Topological grading of semigroup $C^*$-algebras and its applications

The main object of the talk is the $C^*$-algebra generated by the left regular isometric representation of a left cancellative semigroup. This $C^*$-algebra is called the reduced semigroup $C^*$-algebra. We describe a method for constructing a topological grading of the reduced semigroup $C^*$-algebra. The method is based on the notion of the index of operator monomial which has been introduced by the author earlier.
The topological grading is applied to the study of structures and properties of Banach and Hilbert modules on the underlying space of the reduced semigroup $C^*$-algebra. In particular, we formulate conditions under which these modules are free and projective.
To demonstrate the results, we give examples.