The projective representation of groups was introduced in 1904 by Issai Schur. It differs from the normal representation of groups by a twisting factor, which we call Schur function in this book and which is called sometimes normalized factor set in the literature (other names are also used). It starts with a discret group T and a Schur function f for T. This is a scalar valued function on T^2 satisfying the conditions f(1,1)=1 and
|f(s,t)|=1, f(r,s)f(rs,t)=f(r,st)f(s,t)
for all r,s,t in T. The projective representation of T twisted by f is a unital C*-subalgebra of the C*-algebra L(l^2(T)) of operators on the Hilbert space l^2(T). This reprezentation can be used in order to construct many examples of C*-algebras. By replacing the scalars R or C with an arbitrary unital (real or complex) C*-algebra E the field of applications is enhanced in an essential way. In this case l^2(T) is replaced by the Hilbert right E-module E tensor l^2(T) and L(l^2(T)) is replaced by the C*-algebra of adjointable operators on E tensor l^2(T). We call Schur product of E and T the resulting C*-algebra (in analogy to cross products which inspired the present construction). It opens a way to creat new K-theories (see the draft "Axiomatic K-theory for C*-algebras").
In a first chapter we introduce some results which are needed for this construction, which is developed in a second chapter. In the third chapter we present examples of C*-algebras obtained by this method. The classical Clifford Algebras (including the infinite dimensional ones) are C*-algbras which can be obtained by projective representations of certain groups. The last chapter of this book is dedicated to the generalization of these Clifford Algebras as an example of Schur products.
Projective representations of reflection groups