Let $A = C_r^*(S_\infty)$ where $S_\infty$ - is permutation group of natural numbers fixing all but a finite number of element. Let $A_n = C_r^*(S_n)$ - subspaces of $A$ and $P_n : A \to A$ is projector on $A_n$.
I have sequence $x_n \in A_n$ for which $P_{n-1}(x_n) = 0$, $P_{n+1} (x_n) = x_n$.
Is it true, that convergence of norms of partial sums $\|x_1 + ... + x_n\|$ implies convergence of partial sums $x_1 + ... + x_n$?
I have similar question before. But that question is more specific and more nontrivial (maybe). Any remarks, allusions and intuitive arguments are welcome!
You have proper inclusions of finite-dimensional C $^*$-algebras $$\tag{1}A_1\subset A_2\subset\cdots $$ embedded in a unital way.
Each of these is a direct sum of full matrix algebras, and the embeddings are tracked by the Bratelli diagram. By looking at a single path in the diagram, we get an inclusion $$\tag{2} M_{n_1}(\mathbb C)\hookrightarrow M_{n_2}(\mathbb C)\hookrightarrow \cdots$$ with $n_k|n_{k+1} $ and the embeddings of the form $$\tag{3} a\longmapsto a\oplus\cdots \oplus a . $$ The restriction of the canonical conditional expectation is "compression to the (block) diagonal".
Note that because the dimensions of $A_1,A_2,\ldots$ are always properly increasing, we may assume that the path chosen in $(2)$ produces increasing sizes of the blocks ($n_k<n_{k+1}$), if necessary by taking a subsequence; if all paths were bounded in dimension, then the direct limit $\varinjlim A_n$ would be finite-dimensional.
With $e_{kj}^{m} $ the canonical matrix units of $M_m(\mathbb C)$, let $x_k=e_{k,n_k}^k $. Because of the form of the embeddings $(3)$, the first row elements are shared between the algebras, i.e. $$ e^k_{k,j}=e^h_{k,j},\ \ \ \text{ if }\ h\geq k. $$ As the conditional expectation is compression, $P_{n-1}(e_{k,n_k}^k)=0$ if $k> n-1$.
Now, \begin{align} \|x_1+\cdots+x_n\|^2&=\|(x_1+\cdots+x_n)^*(x_1+\cdots+x_n)\|\\ \ \\ &=\left\|\sum_{k,j=1}^n x_k^*x_j\right\| =\left\|\sum_{k,j=1}^n e_{n_k,k}e_{j,n_j}\right\|\\ \ \\ &=\left\| \sum_{k=1}^n e_{n_k,n_k}\right\|=1. \end{align} Finally, $$ \|(x_1+\cdots+x_{n+1})-(x_1+\cdots+x_n)\|=\|x_{n+1}\|=1, $$ so the sequence of partial sums is not Cauchy.