I would like to know if the following is true :


Let $\mathcal{H}$ be the complex Hilbert space $L^2([0,1])$ for the Lebesgue measure.
Let $q$ be the orthogonal projection on the subspace of $\mathcal{H}$ spammed by the $(\exp(\pm 2 i \pi 2^k x ))\_{k \in \mathbb{N}}$.


Let $f\_n$ be a sequence of elements of $\mathcal{H}$ such that :


* $\Vert f\_n \Vert\_2 =1$
* $\displaystyle \forall \varepsilon >0, \lim\_{n \rightarrow \infty} \int\_0^\varepsilon |f\_n|^2 = 1$.


Then :


$\lim\_{n \rightarrow \infty} \Vert q f\_n \Vert\_2 = 0$


In the first place I thought it will be false, but I haven't been able to found a counterexample and in second thought It is reasonable because functions in the image of $q$ have quasi-periodicity property, and hence it shouldn't be possible to produce of sequence with the desired properties inside the image of $q$.


**The motivation** for the question lie in the paper of F.W. Shultz "[Pure states as a dual object for C-algebras](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-82/issue-4/Pure-states-as-a-dual-object-for-C-algebras/cmp/1103920655.full)". In the end of this paper, the author give an exemple of a non perfect type one $C^\*$-algebra : the $C^\*$-algebra generated in $B(\mathcal{H})$ by the compact operator and $\mathcal{C}([0,1])$ acting on $\mathcal{H}$ by multiplication.
The pure state of this algebra are :


* The vectors state from the action on $\mathcal{H}$
* the character of $\mathcal{C}([0,1])$ extended to $C$ by sending all the compact operator to $0$.


Hence the atomic part of the enveloping algebra is $B(\mathcal{H}) \oplus l^{\infty}([0,1])$ (where $l^2$ is for the counting measure on the discret set.) and (if I'm understanding well) the author affirm without proof that $(q,0)$, is continuous on the stat space. I'm not convinced by this fact, and I would like to understand this point better, and my question is (if I'm not mistaken) equivalent to this continuity at the character $\mathrm{ev}\_0$ (the sequence $f\_n$ is exactly a sequence of vector state converging weakly to the character $\mathrm{ev}\_0$, and $\Vert f\_n q \Vert\_2$ is the evaluation of $q$ on the state corresponding to $f\_n$. )


Thanks !



Non-perfect type one C^*-algebra, and a lemma in Fourier analysis
It is true. More precisely, if $P\_n$ is the orthogonal projection on the space of functions supported in $[0,1/2^n]$, I claim that $\| P\_n q P\_n \| \leq (2n+2) 2^{-n}$. This implies what you are asking for.


Here is a proof. For $k \in \mathbb Z$, let $e\_k$ be the function $e^{sgn(k) 2i\pi 2^{|k|}}$ and $q(k)$ the orthogonal projection on $\mathbb C e\_k$, so that $q=\sum\_k q(k)$. $P\_n e\_k$ has norm $2^{n/2}$, and the family $(P\_n e\_k)\_{|k|>n}$ is orthogonal. Hence,
$$\| P\_n q P\_n \| \leq \sum\_{|k|\leq n} \| P\_n q(k) P\_n\| + \|\sum\_{|k|>n} P\_n q(k) P\_n\| = (2n+1) 2^{-n} + 2^{-n}.$$


