Let G be a discrete group and (G, G+) be a quasi-ordered group.
Set G0+ = G+∩(G+)−1 and
G1 = (G+[setmn ]G0+)∪{e}. Let
[Fscr ]G1(G) and [Fscr ]G+(G) be the
corresponding Toeplitz algebras. In the paper, a necessary and sufficient condition for a representation of
[Fscr ]G+(G) to be faithful is given. It is proved that when G is
abelian, there exists a natural C*-algebra morphism from
[Fscr ]G1(G) to [Fscr ]G+(G).
As an application, it is shown that when G = ℤ2 and
G+ = ℤ+ × ℤ, the K-groups
K0([Fscr ]G1(G)) ≅ ℤ2,
K1([Fscr ]G1(G)) ≅ ℤ and all Fredholm
operators in [Fscr ]G1(G) are of index zero.