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{\bf ERRATA for LIMIT ALGEBRAS}\\
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\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.
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