  We present a general approach to quantum entanglement and entropy that is
based on algebras of observables and states thereon. In contrast to more
standard treatments, Hilbert space is an emergent concept, appearing as a
representation space of the observable algebra, once a state is chosen. In this
approach, which is based on the Gelfand-Naimark-Segal construction, the study
of subsystems becomes particularly clear. We explicitly show how the problems
associated with partial trace for the study of entanglement of identical
particles are readily overcome. In particular, a suitable entanglement measure
is proposed, that can be applied to systems of particles obeying Fermi, Bose,
para- and even braid group statistics. The generality of the method is also
illustrated by the study of time evolution of subsystems emerging from
restriction to subalgebras. Also, problems related to anomalies and quantum
epistemology are discussed.
