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

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

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

References

• 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