Abelian von Neumann Algebras

Prove the following proposition (Proposition 7.3).

Prove the following proposition (Proposition 7.4). Let $\mathcal{A}_j$ be a von Neumann algebra on $\mathcal{H}_j$, $j = 1, 2$. If $U: \mathcal{H}_1 \to \mathcal{H}_2$ is an isomorphism such that $U \mathcal{A}_1 U^{-1} = \mathcal{A}_2$, then $U \mathcal{A}_1' U^{-1} = \mathcal{A}_2'$.

Why are $\mathcal{A}_\mu$ and $\mathcal{A}_\mu^{(2)}$ not spatially isomorphic?

Show that if $X$ is any compact metric space, there is a separable Hilbert space $\mathcal{H}$ and a $*$-monomorphism $\tau : C(X) \to \mathcal{B}(\mathcal{H})$. Find the spectral measure for $\tau$.

Let $\mathcal{A}$ be an abelian $C^*$-subalgebra of $\mathcal{B}(\mathcal{H})$ such that $\mathcal{A}'$ contains no uncountable collection of pairwise orthogonal projections. Show that the following statements are equivalent:

1. $\mathcal{A}$ is a maximal abelian von Neumann algebra;

2. $\mathcal{A} = \mathcal{A}'$;

3. $\mathcal{A}$ has a cyclic vector and is SOT closed;

4. there is a finite measure space $(X, \Omega , \mu )$ and an isomorphism $U: L^2(\mu ) \to \mathcal{H}$ such that $U \mathcal{A}_\mu U^{-1} = \mathcal{A}$.

Let $\{ P_n\}$ be a sequence of commuting projections in $\mathcal{B}(\mathcal{H})$ and put $A = \sum_{n=1}^\infty 3^{-n}(2P_n - 1)$. Show that $C^*(A)$ is the $C^*$-algebra generated by $\{ P_n\}$. (Do you see a connection between $A$ and the Cantor--Lebesgue function?)

If $\mathcal{A}$ is an abelian von Neumann algebra on a separable Hilbert space $\mathcal{H}$, show that there is a hermitian operator $A$ such that $\mathcal{A}$ equals the smallest von Neumann algebra containing $A$. (Hint: Let $\{ P_n\}$ be a countable WOT dense subset of the set of projections in $\mathcal{A}$ and use Exercise 6. This proof is due to Rickart [1960], pp. 293--294. Also see Jenkins [1972].)

If $X$ is a compact space, show that $C(X)$ is generated as a $C^*$-algebra by its characteristic functions if and only if $X$ is totally disconnected. If $A$ is as in Exercise 6, show that $\sigma (A)$ is totally disconnected.

If $X$ and $Y$ are compact spaces and $\tau : C(X) \to C(Y)$ is a homomorphism with $\tau (1) = 1$, show that there is a continuous function $\phi : Y \to X$ such that $\tau (u) = u \circ \phi$ for every $u$ in $C(X)$. Show that $\tau$ is injective if and only if $\phi$ is surjective, and, in this case, $\tau$ is an isometry. Show that $\tau$ is surjective if and only if $\phi$ is injective.

Let $X$ and $Z$ be compact spaces, $Y = X \times Z$, and let $\phi : Y \to X$ be the projection onto the first coordinate. Define $\tau : C(X) \to C(Y)$ by $\tau (u) = u \circ \phi$. Describe the range of $\tau$.

Adopt the notation of Exercise 9. Define an equivalence relation $\sim$ on $Y$ by saying $y_1 \sim y_2$ if and only if $\phi (y_1) = \phi (y_2)$. Let $q: Y \to Y/{\sim }$ be the natural map and $q^*: C(Y/{\sim }) \to C(Y)$ the induced homomorphism. Show that there is a $*$-epimorphism $\rho : C(X) \to C(Y/{\sim })$ such that the diagram

commutes. Find the corresponding injection $Y/{\sim } \to X$.

If $X$ is any compact metric space, show that there is a totally disconnected compact metric space $Y$ and a continuous surjection $\phi : Y \to X$. (Hint: Start by embedding $C(X)$ into $\mathcal{B}(\mathcal{H})$ and use Exercises 7 and 8.)

Show that every totally disconnected compact metric space is the continuous image of the Cantor ternary set. (Do this directly; do not try to use $C^*$-algebras.) Combine this with Exercise 12 to get that every compact metric space is the continuous image of the Cantor set.

If $\mathcal{A}$ is an abelian von Neumann algebra and $X$ is its maximal ideal space, then $X$ is a Stonean space; that is, if $U$ is open in $X$, then $\operatorname {cl} U$ is open in $X$.

This exercise assumes Exercise 2.21, where it was proved that $\mathcal{B}_1^* = \mathcal{B}$. When referring to the weak* topology on $\mathcal{B} = \mathcal{B}(\mathcal{H})$, we mean the topology $\mathcal{B}$ has as the Banach dual of $\mathcal{B}_1$.

Prove the converse of the following proposition (Proposition 7.6: if $e_0$ is a cyclic vector for $\mathcal{A}$, then $e_0$ is a separating vector for $\mathcal{A}'$).