Local Limit Theorems and Unitary Representations

The unitary representations of free groups, introduced in the last chapter, are now used to prove estimates on the asymptotic behavior of the sequence of convolution powers of nontrivial radial probability measures on free groups. These results are called local limit theorems. They can also be thought of as results concerning the limit points, in the weak operator topology, of the sequence of iterates of the operator λ(μ₁), where, as usual, λ is the left regular representation of F r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071747/9455763b-cb5e-4c79-bdd6-3c0df1a0015b/content/eq1336.tif"/> . From this point of view it is natural to ask the same questions of the sequence of iterates of π(μ₁), where π is any unitary representation having a radial coefficient. This question is considered in Section 3, where results are obtained for the operators π₁ (μ₁) with γ(z) real and 2 r − 1 / r ≤ | γ(z) | < 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071747/9455763b-cb5e-4c79-bdd6-3c0df1a0015b/content/eq1337.tif"/> . It is easy to see, by (3.1), (3.2) and Chapter 5, 3–13), that no similar result can hold for π_z(μ₁) with − 2 r − 1 / r < γ ( z ) < 2 r − 1 / r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071747/9455763b-cb5e-4c79-bdd6-3c0df1a0015b/content/eq1338.tif"/> .