The aim of this lecture is to find mathematical models for the
observable algebras of quantum systems. We will take the necessary
requirements from physics as orientation to pass from general
algebraic structures, *-algebras and their states, to more analytic
versions. Analysis will be needed to arrive at meaningful and
manageable algebras. This will include among others the class of
C*-algebras. Having found this class as physically very appealing
algebras we will study their properties in detail. Here the spectral
calculus plays a dominant role. While having already very nice
properties, an abstract C*-algebra is not yet enough for physical
models: one needs to implement the abstract algebras as an algebra of
operators on a Hilbert space. Thus we will investigate the bounded
operators on a Hilbert space from a more conceptual point of view to
establish the bounded-measurable functional calculus for Hermitian
operators. If time permits, we will extend our discussion then to
unbounded operators and investigate the spectral theory of possibly
unbounded self-adjoint operators in Hilbert spaces.