Invertibility in G-space groupiod C^*-algebras

Abstract

Consider a second-countable, Hausdorff, etale, amenable groupoid
 with compact unit space X=  and a G-space (Y, p), we indicate that for every
(a; y)  C*(Y × ) is invertible if and only if  ×  is invertible for all

(y; x)   , where  × refers to the regular representation of C*(Y × )
on  (Y × ). We also prove thise where for every (y; a) in C*( ), there are
some (y; a)   such that ǁ(y; a) ǁ = ǁ  ×  (y; a) ǁ.