I was reading Gerald Folland's A Course in Abstract Harmonic Analysis where the involution operation $f^*(x)=\overline{f(-x)}$ is defined over functions in $L^1(\mathbb{R})$. This is done in order to make $L^1(\mathbb{R})$ a $*$-algebra with convolution being the product operation.
But, I am unable to understand why the inner negative sign in $\overline{f(-x)}$ is necessary. I was able to verify that $f^*(x)=\overline{f(x)}$ satisfies all the properties required in the definition of involution (These are $(x+y)^*=x^*+y^*$, $(\lambda x)^*=\overline{\lambda}x^*$, $(xy)^*=y^*x^*$ and $x^{**}=x$).
Is there a good reason why the involution operation should be defined as $f^*(x)=\overline{f(-x)}$ instead of $f^*(x)=\overline{f(x)}$? Are there any important properties of $*$-algebras that are lost if the latter definition is used instead of the former?