Given an inclusion $N\subset M$ of von Neumann algebras, a conditional expectation is a map $E:M\to N$ that is a projection ($E^2=E$) and it has $\|E\|=1$. This automatically implies that $E$ is completely positive, and that $$\tag1 E(axb)=aE(x)b,\ \ \ a,b\in N,\ x\in M. $$ Conversely, a completely positive idempotent satisfies $\|E\|=1$.

In the particular case where $N=A$ is a masa (maximal abelian subalgebra), is it true that a conditional expectation exists for any masa $A\subset M$? Is it always normal?


There are cases where a normal conditional expectation does not exist. By Exercise IX.4.1 in Takesaki's Theory of Operator Algebras, volume 2, if $E:M\to A$ is a normal conditional expectation and $M$ is atomic, then $A$ is atomic. So, for instance, if $M=B(L^2(X,\mu))$ and $A=L^\infty(X,\mu)$ for a diffuse measure $\mu$, then no normal conditional expectation exists.

Without the requirement of normality, a conditional expectation onto a masa always exists. For this consider the collection $\mathcal F$ of finite partitions of 1 in $A$ (that is, projections $p_1,\ldots,p_n\in A$ with $\sum p_j=1$), ordered by inclusion; this makes it a net. Because $A$ is a masa, $\mathcal F'\cap M=A$.

Given any $P=\{p_1,\ldots,p_n\}\in\mathcal F$, define $\phi_P:M\to M$ by $$ \phi_P(x)=\sum_{j=1}^n p_jxp_j. $$ Note that $\phi_P(x)p_k=p_k\phi_P(x)$ for all $k=1,\ldots,n$. The maps $\phi_P$ are unital and completely positive. In particular the net $\{\phi_P\}_{P\in\mathcal F}$ is bounded, and so it has cluster points in the BW-topology (pointwise wot convergence). Let $\phi$ be a cluster point. Given any projection $p\in A$, we may consider partitions of unity that refine $p$, so $p\phi_P(x)=\phi_P(x)p$ for all $x\in M$ and all $P\in\mathcal F$ that refine $p$. The subnet we used to obtain $\phi$ will eventually contain these partitions, and so $p\phi(x)=\phi(x)p$. As we can do this for any projection $p\in A$, we get that $\phi(x)\in A'\cap M=A$. Note also that for $a\in A$ and any $P\in\mathcal F$, we have $\phi_P(a)=a$, so $\phi(a)=a$. Finally, each $\phi_P$ is unital and completely positive, so the same is true for $\phi$, which is then a conditional expectation.

Akemann and Sherman have proven that if $A$ is singly generated (always true when $H$ is separable) and the construction above yields a unique conditional expectation, then said expectation is normal. This shows that when $M=B(L^2(X,\mu))$ and $A=L^\infty(X,\mu)$ for a diffuse measure $\mu$, the above construction yields more than one expectation. In this case the construction can be done more or less explicitly, and one can then see that there are actually infinitely many.

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  • $\begingroup$ I do not understand the convergence part. Could you please explain me what is BW topology and its convergence? $\endgroup$ – rkmath Jan 31 '19 at 10:00
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    $\begingroup$ $\phi_j\to\phi$ means $\phi_j(x)\to\phi(x)$ wot for every $x$. It is known that this topology has an Alaoglu theorem. $\endgroup$ – Martin Argerami Jan 31 '19 at 14:50

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