  It is shown that if a representation of a *-algebra on a vector space $V$ is
an irreducible *-representation with respect to some inner product on $V$ then
under appropriate technical conditions this property determines the inner
product uniquely up to a constant factor. Ashtekar has suggested using the
condition that a given representation of the algebra of quantum observables is
a *-representation to fix the inner product on the space of physical states.
This idea is of particular interest for the quantisation of gravity where an
obvious prescription for defining an inner product is lacking. The results of
this paper show rigorously that Ashtekar's criterion does suffice to determine
the inner product in very general circumstances. Two versions of the result are
proved: a simpler one which only applies to representations by bounded
operators and a more general one which allows for unbounded operators. Some
concrete examples are worked out in order to illustrate the meaning and range
of applicability of the general theorems.
