Let $R$ and $L$ be the right and left regular representations of an arbitrary locally compact group G, respectively. We know that $R$ and $L$ are unitary with respect to the right and left Haar measure on $G$, respectively.
My question: Are they equivalent representations for all locally compact groups?
If we take the linear operator $A:L^2(G)\to L^2(G)$ defined by (Af)$(x)=f(x^{-1})$, then we have $AR(t)=L(t)A$ for all $t\in G$. But, to show that the unitarity of $A$, for any $f,h\in L^2(G)$ we need to have that \begin{align} \int_G f(x^{-1})\overline{h(x^{-1})}dx=\int_Gf(x)\overline{h(x)}dx \end{align} I am not sure that the above equation is true for all locally compact groups. Also it is possible that there is another intertwining operator $A$ that makes $R$ and $L$ equivalent. Thanks.