The study of relation between existence of admissible vectors and amenability and compactness of a locally compact group

The existence of admissible vectors for a locally compact group is closely related to the group's profile. In the compact groups, according to Peter-weyl theorem, every irreducible representation has admissible vector. In this paper, the conditions under which the inverse of this case is being investigated has been investigated. Conditions such as views that are admissible and stable will get compactness results for the group. SIN-groups are also compact with irreducible representations that have admissible vectors. Study of this part of harmonic analysis results in the study of the properties of the rajhman algebra , AR-groups, and SIN-groups. Given that with an admissible algorithm, an isometry of the Hilbert space is displayed in L ^ 2 (G), and this is necessary to look at the corresponding representation assubrepresentation of λ_G. Examining the properties of the subrepresentation of λ_G of the consequences of the admissible vector's existence. One of the important issues in the harmonic analysis is the writing of a left regular representation in the form of direct sum of irreducible representations. In this article, we use this important issue with the existence of admissible vectors composition and use it for the compactness properties of the group. According to the reference which evaluates the amenability display abilities, the poor appearance of the views is presented, so the first result is the results that are admissibility for the unpublished show.