Kadison transitivity 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 $𝔄$ 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 $𝔄$ 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 $𝔄$ acts topologically irreducibly on the Hilbert space $\mathscr{H}$, then it acts algebraically irreducibly.

References

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• Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, Template:ISBN