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]- ^ Theorem 5.4.3; Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, ISBN 978-0821808191
- Kadison, Richard (1957), "Irreducible operator algebras", Proc. Natl. Acad. Sci. U.S.A., 43 (3): 273–276, Bibcode:1957PNAS...43..273K, doi:10.1073/pnas.43.3.273, PMC 528430, PMID 16590013.
- Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, ISBN 978-0821808191