Mathematics

Dr. Rohit D. Holkar
IISER Pune

A complex Banach *-algebra satisfying the C-condition, jjxxjj = jjxjj2, is
called a C-algebra. Topological spaces and groups have their naturally associ-
ated C-algebras; the former is the space of continuous functions vanishing at
innity and the latter is a completion of L1 of a group in a certain norm, re-
spectively. When equipped with the *-homomorphisms, the category of abelian
C-algebras is isomorphic to that of the locally compact spaces. This result
establishes a connection between topology and C-algebras which is further
strengthened by the equivalence between the K-theory of C-algebras and the
K-theory of Atiyah. We shall introduce C-algebra: discuss the basics, topolog-
ical and geometrical counterparts of C-algebras, tools to work in this area and
some current research topics. Groupoids play a major role in the topological
counterpart of C-algebras and they are main tools for our research. Hence we
shall also discuss groupoids and their role in this area