The theory of unitary representations of groups is closely connected with harmonic analysis.

Conversely, given a positive-definite function, one can define a unitary representation of G in a natural way.

Note that is a representation, but not a unitary representation.

Now, one looks for unitary representations of this quantum algebra.

It does however have a well-known family of infinite-dimensional unitary representations.

Another approach involves considering all unitary representations, not just the irreducible ones.

Remember we are talking about unitary representations here, which means these values have to be real.

It can be checked that the mapping is a projective unitary representation.

For a discussion of such unitary representations, see Wigner's classification.

We know that any irreducible representation can be turned into a unitary representation.