ACTA issues

Von Neumann algebras associated with composition operators

Charles Burnap, Thomas B. Hoover, Alan Lambert$^*$

Acta Sci. Math. (Szeged) 62:3-4(1997), 565-582

Abstract. Let $(X,{\cal A},\mu )$ be a complete finite measure space and let $T\colon X\to X$ be a measurable transformation for which $\mu\circ T^{-1}\ll\mu $ and ${d\mu\circ T^{-1}\over d\mu } \in L^\infty(X,{\cal A},\mu )$. Then $C\colon f\mapstochar\rightarrow f\circ T$ is a bounded linear operator on $L^2$. We examine the von Neumann algebra $\cal W$ generated by $C$. Two sigma algebras $\cal M$ and ${\cal A}^\prime $ play a key role in the investigation. The orthogonal projection $E^{{\cal A}^\prime }$ onto the space of ${\cal A}^\prime $ measurable functions is given by the conditional expectation with respect to the sigma algebra ${\cal A}^\prime $. This projection is always abelian in $\cal W$ and it is minimal in $\cal W$ whenever ${\cal M}\cap{\cal A}^\prime ={\cal T}$ ($\cal T$ is the sigma algebra generated by $\emptyset $ and $X$). Similarly, the orthogonal projection $E^{\cal M}$ onto the space of $\cal M$ measurable functions is always abelian in ${\cal W}^\prime $ and minimal whenever ${\cal M}\cap{\cal A}^\prime ={\cal T}$. Moreover, both $E^{\cal M}$ and $E^{{\cal A}^\prime }$ have central carrier $E^{{\cal M}\vee{\cal A}^\prime }$ (${\cal M}\vee{\cal A}^\prime $ denotes the smallest $\sigma $-algebra containing both $\cal M$ and ${\cal A}^\prime $). If ${\cal M}\vee{\cal A}^\prime ={\cal A}$ then $E^{{\cal A}^\prime }$ and $E^{\cal M}$ are faithful abelian projections and $\cal W$ is a type I von Neumann algebra. In this case, the center of $\cal W$ consists of all $L^\infty({\cal M} \cap{\cal A}^\prime )$ multiplication operators, hence $\cal W$ is a factor if and only if ${\cal M}\cap{\cal A}^\prime ={\cal T}$. Furthermore, there is an apparent symmetry involving ${\cal W},{\cal W}^\prime $ (the commutant of $\cal W$), $\cal M$ and ${\cal A}^\prime $.

AMS Subject Classification (1991): 47C15, 47B38

Received January 9, 1996 and in revised form July 15, 1996. (Registered under 6136/2009.)