Let $G$ be a locally compact group. Then the space $L^1(G)$ with the convolution $$f\star g = \int_G\ f(xy^{-1}) g(y)\ d\mu(y)$$
is a $*$-algebra ($\mu$ is the left Haar measure). The involution is given by $$f^*(x) = \Delta^{-1}(x) \overline{f}(x^{-1}).$$
Following the paper (page 6, at the bottom):
https://dmitripavlov.org/scans/terp-fourier.pdf
let $VN^1(G)$ denote the smallest von Neumann algebra generated by $$\bigg\{L_f \ : \ f\in L^1(G)\bigg\},$$
where $L_f(g) = f\star g$. It would be beautiful if $(L_f)^* = L_{f^*}$, but that does not seem to be the case (for non-unimodular groups). Here is my reasoning:
\begin{gather*} \langle g| (L_f)^*h\rangle = \langle L_fg|h \rangle = \int_G\ f\star g(x)\ \overline{h}(x)\ dx \\ = \int_G\ \int_G\ f(xy^{-1})g(y)\ dy\ \overline{h}(x)\ dx \stackrel{Fubini}{=} \int_G\ g(y)\ \int_G\ f(xy^{-1})\ \overline{h}(x)\ dx\ dy \\ = \int_G\ g(y)\ \overline{\int_G\ \overline{f}(xy^{-1})\ h(x)\ dx}\ dy = \int_G\ g(y)\ \overline{\int_G\ \widehat{f}(yx^{-1})\ h(x)\ dx}\ dy \\ = \int_G\ g(y)\ \overline{L_{\widehat{f}}h}\ dy , \end{gather*}
where $\widehat{f} = \overline{f(x^{-1})}$. Hence $(L_f)^* = L_{\widehat{f}}$, which is kind of ugly. My question is: does it have to be that way or am I stubborn making some stupid computational mistake i.e. missing $\Delta^{-1}(x)$? Thank you in advance for any tips!