Hilbert algebra

An algebra with involution (cf. Involution algebra) over the field of complex numbers, equipped with a non-degenerate scalar product , for which the following axioms are satisfied: 1) for all ; 2) for all ; 3) for all the mapping of into is continuous; and 4) the set of elements of the form , , is everywhere dense in . Examples of Hilbert algebras include the algebras (with respect to convolution), where is a compact topological group, and the algebra of Hilbert–Schmidt operators (cf. Hilbert–Schmidt operator) on a given Hilbert space.

Let be a Hilbert algebra, let be the Hilbert space completion of and let and be the elements of the algebra of bounded linear operators on which are the continuous extensions of the multiplications from the left and from the right by in . The mapping (respectively, ) is a non-degenerate representation of (respectively, of the opposite algebra), on . The weak closure of the family of operators (respectively, ) is a von Neumann algebra in ; it is called the left (respectively, right) von Neumann algebra of the given Hilbert algebra and is denoted by (respectively, ); and are mutual commutators; they are semi-finite von Neumann algebras. Any Hilbert algebra unambiguously determines some specific normal semi-finite trace on the von Neumann algebra (cf. Trace on a -algebra). Conversely, if a von Neumann algebra and a specific semi-finite trace on are given, then it is possible to construct a Hilbert algebra such that the left von Neumann algebra of this Hilbert algebra is isomorphic to and the trace determined by the Hilbert algebra on coincides with the initial one [1]. Thus, a Hilbert algebra is a means of studying semi-finite von Neumann algebras and traces on them; a certain extension of the concept of a Hilbert algebra makes it possible to study by similar means von Neumann algebras that are not necessarily semi-finite [2].

References

[1]	J. Dixmier, "Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann" , Gauthier-Villars (1957)
[2]	M. Takesaki, "Tomita's theory of modular Hilbert algebras and its applications" , Lect. notes in math. , 128 , Springer (1970)