RIGHT CANCELLATION IN THE ${\cal L}{\cal U}{\cal C}$-COMPACTIFICATION OF A LOCALLY COMPACT GROUP

Abstract

Write $G^{\ast} = G^{{\cal L}{\cal U}{\cal C}}\setminus G$ where $G^{{\cal L}{\cal U}{\cal C}}$ is the largest semigroup compactification of the locally compact group $G$ . Then the set of points of $G^{\ast}$ which are right cancellable in $G^{{\cal L}{\cal U}{\cal C}}$ is large; in fact it has an interior in $G^{\ast}$ which is dense in $G^{\ast}$ . Corollaries are given about the number of left ideals in $G^{{\cal L}{\cal U}{\cal C}}$ and the size of right ideals in the algebra ${{\cal L}{\cal U}{\cal C}}(G)^{\ast}$ .