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.
Spectrum (functional analysis)
For example, consider the Hilbert space formula_24, that consists of all bi-infinite sequences of real numbers
that have a finite sum of squares formula_26. The bilateral shift operator formula_2 simply displaces every element of the sequence by one position; namely if formula_28 then formula_29 for every integer formula_30. The eigenvalue equation formula_31 has no solution in this space, since it implies that all the values formula_32 have the same absolute value (if formula_33) or are a geometric progression (if formula_34); either way, the sum of their squares would not be finite. However, the operator formula_20 is not invertible if formula_36. For example, the sequence formula_37 such that formula_38 is in formula_24; but there is no sequence formula_40 in formula_24 such that formula_42 (that is, formula_43 for all formula_30).