Hilbert space
The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.
Geometric quantization
The next step in the process of geometric quantization is the choice of a polarization. A polarization is a choice at each point in formula_6 a Lagrangian subspace of the complexified tangent space of formula_6. The subspaces should form an integrable distribution, meaning that the commutator of two vector fields lying in the subspace at each point should also lie in the vector field at each point. The "quantum" (as opposed to prequantum) Hilbert space is the space of sections of formula_14 that are covariantly constant in the direction of the polarization. The idea is that in the quantum Hilbert space, the sections should be functions of only formula_27 variables on the formula_28-dimensional classical phase space.