Kadison transitivity theorem explained

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

l{F} of bounded operators on a Hilbert space l{H} is said to act topologically irreducibly when \{ 0 \} and l{H} are the only closed stable subspaces under l{F} . The family l{F} is said to act algebraically irreducibly if \{ 0 \} and l{H} are the only linear manifolds in l{H} stable under l{F} .

Theorem. ^[1] If the C*-algebra

ak{A} acts topologically irreducibly on the Hilbert space l{H}, \{ y _{1,} … , y _{n} \} is a set of vectors and \{x _{1,} … , x _{n} \} is a linearly independent set of vectors in l{H} , there is an A in ak{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

ak{A} acts topologically irreducibly on the Hilbert space l{H} , then it acts algebraically irreducibly.

References

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

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