Closure of Recognizable Languages under Kleene Star: Algebraic Proof?
For a pointer on concatenation-like operations (including Kleene star) and their impact on the algebraic structure, you can look to Schützenberger products (see here [1] for instance).

[1] Pin, Jean-Éric. "Theme and variations on the concatenation product." International Conference on Algebraic Informatics. Springer, Berlin, Heidelberg, 2011, https://www.irif.fr/~jep/PDF/ThemeVariations.pdf