Minor corrections to published papers

These are listed in the approximate order in which the errors were found. My thanks to those who have written to me pointing some of these out.

  1. [GMJ '06] Simplicial homology and Hochschild cohomology of Banach semilattice algebras
  2. [HJM '10] Simplicial homology of strong semilattices of Banach algebras
  3. [JFA '09] Approximate and pseudo-amenability of various classes of Banach algebras
  4. [QJM '11] Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras
  5. [QJM '10] Hochschild homology and cohomology of $\ell^1({\mathbb Z}_+^k)$
  6. [IEOT '10] Group representations with empty residual spectrum
  7. [PAMS '11] Characterizing derivations from the disk algebra to its dual

1. Simplicial homology and Hochschild cohomology of Banach semilattice algebras

The remark at the bottom of page 232 (on decomposing commutative semigroups) is not quite right, as is clear from considering the commutative semigroup ${\mathbb N}$ with usual addition. This can be fixed by replacing the words "where $L$ is the set of idempotents in $S$" with "where $L$ is a semilattice". The error does not affect any of the other results stated or proved in the paper.

There is a minor typographical error on p. 237, line 16, where $\operatorname{Ext}_{R^e}$ should have a superscript $n$.

2. Simplicial homology of strong semilattices of Banach algebras

Halfway down page 242 in the published version (the paragraph before Lemma 2.9), the sentence beginning "If the semilattice L..." should be prefaced by the extra hypotheses

If $A_1$ is unital and every transition homomorphism is unital...

(This does not affect Lemma 2.9 itself.)

3. Approximate and pseudo-amenability of various classes of Banach algebras

The final remark in Section 3 is not quite right. It is true that for $1/2 \lt \alpha \lt 1$, the algebra $\operatorname{lip}_\alpha({\mathbb T})$ is known not to be weakly amenable; but no such general result is true if we replace ${\mathbb T}$ by an arbitrary infinite compact metric space. (For instance, consider the Cantor space inside $[0,1]$, equipped with the subspace metric.) Note, however, that this mis-statement does not affect the validity of Corollary 3.8.

4. Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras

There is a typo near the bottom of page 47 in the published version: in the definition of the function $b_r$, we should have

$$b_r(x) = 2- r^{-1}\vert x\vert \quad\text{if $r\leq \vert x \vert \leq 2r$} $$

5. Hochschild homology and cohomology of $\ell^1({\mathbb Z}_+^k)$

Certain "corrections" were made by sub-editors at the proof stage, which I failed to pick up on. For the record: throughout the paper there is a tendency to refer to "the Hochschild (co)homology" where I would personally say "Hochschild (co)homology". The definite article seems more appropriate when one is referring to the Hochschild cohomology of a particular algebra; whereas in other contexts, "Hochschild cohomology" is an abstract concept that does not need a definite article, like "solar radiation" or "gravitational lensing". See the last line of page 9 for a similar example.

While Proposition 7.2 is correct, the "proof" given in the paper is not: it only shows that $N'$ is injective as a left $A$-module, not as an $A$-bimodule. However, a small modification of the argument is enough to obtain a valid proof.

6. Group representations with empty residual spectrum

There is an official erratum for this paper, which appeared as Int. Eq. Op. Th. 69 (2011), no. 1, 149--150.

7. Characterizing derivations from the disk algebra to its dual

In Proposition 2.5, the hypothesis in part (i) is mis-stated. What is actually proved is the following statement:

If $f\in A$ and $f$ is real-valued then $D(f)=0$.
Correspondingly, in the 4th line after Equation (2.2), it should read
If $\operatorname{Re} h = 0$, then $ih$ is a real-valued function in $A$, and hence by part $(i)\dots$