  The treatment of supersymmetry is known to cause difficulties in the
C*-algebraic framework of relativistic quantum field theory; several no-go
theorems indicate that super-derivations and super-KMS functionals must be
quite singular objects in a C*-algebraic setting. In order to clarify the
situation, a simple supersymmetric chiral field theory of a free Fermi and Bose
field defined on $\R$ is analyzed. It is shown that a meaningful C*-version of
this model can be based on the tensor product of a CAR-algebra and a novel
version of a CCR-algebra, the "resolvent algebra". The elements of this
resolvent algebra serve as mollifiers for the super-derivation. Within this
model, unbounded (yet locally bounded) graded KMS-functionals are constructed
and proven to be supersymmetric. From these KMS-functionals, Chern characters
are obtained by generalizing formulae of Kastler and of Jaffe, Lesniewski and
Osterwalder. The characters are used to define cyclic cocycles in the sense of
Connes' noncommutative geometry which are "locally entire".
