Kadison transitivity theorem - Alchetron, the free social encyclopedia

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

Theorem. If the C*-algebra A acts topologically irreducibly on the Hilbert space H , { y 1 , ⋯ , y n } is a set of vectors and { x 1 , ⋯ , x n } is a linearly independent set of vectors in H , there is an A in A such that A x j = y j . If B x j = y j for some self-adjoint operator B , then A can be chosen to be self-adjoint.

Corollary. If the C*-algebra A acts topologically irreducibly on the Hilbert space H , then it acts algebraically irreducibly.

References

Kadison transitivity theorem Wikipedia

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