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

Theorem. If the C*-algebra $\mathfrak{𝔄}$ acts topologically irreducibly on the Hilbert space ${\mathscr{H}},\{y_1,\cdots ,y_n\}$ is a set of vectors and $\{x_1,\cdots ,x_n\}$ is a linearly independent set of vectors in ${\mathscr{H}}$, there is an $A$ in $\mathfrak{𝔄}$ such that $A{x}_j=y_j$. If $B{x}_j=y_j$ 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 ${\mathscr{H}}$, then it acts algebraically irreducibly.

References

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