xl1 = k2 yl2 for some k1 , k2 , l1 , l2 in K . It follows that x is in K y K , hence K x K ⊆ K y K . Similarly, we have K y K ⊆ K x K , hence the two double cosets coincide. We equip the set of all double cosets with the factor topology arising from the equivalence relation corresponding to the partition formed by all double cosets. The resulting topological space G//K is locally compact and it is called the double coset space corresponding to the compact subgroup K . The natural mapping defined by (x) = K x K from G onto G//K is a continuous and open mapping. We call the function f : G → C K -invariant if f (kxl) = f (x) holds for each k, l in K and x in G. Clearly, all continuous K -invariant functions on G form a closed subspace in C (G), which is topologically isomorphic to the space C (G//K ), and the topological isomorphism is given by the mapping f → f , where f (K x K ) = f (x) for each x in G, where f is in C (G). In other words, we have f ◦ = f. For the sake of simplicity we shall identify f and f. The K -projection of each f in C (G) is defined as the function f (kxl) dω(k) dω(l) f # (x) = K
K
for each x in G. Then f → f # is a continuous linear mapping of C (G) onto C (G//K ). Clearly, f is K -invariant if and only if f = f # . Let Mc (G) denote the space of all compactly suppo
Data Loading...
