Jump to content

Kadison transitivity theorem

From Wikipedia, the free encyclopedia

In mathematics, Kadison transitivity theorem is a result in the theory of C*-algebras that, in effect, asserts the equivalence of the notions of topological irreducibility and algebraic irreducibility of representations of C*-algebras. It implies that, for irreducible representations of C*-algebras, the only non-zero linear invariant subspace is the whole space.

The theorem, proved by Richard Kadison, was surprising as a priori there is no reason to believe that all topologically irreducible representations are also algebraically irreducible.

Statement

[edit]

A family of bounded operators on a Hilbert space is said to act topologically irreducibly when and are the only closed stable subspaces under . The family is said to act algebraically irreducibly if and are the only linear manifolds in stable under .

Theorem. [1] If the C*-algebra acts topologically irreducibly on the Hilbert space is a set of vectors and is a linearly independent set of vectors in , there is an in such that . If for some self-adjoint operator , then can be chosen to be self-adjoint.

Corollary. If the C*-algebra acts topologically irreducibly on the Hilbert space , then it acts algebraically irreducibly.

References

[edit]
  1. ^ Theorem 5.4.3; Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, ISBN 978-0821808191