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- W3205616919 abstract "We study a finite (discrete) group G through the information we obtain fromP1(G)={〈π(⋅)ξ,ξ〉:π:G→U(H)is unitary,ξ∈H,‖ξ‖=1} of norm one positive definite functions of G, arising from matrix coefficients of any unitary representation π. Below we summarize our results. Knowing P1(G) as a set of functions, when G is a finite abelian group we can determine G≅∏jZpjrj as a direct product of its cyclic subgroups of prime power orders. Knowing P1(G) as a multiplicative semigroup, we can construct the subgroup lattice L(G) of G. With L(G) in stock, we can tell if G is cyclic, simple, perfect, solvable, supersolvable, or nilpotent. When G′ is a finite simple group with P1(G′)≅P1(G) as multiplicative semigroups, we show that G′≅G as groups. Knowing P1(G) as a compact convex set, we can construct the group von Neumann algebra vN(G) of G. Consequently, when G′ is another finite group with P1(G)≅P1(G′) as convex sets, we show that vN(G)≅vN(G′) as von Neumann algebras. In particular, we can tell if G is abelian." @default.
- W3205616919 created "2021-10-25" @default.
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- W3205616919 date "2022-01-01" @default.
- W3205616919 modified "2023-09-25" @default.
- W3205616919 title "How the norm one positive definite functions determine a finite group" @default.
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- W3205616919 doi "https://doi.org/10.1016/j.laa.2021.10.004" @default.
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