Complete regularity in C*-algebras

Question

It is clear that commutative C*-algebras correspond to locally compact Hausdorff spaces. And locally compact Hausdorff spaces are completely regular. Now, does the complete regularity statement have some kind of generalization to general C*-algebras. In particular, I am looking for a statement using pure states of the C*-algebra. So something of the form:

If $\tau$ is a pure state on the C*-algebra and if $T$ is a collection of pure states satisfying some conditions (this should correspond to the closed set condition in commutative cases), then there exists an element $a$ of the $C^*$-algebra such that $\tau(a) = 1$ and $\sigma(a)=0$ for each $\sigma\in T$.

Is is a statement like this true? Are there other interesting generalizations of complete regularity?

Answer 1

You may be looking for the noncommutative Urysohn lemma: if $A$ is a C*-algebra and $p,q \in A^{**}$ are projections, $p \leq q$, with $p$ compact and $q$ open, then there is a positive element $x \in A$ such that $p \leq x \leq q$. It is due to Akemann, The general Stone-Weierstrass problem, J. Funct. Anal. 4 (1969), 277-294. Or just Google "noncommutative Urysohn lemma".

Answer 2

Nik's answer seems perfectly suited to what the OP was looking for, but there are indeed other generalizations of (complete) regularity to C*-algebras, as shown in Rosický's "Multiplicative lattices and C*-algebras" Cahiers Top. Geom. Diff. Cat. tome 30 no 2 (1989), 95-110. To paraphrase part of that paper, note first that regularity can be stated in terms of open and closed sets, without reference to points. Specifically, a topological space $X$ is regular iff, for every open $O\subseteq X$, $$O=\bigcup\{N:N\text{ is open and }\overline{N}\subseteq O\}.$$ So a natural question to ask is whether, for any C*-algebra $A$ and open projection $q\in A^{**}$, $$q=\bigvee\{p\in A^{**}:p\text{ is an open projection and }\overline{p}\leq q\}?$$ The answer is yes. To see this, take any $a\in A^1_+$ with $a\leq q$ and note that $\overline{a_{(\epsilon,1]}}\leq a_{[\epsilon,1]}\leq a_{(0,1]}\leq q$, for any $\epsilon>0$, where $a_S$ denotes the spectral projection of $a$ in $A^{**}$ corresponding to $S\subseteq\mathbb{R}$. The result now follows because $q$ being open means $$q=\bigvee\{a_{(0,1]}:a\in A^1_+\text{ and }a\leq q\}.\\$$

As $\overline{N\cup M}=\overline{N}\cup\overline{M}$, for open subsets $N$ and $M$, above we actually have a directed union. On the other hand, we can have $\overline{p\vee r}\neq\overline{p}\vee\overline{r}$ for open projections $p$ and $r$, as in Example II.6 in Akemann's paper referred to by Nik, which also shows that the supremum above is not always directed. However, we might ask:

Can the open projections above be replaced by a directed subset?

In general I do not know. If $A$ is separable then the answer is yes, as then every open $p$ will be of the form $a_{(0,1]}$, for some $a\in A^1_+$, and $a_{(\epsilon,1]}$ is directed, for $\epsilon>0$. More generally, the answer is yes if $A$ has an "almost idempotent" approximate unit $(a_\lambda)\subseteq A^1_+$, i.e. such that $a_\lambda a_\gamma=a_\lambda$ whenever $\lambda<\gamma$. Wondering if, in fact, all C*-algebras have such an approximate unit led me to post this question.

Regardless, in general we can still get a weaker iterated form of directedness. Specifically, note that $(a+b)_{(0,1]}=a_{(0,1]}\vee b_{(0,1]}$, for all $a,b\in A^1_+$, so every open projection $r$ is a directed supremum of open projections $q$, which are all themselves directed supremums of open projections $p$ with $\overline{p}\leq q$.

Also note the relation $\overline{N}\subseteq O$ can be expressed purely in terms of open sets as $$\text{there exists open $M$ disjoint from $N$ such that }X=M\cup O.$$ Wondering if, for open projections $p,q\in A^{**}$, $\overline{p}\leq q$ can also be expressed similarly in terms of open projections just led me to post this question. In fact, it is really relation 2. of that question, used with iterated directness as above, that Rosický considers as the appropriate notion of regularity for C*-algebras in the paper mentioned above.

For complete regularity one replaces $\overline{N}\subseteq O$ with the statement that there exist open $(O_r)$ for $r\in[0,1]$ (or $[0,1]\cap\mathbb{Q}$, as in Rosický's paper) such that $N\subseteq O_0$, $O_1\subseteq O$ and $\overline{O}_s\subseteq O_t$ whenever $s<t$. Again, the analogous statement for complete regularity holds for open projections in C*-algebras although again it is not clear if the supremum can always be replaced with a directed supremum.