Endowing the vector space of integrable functions with a product given by the convolution turns it into a commutative associative algebra without identity called the "convolution algebra", which has been studied extensively.
It seems to me that the vector space of real-valued integrable functions forms a very similar "cross-correlation algebra" with the product given by the cross-correlation. By basically the exact same reasoning as for the convolution, this is also an associative algebra over the reals. The only thing that makes it a little less "nice" is that it isn't commutative (although it is if we restrict to the subspace of even functions). However, I can't find a single mention of any "cross-correlation algebra" anywhere online. Is there any reason why this algebra is less natural or interesting to study than the convolution algebra?
(The cross-correlation also gives you an algebraic structure over the vector space of complex integrable functions that's pretty close to being an algebra, except that the product operation is sesquilinear instead of bilinear. Are "sesquilinear algebras" ever studied, or is there some reason why they're less interesting/natural than the standard bilinear algebras?)
Edit: On second thought, I'm not sure that this algebra is associative because of the complex conjugation. I think it's still an algebra, but I see why the fact that it (may not be) either commutative or associate makes it a lot less "nice" than the convolution algebra.