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.
Statement
A family [math]\displaystyle{ \mathcal{F} }[/math] of bounded operators on a Hilbert space [math]\displaystyle{ \mathcal{H} }[/math] is said to act topologically irreducibly when [math]\displaystyle{ \{0\} }[/math] and [math]\displaystyle{ \mathcal{H} }[/math] are the only closed stable subspaces under [math]\displaystyle{ \mathcal{F} }[/math]. The family [math]\displaystyle{ \mathcal{F} }[/math] is said to act algebraically irreducibly if [math]\displaystyle{ \{0\} }[/math] and [math]\displaystyle{ \mathcal{H} }[/math] are the only linear manifolds in [math]\displaystyle{ \mathcal{H} }[/math] stable under [math]\displaystyle{ \mathcal{F} }[/math].
Theorem. [1] If the C*-algebra [math]\displaystyle{ \mathfrak{A} }[/math] acts topologically irreducibly on the Hilbert space [math]\displaystyle{ \mathcal{H}, \{ y_1, \cdots, y_n \} }[/math] is a set of vectors and [math]\displaystyle{ \{x_1, \cdots, x_n \} }[/math] is a linearly independent set of vectors in [math]\displaystyle{ \mathcal{H} }[/math], there is an [math]\displaystyle{ A }[/math] in [math]\displaystyle{ \mathfrak{A} }[/math] such that [math]\displaystyle{ Ax_j = y_j }[/math]. If [math]\displaystyle{ Bx_j = y_j }[/math] for some self-adjoint operator [math]\displaystyle{ B }[/math], then [math]\displaystyle{ A }[/math] can be chosen to be self-adjoint.
Corollary. If the C*-algebra [math]\displaystyle{ \mathfrak{A} }[/math] acts topologically irreducibly on the Hilbert space [math]\displaystyle{ \mathcal{H} }[/math], then it acts algebraically irreducibly.
References
- "Irreducible operator algebras", Proc. Natl. Acad. Sci. U.S.A. 43 (3): 273–276, 1957, doi:10.1073/pnas.43.3.273, PMID 16590013, Bibcode: 1957PNAS...43..273K.
- Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, ISBN:978-0821808191
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