  In the theory of C*-algebras, interesting noncommutative structures arise as
deformations of the tensor product. For instance, the rotation algebra may be
seen as a scalar twist deformation of the tensor product of the functions on
the circle with itself. We deform the tensor product of two Toeplitz algebras
in the same way, introducing the universal C*-algebra generated by two
isometries u and v such that uv=e^{it}vu and u*v=e^{-it}vu*, for a fixed real
parameter t. Since the second relation implies the first one, we also consider
the universal C*-algebra generated by two isometries u and v with the weaker
relation uv=e^{it}vu. Such a "weaker case" does not exist in the case of
unitaries, and it turns out to be much more interesting than the twisted
"tensor product case" of two Toeplitz algebras. We show that the C*-algebra in
the "tensor product case" is nuclear, whereas in the "weaker case" it is not
even exact. Also, we compute the K-groups and we obtain K_0 = Z and K_1 = 0 for
both C*-algebras. This answers a question raised by Murphy in 1994 concerning
the K-theory of the C*-algebra associated to N^2.
