Fourier Algebras of finite groups

• Main Authors: Yau-Lun Wong, 黃有鄰
• Other Authors: Ya-Shu Wang
• Format: Others
• Language: en_US
• Published: 2019
• Online Access: http://ndltd.ncl.edu.tw/cgi-bin/gs32/gsweb.cgi/login?o=dnclcdr&s=id=%22107NCHU5507013%22.&searchmode=basic

碩士 === 國立中興大學 === 應用數學系所 === 107 === Let G be a locally compact group and A(G) be the Fourier algebra consisting of matrix coefficients f(a)= ＜ π(a)h, k>, ∀ a∈ G, of any continuous unitary representation π: G→U(H) of G vanishing at infinity, where U(H) is the topological group of unitary operators on the Hilbert space H and h,k are vectors in H. For a finite (discrete) group G, the Fourier algebra A(G) consists of all functions from G into the complex field. Being a finite dimensional algebra, A(G) only determines the order o(G) of G; namely, dim A(G) = o(G). Nevertheless, the function algebra A(G) does not encode other information of G. However, we can further equip A(G) with the norm and order structure. In this thesis, we study the case when G is a finite group, through the information we obtain from the set P1(G) of norm one positive definite functions on G. Here, P1(G) = {＜ π(.)h,h> ∈ A(G) | π:G → U(H)is a unitary representation, h∈ H, |h|=1}. We will see that P1(G) determines G in many situations, especially when G is a finite abelian group. We summarize below our, seemly new, findings. Assume we are given the convex set P1(G) of positive definite functions of a finite group G. (a) We can construct both the group von Neumann algebra vN(G) and the lattice L(G) of the subgroups of G. (b) We can tell if G is abelian, cyclic, simple, perfect, solvable, supersolvable, or nilpotent. In the case when G is abelian, we can determine G as a direct product of its cyclic subgroups of prime power orders. (c) Let G'' be a finite simple group such that P1(G'') and P1(G) are isomorphic as partially ordered sets. Then G'' and G are isomorphic as groups.