Topologies and Density Theorems in Operator Algebras

  • Masamichi Takesaki


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.


Hilbert Space Banach Space Unit Ball Operator Algebra Weak Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Masamichi Takesaki
    • 1
  1. 1.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA

Personalised recommendations