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 $$\mathcal{F}$$ 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.