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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeAug 22nd 2023
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIf $G$ is an arbitrary monoid with multiplication $m:G\times G\to G$ then $m$ induces a map $m^*:Fun(G,k)\to Fun(G\times G,k)$, $m^*(f):f\mapsto f\circ m$. We say that $f\in Fun(G,k)$ is representative if $m^*(f)$ is in the image of the canonical map $Fun(G,k)\otimes Fun(G,k)\hookrightarrow Fun(G\times G,k)$. Equivalently, $f$ is representative if the span of all functions $g\cdot f : h\mapsto f(h\cdot g)$ is finite dimensional. It follows then that $m^*(f)$ is in fact in (the image of) $R(G)\otimes R(G)$ where $R(G)$ is the space of all representative functions on $G$. [[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 $G$. <a href="https://ncatlab.org/nlab/revision/diff/representative+function/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/representative+function/4">v4</a>, <a href="https://ncatlab.org/nlab/show/representative+function">current</a>

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

   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 $G$.

   diff, v4, current