Convergence of specific sequence in reduced group algebra

Question

I have $B = C_r^* (S_\infty)$ - reduced group algebra of permutation group of naturam numbers fixing all but a finite number of element. $B$ have countable family of subalgebras: $B_n = C_r^* (S_n)$ and I have sequence of elements $x_n \in B_n \setminus B_{n-1}$. Is it true that convergence of norms of partial sums $||x_1 + ... + x_n||$ implies convergence of series $x_1+x_2+x_3+...$ ?

It may be true because $x_n/||x_n||$ is Schauder basis of linear span of $x_1,x_2,x_3,...$ What are you think about that? Any remarks, allusions and intuitive arguments are welcome!

Thanks!

Answer 1

Each $B_n$ can be seen as $M_n(\mathbb C)$, with the embeddings $$\gamma_{n,n+1}:X\longmapsto \begin{bmatrix}B&0\\0&1\end{bmatrix}.$$ Let $\{e_{kj}^{(n)}\}$ denote the matrix units of $M_n(\mathbb C)$, and define $x_n=e_{nn}$. Then, for all $n$, $$ \|x_1+\cdots+x_n\|=1, $$ while $$\|(x_1+\cdots+x_m)-(x_1+\cdots+x_n)\|=1$$ if $m\ne n$.