Kadison transitivity theorem

Not to be confused with Kadison's theorem generalizing Wigner's theorem.

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

A family ${\displaystyle {\mathcal {F}}}$  of bounded operators on a Hilbert space ${\displaystyle {\mathcal {H}}}$  is said to act topologically irreducibly when ${\displaystyle \{0\}}$  and ${\displaystyle {\mathcal {H}}}$  are the only closed stable subspaces under ${\displaystyle {\mathcal {F}}}$ . The family ${\displaystyle {\mathcal {F}}}$  is said to act algebraically irreducibly if ${\displaystyle \{0\}}$  and ${\displaystyle {\mathcal {H}}}$  are the only linear manifolds in ${\displaystyle {\mathcal {H}}}$  stable under ${\displaystyle {\mathcal {F}}}$ .

Theorem. ^[1] If the C*-algebra ${\displaystyle {\mathfrak {A}}}$  acts topologically irreducibly on the Hilbert space ${\displaystyle {\mathcal {H}},\{y_{1},\cdots ,y_{n}\}}$  is a set of vectors and ${\displaystyle \{x_{1},\cdots ,x_{n}\}}$  is a linearly independent set of vectors in ${\displaystyle {\mathcal {H}}}$ , there is an ${\displaystyle A}$  in ${\displaystyle {\mathfrak {A}}}$  such that ${\displaystyle Ax_{j}=y_{j}}$ . If ${\displaystyle Bx_{j}=y_{j}}$  for some self-adjoint operator ${\displaystyle B}$ , then ${\displaystyle A}$  can be chosen to be self-adjoint.

Corollary. If the C*-algebra ${\displaystyle {\mathfrak {A}}}$  acts topologically irreducibly on the Hilbert space ${\displaystyle {\mathcal {H}}}$ , then it acts algebraically irreducibly.

References

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

• 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