By Anton Deitmar, Siegfried Echterhoff (auth.)

This publication bargains an entire and streamlined therapy of the relevant rules of abelian harmonic research: Pontryagin duality, the Plancherel theorem and the Poisson summation formulation, in addition to their respective generalizations to non-abelian teams, together with the Selberg hint formulation. the foundations are then utilized to spectral research of Heisenberg manifolds and Riemann surfaces. This new version encompasses a new bankruptcy on p-adic and adelic teams, in addition to a complementary part on direct and projective limits. the various assisting proofs were revised and subtle. The publication is a wonderful source for graduate scholars who desire to research and comprehend harmonic research and for researchers trying to follow it.

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For the last assertion, we need a lemma. 3 (a) Let X be a locally compact Hausdorff space and let x1 , . . , xn ∈ X pairwise different points. Let λ1 , . . , λn ∈ C any given numbers. Then there exists f ∈ Cc (G) with f (xj ) = λj for each j = 1, . . , n. (b) Let G be a locally compact group and g ∈ Cc (G) with the property that G g(y) dy = G |g(y)| dy. Then there is θ ∈ T such that g(x) ∈ θ [0, ∞) for every x ∈ G. Proof (a) By the Hausdorff property there are open sets Ui,j and Vi,j for i = j with xi ∈ Ui,j , xj ∈ Vi,j and Ui,j ∩ Vi,j = ∅.

Therefore, G is Hausdorff. (c) Now suppose the system is a Mittag-Leffler direct system and that all Gi are Hausdorff. Let i ∈ I and fix k0 ∈ I such that Hi = ker φik = ker φik0 holds for every k ≥ k0 . Then the closed subgroup Hi is also the kernel of ψi , so G is Hausdorff by part (b). (d) Finally, suppose that all Gi are locally compact groups and the kernels ker(ψi ) are closed. Then G is Hausdorff by (b), further, as each ψi : Gi → G is open as well, a compact unit neighborhood U inside Gi maps to a compact unit neighborhood in G, which therefore is locally compact.

9 Exercises 35 (c) Show that H f ( g x) dμH (x) = δ(g) that δ is continuous. (d) Show that a Haar integral on H H f (x) dμH (x) for f ∈ Cc (H ) and deduce G is given by f (h, g)δ(g) dμH (h) dμG (g). 12 For a finite group G define the group algebra C[G] to be a vector space of dimension equal to the group order |G|, with a special basis (vg )g∈G , and def equipped with a multiplication vg vg =vgg . Show that C[G] indeed is an algebra over C. Show that the linear map vg → 1{g} is an isomorphism of C[G] to the convolution algebra L1 (G).

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