Topologies and Density Theorems in Operator Algebras

By nature, our objects in this book are infinite dimensional, which makes topological and approximation arguments indispensable. In Section 1, we first study the Banach spaces of operators on a Hilbert space ℌ. It is proved that the second conjugate space ℒℭ(ℌ)** of the C*-algebra of all compact operators on ℌ as a Banach space is naturally identified with the Banach space ℒ(ℌ) of all bounded operators on ℌ. This result allows us to introduce, in Section 2, various kinds of locally convex topologies in ℒ(ℌ) related to the duality of ℒ(ℌ) and ℒℭ(ℌ)* as well as to the algebra structure of ℒ(ℌ). In Section 3, the fundamental theorem of operator algebras (the double commutation theorem), due to J. von Neumann, is proved and a few of its immediate consequences are drawn. Section 4 is devoted to various approximation theorems. Among them, Theorem 4.8 is most important. It may be called the fundamental approximation theorem. In this section, the strong continuity of functional calculus is also shown. A striking consequence of this section, Theorem 4.18, is the algebraic irreducibility of an irreducible representation of a C*-algebra. The proof presented here is not the most economic, it is drawn as a consequence from a more powerful result, the noncommutative Lusin’s theorem, Theorem 4.15, which is somewhat technical.