  These notes provide an explanation of the type classification of von Neumann
algebras, which has made many appearances in recent work on entanglement in
quantum field theory and quantum gravity. The goal is to bridge a gap in the
literature between resources that are too technical for the non-expert reader,
and resources that seek to explain the broad intuition of the theory without
giving precise definitions. Reading these notes will provide you with: (i) an
argument for why "factors" are the fundamental von Neumann algebras that one
needs to study; (ii) an intuitive explanation of the type classification of
factors in terms of renormalization schemes that turn unnormalizable positive
operators into "effective density matrices;" (iii) a mathematical explanation
of the different types of renormalization schemes in terms of the allowed
traces on a factor; (iv) an intuitive characterization of type I and II factors
in terms of their "standard forms;" and (v) a list of some interesting
connections between type classification and modular theory, including the
argument for why type III$_1$ factors are believed to be the relevant ones in
quantum field theory. None of the material is new, but the pedagogy is
different from other sources I have read; it is most similar in spirit to the
recent work on gravity and the crossed product by Chandrasekaran, Longo,
Penington, and Witten.
