 In the late 1920s, von Neumann developed the separable Hilbert space formulation of quantum mechanics, which later became the definitive one (from the standpoint of mathematical rigor, at least). In the mid-1930s, he worked extensively on lattice theory (see the entry on  quantum logic),  rings of operators, and continuous geometries. Part of his expressed motivation for developing these mathematical theories was to develop an appropriate framework for QFT and a better foundation for quantum mechanics. During this time, he noted two closely related structures, modular lattices and finite type-II factors (a special type of ring of operators), that have what he regarded as desirable features for quantum theory. These observations led to his developing a more general framework, continuous geometries, for quantum theory. Matters did not work out as von Neumann had expected. He soon realized that such geometries must have a transition probability function, if they are to be used to describe quantum mechanical phenomena, and that the resulting structure is not a generalization at all beyond the operator rings that were already available. Moreover, it was determined much later that the type-III factors are the most important type of ring of operators for quantum theory. In addition, a similar verdict was delivered much later with regards to his expectations concerning lattice theory. The lattices that are appropriate for quantum theory are orthomodular – a lattice is orthomodular only if it is modular, but the converse is false. Of the three mathematical theories, it is the rings of operators that have proven to be the most important framework for quantum theory. It is possible to use a ring of operators to model key features of physical systems in a purely abstract, algebraic setting (this is discussed in section 4.1). A related issue concerns whether it is necessary to choose a representation of the ring in a Hilbert space; see Haag and Kastler (1964), Ruetsche (2003), and Kronz and Lupher (2005) for further discussion of this issue. In any case, the separable Hilbert space remains a crucial framework for quantum theory. The simplest examples of separable Hilbert spaces are the finite dimensional ones, in which case the algebra of operators is a type-I\(_n\) factor (n is a positive integer). The operators are n-by-n complex matrices, which are typically used to describe internal degrees of freedom such as spin. Readers wanting to familiarize themselves with these basic examples should consult the entry on  quantum mechanics.