\documentstyle{article} \parskip 1ex \pagestyle{empty} \begin{document} \begin{center} {\bf ERRATA for LIMIT ALGEBRAS}\\ \end{center} \begin{enumerate} \item[\bf p18] :\ \ $v=x(x^*x)^{-1/2}$ \item[\bf p19] Line 7: Lemma 3.6 \item[] Notes:\ \ ...\ see E. Christensen, {\it Near inclusions of C*-algebras,} Acta Math. 144 (1980), 249-265. \item[\bf p27] :\ The condition (iii) is equivalent for block upper triangular digraph algebras but is not equivalent in general. \item[\bf p32] :\ $R^\prime$ \item[\bf 6.8] : \ Condition (ii)\ \ in the definition of a strongly regular embedding should be replaced by $\phi(N^s_{C_1}(A_1)) \subseteq N^s_{C_2}(A_2)$ where $N^s_{C}(A)$ is the subsemigroup of $\prec$-preserving elements. The given local reformulation of strong regularity is different and the resulting class is not closed under compositions. (This has implications for the proof of Theorem 11.11.) \item[\bf p46] Line -10:\ \ ...extensions \ $M_{2^{k+1}} \otimes C(S^1) \to M_{2^{k+2}} \otimes C(S^1)$\ for the images in $A$ of the inclusions $A(G_k) \to A(G_{k+1})$, and... \item[\bf p60] Line 4:\ \ .. such that $x \ne y$ and $(x,y) \in R(A)$ or $x = y$ and $(x^\prime,y^\prime) \in R(A^\prime)$.\\ Line 7:\ \ .. it is.. \item[\bf p61] Line -7:\ \ $\rho_k$ \item[\bf p63] :\ Lemma 8.1 is false as stated (even in for finite dimensional algebras) but remains true for many familes of digraph algebras including elementary algebras. \item[\bf p73] Par 2:\ \ appear\\ Line 9:\ \ principle \item[\bf p84] Prop. 9.5:\ \ $y_j$ and $x_j$ in place of $y_k$ and $x_{j-1}$ \item[\bf p89] Line 3: \ \ $A_V^-$ \item[\bf p91] Line 6: \ \ $q_3$ \item[\bf Proof 9.5] : replace \ \ \ $\overline{{\cal O}(x^+)}\cap \overline{{\cal O}(x)} = {x^+}$\ \ \ by \ \ \ $\overline{{\cal O}(x^+)}\setminus\overline{{\cal O}(x)} = \{x^+\}$ \item[\bf Proof 9.6] : The proof is incomplete since it is not clear, in the last paragraph, whether $q$ can be chosen with the desired properties. A complete proof is in {\it Lexicographic semigroupoids, } J. of Ergodic Th. and Dynamical Systems, to appear. \item[\bf p86] Line -3,-4: $B$ for $A$ and $A$ for $B$ \item[\bf p89] Lines 1-4: Confused argument. Instead note that one algebra has one maximal point and the other has two maximal points. \item[\bf p109] Line 10: \ \ $x_1$ not $x_1'$ \item[{\bf p114}] Last line: \ \ $D$ not $B$ \item[\bf p128] Line 5: (see (6.8))\\ Line 16: $M_\infty(A^\prime)$ \item[\bf Ex. 11.3] : \ \ ... \underline{triangular} elementary algebra .... \item[\bf 11.11] : The proof is incorrect partly because of the appeal to Exercise 11.3, which is only correct in the triangular case, but mainly because a composition of two embeddings which map matrix units into the strong normaliser need not have this property. Thus Lemma 11.13 cannot be used, nor a local version of it. The theorem is certainly true for triangular algebras defined by ordered Bratteli diagrams ([{\bf P11}]) and also true in various nontriangular contexts, notably the case of $2$ by $2$ block upper triangular building blocks - see [{\bf P11}] \ and D. Heffernan, {\it Uniformly $T_2$ algebras in approximately finite-dimensional C*-algebras} J. London Mathematical Soc., 44 (1997), 181-192. (The strong regularity assumption (order preserving morphisms) is not needed here.) I don't yet know of a counterexample to the general theorem (which should now be adorned with a $\dagger$). For related results see A.P. Donsig and A. Hopenwasser, {\em Order preservation in limit algebras}, J. Funct. Anal. 133 (1995), 342-394. \item[\bf p130] Line -4: \ \ ...a regular star-extendible... \item[\bf %Definition 11.19 (ii)] : \rm \ \ $h_{min}(\phi) \le -2|m|$ \item[\bf p137] \ \ $A(D_4)$ \item[\bf p140] ...is that, given the $K_0$ map, the $H_1$ map should have the correct congruence class mod 4. \item[\bf p141] Par -2: $X^\prime_1 = (\phi^\prime_j)_* \circ \dots \circ (\phi^\prime_1)_* \circ X_1$ \item[\bf p141] Line -2:\ \ .., and that $m_0$ in... \item[\bf p143] Definition 11.24: replace $"\le \epsilon"$ by $"\le \delta"$ and $"\le C \epsilon"$ by $"\le \epsilon"$ and rephrase accordingly.\\ Lemma 11.25: replace $"\le c \epsilon"$ by $\le c \epsilon^{1/2}$. \item[\bf p150] Line 12:\ \ \it principle \rm \item[\bf p153] (iv): $|y| < \alpha x$ \item[\bf p155] Line 5: between \ $\displaystyle{\lim_\rightarrow (A(D_4 \times K_{7^{2k}}), \phi_k)}$ \ and \ $\displaystyle{\lim_\rightarrow (A(D_4 \times K_{7^{2k+1}}), \phi^\prime_k)}$ \item[\bf p176] Line -5: 11.22 \end{enumerate} \vfill \noindent Thanks to Allan Donsig, David Heffernan, Alan Hopenwasser, Timothy Hudson and John Orr for spotting errors. \end{document}