On the commutant of $C(X)$ in $C^{\ast }$-crossed products by $\mathbb{Z}$ and their representations

Abstract

For the $C^{\ast }$-crossed product $C^{\ast }(\Sigma )$ associated with an arbitrary topological dynamical system $\Sigma =(X,\sigma )$, we provide a detailed analysis of the commutant, in $C^{\ast }(\Sigma )$, of $C(X)$ and the commutant of the image of $C(X)$ under an arbitrary Hilbert space representation $\tilde{\pi }$ of $C^{\ast }(\Sigma )$. In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system $\Sigma$, the commutant of $C(X)$ has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of $C^{\ast }(\Sigma )$. We also show that the corresponding statement holds true for the commutant of $\tilde{\pi }(C(X))$ under the assumption that a certain family of pure states of $\tilde{\pi }(C^{\ast }(\Sigma ))$ is total. Furthermore we establish that, if $C(X)⊊C(X{)}^{\prime }$, there exist both a $C^{\ast }$-subalgebra properly between $C(X)$ and $C(X{)}^{\prime }$ which has the aforementioned intersection property, and such a $C^{\ast }$-subalgebra which does not have this property. We also discuss existence of a projection of norm one from $C^{\ast }(\Sigma )$ onto the commutant of $C(X)$.

arXiv comment19 pages. Minor clarifications and improvements of exposition made. Some typos corrected

Cited by in corpus (1)

• Dynamical systems associated with crossed products