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.
Basil Hiley
In 2000, Brown and Hiley showed that the Schrödinger equation can be written in a purely algebraic form that is independent of any representation in a Hilbert space. This algebraic description is formulated in terms of two operator equations. The first of these (formulated in terms of the commutator) represents an alternative form of the quantum Liouville equation, which is well known to describe the conservation of probability, the second (formulated in terms of the anticommutator), which they dubbed the "quantum phase equation", describes the conservation of energy. This algebraic description in turn gives rise to descriptions in terms of multiple vector spaces, which Brown and Hiley call "shadow phase spaces" (adopting the term "shadow" from Michał Heller). These shadow phase space descriptions include the descriptions in terms of the "x"-space of the Bohm trajectory description, of the quantum phase space, and of the "p"-space. In the classical limit, the shadow phase spaces converge to one unique phase space. In their algebraic formulation of quantum mechanics the equation of motion takes on the same form as in the Heisenberg picture, except that the "bra" and "ket" in the bra–ket notation each stand for an element of the algebra and that the Heisenberg time evolution is an inner automorphism in the algebra.