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It is shown that if a representation of a *-algebra on a vector space $V$ isan irreducible *-representation with respect to some inner product on $V$ thenunder appropriate technical conditions this property determines the innerproduct uniquely up to a constant factor. Ashtekar has suggested using thecondition that a given representation of the algebra of quantum observables isa *-representation to fix the inner product on the space of physical states.This idea is of particular interest for the quantisation of gravity where anobvious prescription for defining an inner product is lacking. The results ofthis paper show rigorously that Ashtekar's criterion does suffice to determinethe inner product in very general circumstances. Two versions of the result areproved: a simpler one which only applies to representations by boundedoperators and a more general one which allows for unbounded operators. Someconcrete examples are worked out in order to illustrate the meaning and rangeof applicability of the general theorems.Comment: 10 page

Unique determination of an inner product by adjointness relations in the algebra of quantum observables