Reference for Ideals, Projections, Support Projections, etc.

Question

I have a number of confusions that I think a good reference might be able to sort out.

Let $A$ be a von Neumann Algebra. At the moment I am mostly concerned with $\dim A<\infty$, but am interested in more general results.

Consider these three ideas:

1. Let $\nu\in S(A)$ be a state on $A$. Define the following null space:

$$N_\nu=\left\{b\in A\,|\,\nu(|b|^2)=0\right\}.$$

This set is a $\sigma$-weakly closed left ideal. Therefore there exists a projection $Q_\nu$ such that $N_\nu=AQ_\nu$. Some properties include the fact that $b\in N_\nu$ if and only if $bQ_\nu=b$. Also, for all $a\in A$ we have $$\nu(Q_\nu)=\nu(aQ_\nu)=\nu(Q_\nu a)=0.$$

Question 1: Is the map $a\mapsto aQ_\nu$ the projection onto $N_\nu$?

Define the projection $P_\nu:=1_A-Q_\nu$. We have $$\nu(a)=\nu(aP_\nu)=\nu(P_\nu a)=\nu(P_\nu aP_\nu),$$ and $\nu(P_\nu)=1$.

Question 2: Is there a projection $P\leq P_\nu$ such that $\nu(P)=1$?

2. Following this answer, consider the projection $$\operatorname{supp}\,\nu=1_A-\bigvee\left\{p\in A,\,\text{a projection such that }\nu(p)=0\right\}.$$

Connecting with the above:

Question 3: Does $\operatorname{supp}\,\nu$ coincide with $P_\nu$ (from above)?

3. I cannot find the definition of the support projection of $\nu$. My naive definition is that the support projection of $\nu$ would be the smallest projection $p$ such that $\nu(p)=1$. However my understanding is that the support projection would usually live in the same space as $\nu$ (and that is not what I am trying to model).

Question 4: Does the definition of $\operatorname{supp}\,\nu$ (above) coincide with "the smallest projection $p$ such that $\nu(p)=1$".

An overarching question

Question: Is there a good reference (possibly lecture notes) which can discuss the (non?-)relationships between these concepts?

---

Context: let $F(X)$ be the algebra of functions on a finite set $X$. Let $\nu\in M_p(X)\subset \mathbb{C}X$ be a probability on $X$. There are a number of ways that we can think about the support of $\nu$:

• it is the subset $S\subset X$ comprised of elements $s\in X$ such that $\nu(\delta_s)>0$. Fix $S$ in the remaining.
• Consider the set $S_\nu=\{g\in F(X)\,|\,\nu(|g|^2)>0\}$. This a subspace of $F(X)$. Let $Q$ be the projection in $L(F(X))$ onto $S_\nu$. The projection $P=1_X-Q$ coincides with the map $f\mapsto \mathbf{1}_Sf$.
• Note that there is no projection $P$ in $F(X)$, $P\leq \mathbf{1}_S$ such that $\nu(P)=1$. In this sense $\mathbf{1}_S$ is the smallest projection $p\in F(X)$ such that $\nu(p)=1$.

Answer 1

1. There is no "the" projection (and sometimes not even "a" projection) onto a von Neumann algebra. Your map is an idempotent that maps onto $N_\nu$, but there is nothing interesting about it, as far as I can tell.

2. No. If $\nu(P)=1$, then $\nu(P_\nu-P)=0$, which implies $P_\nu-P\in N_\nu$, so $$ 0=Q_\nu(P_\nu-P)=P_\nu-P. $$

3. Yes, you can prove this using 2. As for "I cannot find the definition..." I have no idea what you mean, as you wrote the definition three lines above.

4. Yes. That's 2.

I think this is in every classic text on von Neumann algebras. The first one that comes to mind is Chapter 7 in Kadison-Ringrose.