  An often used model for quantum theory is to associate to every physical
system a C*-algebra. From a physical point of view it is unclear why operator
algebras would form a good description of nature. In this paper, we find a set
of physically meaningful assumptions such that any physical theory satisfying
these assumptions must embed into the category of finite-dimensional
C*-algebras. These assumptions were originally introduced in the setting of
effectus theory, a categorical logical framework generalizing classical and
quantum logic. As these assumptions have a physical interpretation, this
motivates the usage of operator algebras as a model for quantum theory.
  In contrast to other reconstructions of quantum theory, we do not start with
the framework of generalized probabilistic theories and instead use effect
theories where no convex structure and no tensor product needs to be present.
The lack of this structure in effectus theory has led to a different notion of
pure maps. A map in an effectus is pure when it is a composition of a
compression and a filter. These maps satisfy particular universal properties
and respectively correspond to `forgetting' and `measuring' the validity of an
effect.
  We define a pure effect theory (PET) to be an effect theory where the pure
maps form a dagger-category and filters and compressions are adjoint. We show
that any convex finite-dimensional PET must embed into the category of
Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure,
then we show that it must embed into either the category of real or complex
C*-algebras, which completes our reconstruction.
