j_1 \in \{2, \ldots, \ell+2\} with \mu_1 \in (\overline{Q_B(Z)}^{I_{j_1}})_{I_{j_1-1}}. Similar, to \mu_2 := (\lambda_1, \lambda_2) \in Q_B(Z), there exists j_2 \in \{2, \ldots, \ell+2\} with \mu_2 \in (\overline{Q_B(Z)}^{I_{j_2}})_{I_{j_2-1}}.

Suppose j_2 < j_1. Since Q_B(Z)^{I_{j_2}} is open, by (2.5) there exist open sets V_k \subseteq \mathbb{R}^{H_k} with \lambda_k \in V_k, k = 1, 2, such that V_1 \times V_2 + V_1 \cup V_2 \subseteq Q_B(Z)^{I_{j_2}} \subseteq Q_B(Z)^{I_{j_1-1}} which contradicts our choice of j_1.

Now suppose j_1 = j_2, i.e. \mu_1, \mu_2 \in (\overline{Q_B(Z)}^{I_{j_1}})_{I_{j_1-1}}. By Lemma 3.4 the space (\overline{Q_B(Z)}^{I_{j_1}})_{I_{j_1-1}} is Hausdorff in its relative topology, which is impossible because we have \mu_1 \in W_2 \cap W_1 for any choice of open sets W_k \subseteq Q_B(Z) with \mu_k \in W_k.

Consequently, we have j_1 < j_2, and iterating this procedure leads to a sequence
\[
2 \le j_1 < \cdots < j_{k_0} \le \ell + 2
\]
which gives \ell \ge k_0 - 1, and completes the proof in the case k_0 = n. On the other hand, if k_0 < n note that by Lemma 3.4 we have
\[
\psi\left( \bigoplus_{k=1}^{k_0} (\overline{Q_B(Z)}^{I_{j_k}})_{I_{j_k-1}} \right) = \bigoplus_{k=1}^{k_0} T_{j_k-1} \subseteq \bigoplus_{j=1}^{\ell+1} T_j
\]
where each T_{j_k} corresponds to a \mu_k, i.e. to an infinite-dimensional irreducible representation. Since Q_B(Z) also has one-dimensional irreducible representations by Theorem 2.1 and (2.4), we have k_0 + 1 \le \ell + 1, i.e. the length of Q_B(Z) is at least k_0. \hfill $\square$

Of course, a combination of Lemma 3.1, Lemma 3.2, and Lemma 3.5 completes the proof of Theorem 1.1.

It remains to prove Proposition 1.2. Recall that the composition series (1.1) is locally stratified if for any irreducible representation \pi of B with \pi(J_k) \ne \{0\} we have \pi(J_{k+1}) \ne \pi(J_k) \cite{13}. Moreover, a solving, stratified composition series coincides with the maximal radical series by \cite{13}. If we abbreviate \ell := \mathrm{length}\, Q_B(Z), the series
\begin{equation}
Q_B(Z) \supseteq J_{\ell+1} \supseteq J_\ell \supseteq \cdots \supseteq J_1 = \{0\}
\tag{3.1}
\end{equation}
with J_k given by (2.3) is solving of minimal length by Theorem 1.1 and the proof of Lemma 3.2. Using the description (2.4) of the irreducible representations of Q_B(Z) and the compatibility conditions (2.1) and (2.2), it is straightforward to check that the series (3.1) is stratified. Thus, it is the maximal radical series for Q_B(Z), hence, an application of Lemma 3.1 completes the proof of Proposition 1.2.

\section*{Acknowledgment}

The author would like to thank Professor B. Gramsch for drawing the author's attention to the b-calculus, Professor B. A. Plamenevskij for suggesting the problem of describing the Jacobson topology of the algebra B(Z, b\Omega^{1}_{2}), and Professor A. Yu. Kokotov for raising the question what the length of B(Z, b\Omega^{1}_{2}) is at the Spring School "Operator algebras with symbolic structure on singular manifolds, index theory, and asymptotics" at Potsdam (February, 98).

Finally, the author wishes to thank Professor R. B. Melrose and the Massachusetts Institute of Technology, where this note was written, for the invitation and hospitality.