Let G be a locally compact group with left Haar measure $\mu$. In Principles of Harmonic Analysis, it's affirmed that $L^1(G)$ is a Banach *-Algebra, where the multiplication operation is convolution and the involution is $f^{\ast} (x) := \Delta(x^{-1})\overline{f(x^{-1})}$, where $\Delta$ is the modular function. However I'm particularly struggling to see why the involution should switch the order of convolution:
If $f, g$ are in $L^1(G)$ and x is in G,then:
$$ (f \ast g)^{\ast} (x) = \Delta(x^{-1}) \overline{ \int_G f(y) g(y^{-1}x^{-1})dy} = \overline{\int_G f(yx) g(y^{-1}x^{-1}x)dy} = \overline{\int_G f(yx) g(y^{-1})dy}$$
But $$(g^{\ast} \ast f^{\ast})(x) = \int_G g^{\ast}(y) f^{\ast}(y^{-1}x)dy = \int_G \Delta(y^{-1}) \overline{g(y^{-1})} \Delta(x^{-1}y) \overline{f(x^{-1}y)}dy = \overline{\int_G g(y^{-1}x) f(x^{-1}yx) dy} = \overline{\int_G g(xy^{-1}x) f(yx) dy} $$
The two ending expressions don't seem to have any reason to be equal. So I would like to know where the mistake is.