6. Generalized B*-Algebras: Unbounded *-Representation Theory

In representing a noncommutative C*-algebra as a norm closed *-subalgebra of bounded linear operators on some Hilbert space, positive linear functionals play an important role. In ensuring that the involved *-representation is faithful, one requires that there are enough positive linear functionals in the sense that they separate the points of the C*-algebra. That this is so for C*-algebras is well known, and relies on the fact that the positive cone of a C*-algebra is closed. It is not known if the positive cone is closed for GB*-algebras, in general, but we prove in this chapter that the positive cone of a GB*-algebra \(\mathcal {A}[\tau ]\) is closed in some stronger topology T making \(\mathcal {A}\) a GB*-algebra (Dixon). In Chap. 4 it was established, that there are enough positive linear functionals on a commutative GB*-algebra, that separate its points (Allan). A partial analogue to this in the noncommutative case also holds (Dixon). It is this result that we use to prove a noncommutative algebraic Gelfand-Naimark type theorem for GB*-algebras. Namely, we show that any GB*-algebra can be faithfully represented as an EC*-algebra (Dixon). In the same chapter, a topological analogue of the Gelfand-Naimark type theorem for GB*-algebras is presented.