Is a Banach sublattice also a band?

Question

Let $A$ be a Banach lattice. If $B \subset A$ is a Banach lattice, is $B$ a band?

It is well known that if $C$ is a band in $A$, then $C$ is also a Banach lattice. But is it true that a Banach lattice $B$ is a band?

Answer 1

No. Take $A = C[0,1]$ with the pointwise order, and take $B$ to be the set of functions in $A$ that vanish at $0$. Then $B$ is a Banach lattice, but if $f_n \in B$ are such that $$ f_n(x) = \begin{cases} 0 &: x = 0 \\ 1 &: 1\geq x\geq 1/n \end{cases} $$ Then $\sup\{f_n\} = 1$ (the constant function), which is not in $B$, so $B$ is not a band.

In fact, one can show that a subset of $C[0,1]$ is a band iff it coincides with the set of functions that vanish on a regular closed set.