Topologically irreducible representations of the Banach -algebra associated with a dynamical system

Abstract

We describe (infinite-dimensional) irreducible representations of the crossed product C * -algebra associated with a topological dynamical system (based on Z) and we show that their restrictions to the underling ℓ 1 -Banach * -algebra are not algebraically irreducible under mild conditions on the dynamical system. The above description of irreducible representations has two ingredients, ergodic measures on the space and ergodic extensions for the tensor product with type I factors; the latter which may not have been explicitly taken up before will be explored by examples. A new class of ergodic measures is also constructed for irrational rotations on the circle.

Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system

“…Here is a sample result: If µ is an ergodic σ-invariant non-atomic probability measure on X, then π 2 µ is a * -representation of ℓ 1 (Σ) on L 2 (X, µ) that is topologically irreducible, but not algebraically irreducible. The latter statement is a special case of [14,Theorem 3.4]. We refer the reader to [14] for further material in this direction.…”

“…The latter statement is a special case of [14,Theorem 3.4]. We refer the reader to [14] for further material in this direction.…”

“…For metrizable X, a thorough study of topologically irreducible * -representations of ℓ 1 (Σ) on Hilbert spaces and their algebraic (ir)reducibility is made in [14], where these representations are related to ergodic σ-quasi-invariant probability measures on X. Here is a sample result: If µ is an ergodic σ-invariant non-atomic probability measure on X, then π 2 µ is a * -representation of ℓ 1 (Σ) on L 2 (X, µ) that is topologically irreducible, but not algebraically irreducible.…”