  There are three main results in this paper. First, we find an easily
computable and simple condition which is necessary and sufficient for a
commuting tuple of contractions to possess a non-zero Toeplitz operator. This
condition is just that the adjoint of the product of the contractions is not
pure. On one hand this brings out the importance of the product of the
contractions and on the other hand, the non-pureness turns out to be equivalent
to the existence of a pseudo-extension to a tuple of commuting unitaries. The
second main result is a commutant pseudo-extension theorem obtained by studying
the unique canonical unitary pseudo-extension of a tuple of commuting
contractions. The third one is about the $C^*$-algebra generated by the
Toeplitz operators determined by a commuting tuple of contractions. With the
help of a special completely positive map, a different proof of the existence
of the unique canonical unitary pseudo-extension is given.
