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In Folland's A Course in Abstract Harmonic Analysis, he defines for a function $f$ on a topological group $G$, and $y\in G$,

$$L_{y}f:x\to f(y^{-1}x)\text{ and }R_{y}f:x\to f(xy)$$

He then remarks that the $y$ is inverted so that the map $y\mapsto L_{y}$ is a group homomorphism.

But it seems to me that $L_{yx}f = f((yx)^{-1}\cdot) = f(x^{-1}y^{-1}\cdot) = \left[L_{x}\circ L_{y}\right]f$, making the map an anti-homomorphism.

Am I misunderstanding the remark?

roo
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1 Answers1

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For any $z \in G$, $$ \begin{align} \left( L_{yx}f \right)(z) &= f((yx)^{-1}z) \\ &= f(x^{-1} y^{-1} z) \\ &= \left( L_x f \right)(y^{-1}z) \\ &= \left( L_y L_x f \right)(z). \end{align} $$ So, $$ L_{yx} f = L_y L_x f $$ for all $f \in G^*$.

Sammy Black
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  • Thank you, I don't know what I was thinking here. – roo Nov 26 '13 at 05:38
  • What is $G^*$? Is it the space of all mapping from $G$ to $G$? – Simon SMN Mar 25 '23 at 17:30
  • There are several similar but distinct notions of dual group $G^$, depending on context. But, in every case $G^$ is the space of functions $G \to R$, where $R$ is a fixed algebraic object, sometimes the circle group $T$ of unit modulus complex numbers, sometimes the field $\mathbb{R}$ or $\mathbb{C}$. Notice that the calculation is agnostic to what $R$ actually is. – Sammy Black Mar 25 '23 at 18:22
  • @SammyBlack All right, thank you. Folland defines $L_y$ in his book "Real Analysis" as well, and I had some trouble understanding the notation "$L_{yx}f = L_yL_xf$", but I think I got it now. – Simon SMN Mar 25 '23 at 18:59