Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems

We examine the problem of the linearity of an algebraic torus action in the associative setting. We prove the free algebra analog of a classical theorem of Białynicki-Birula, which establishes linearity of maximal torus action. Additionally, we formulate and prove linearity theorems for specific classes of regular actions, and provide a framework for constructing non-linearizable actions, analogous to the work of Asanuma. This framework has applications in the study of the Associative Cancellation Conjecture. Furthermore, we show the existence of two non-isomorphic algebras, whose free products with a polynomial ring are isomorphic.