Algebry operatorów i ich zastosowania w fizyce

In this talk we will discuss a model for a weighted version of Fourier algebras on locally compact groups. Note that the Fourier algebra is the L1-algebra of a co-commutative (l.c.) quantum group. We first introduce the concept of "weight" in this context and some examples of "weights". By introducing a "weight" we finally get a new commutative Banach algebra. If we recall that the spectrum of the Fourier algebra is nothing but the underlying group itself (as a topological space), then it is natural to be interested in determining the spectrum of weighted algebras. We will demonstrate that the spectrum of the resulting commutative Banach algebra is realized inside the complexification of the underlying Lie group by focusing on the case of E(2) and determine them in some concrete cases. There will be a short discussion for the possible quantum extension of this theory in the end of the talk.