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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

Mathematics Encyclopedia

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