Erratum: Generalized Group Algebras of Locally Compact Groups

An error was detected in “Generalized Group Algebras of Locally Compact Groups,” Communications in Algebra 36(9):3559–3563 (2008).

It has been brought to our attention that Lemma 9 of [1] is incorrect. Lemma 9 is used only in the proof of Proposition 10, but Proposition 10 is used in some of the results that follow it. We withdraw Lemma 9 and give here a new proof of Proposition 10, which is independent of the incorrect Lemma 9 and thus reinforce the correctness of all other results in [1].

Proposition 10 ([1])

If RG is right continuous, then R is right continuous.

Proof

By Lemma 8 [1], R is right quasi-continuous. To prove that R is right continuous, we only need to show that every right ideal of R isomorphic to a direct summand of R is itself a direct summand of R. Let I be a right ideal isomorphic to eR for some e = e ² ∈ R. Let σ: I → eR be a right R-module isomorphism. Then ϕ: IG → eRG given by for a _g  ∈ I and g ∈ G, is a right RG-module isomorphism. But since RG is right continuous, we have IG = fRG for some f = f ² ∈ RG. Suppose . Put . Define ψ: RG → R by . Then ψ(f) = m and m ² = ψ(f)ψ(f) = ψ(f ²) = ψ(f) = m. Thus m is an idempotent in R. Now, I = ψ(IG) = ψ(fRG) = ψ(f)ψ(RG) = mR. This shows that I is a direct summand of R. Therefore, R is right continuous.

In view of the above, the main theorem of [1] for generalized group algebras still holds true.

Theorem 11 ([1])

If L ¹(G, A) is right continuous then G is finite and A is right continuous.

ACKNOWLEDGMENT

We would like to thank Evrim Akalan for pointing out the error.