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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. [1] 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

Theorem 5.4.3; Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, ISBN 978-0821808191

Kadison, Richard (1957), "Irreducible operator algebras", Proc. Natl. Acad. Sci. U.S.A., 43: 273–276, 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

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