Compact quantum subgroups and left invariant C⁎-subalgebras of locally compact quantum groups

We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group G and certain left invariant C⁎-subalgebras of C0(G). We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group G and non-zero, invariant C⁎-subalgebras of the group C⁎-algebra .