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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeAug 22nd 2023

    If GG is an arbitrary monoid with multiplication m:G×GGm:G\times G\to G then mm induces a map m *:Fun(G,k)Fun(G×G,k)m^*:Fun(G,k)\to Fun(G\times G,k), m *(f):ffmm^*(f):f\mapsto f\circ m. We say that fFun(G,k)f\in Fun(G,k) is representative if m *(f)m^*(f) is in the image of the canonical map Fun(G,k)Fun(G,k)Fun(G×G,k)Fun(G,k)\otimes Fun(G,k)\hookrightarrow Fun(G\times G,k). Equivalently, ff is representative if the span of all functions gf:hf(hg)g\cdot f : h\mapsto f(h\cdot g) is finite dimensional. It follows then that m *(f)m^*(f) is in fact in (the image of) R(G)R(G)R(G)\otimes R(G) where R(G)R(G) is the space of all representative functions on GG.

    Peter-Weyl theorem says that the continuous representative functions form a dense subspace of the space of all continuous functions on a compact Lie group GG.

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