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.

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 ${\displaystyle {\mathcal {F}}}$.

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