TY - JOUR
T1 - Corrigendum to “Cartan subalgebras for non-principal twisted groupoid C⁎-algebras” (Journal of Functional Analysis (2020) 279(6), (S0022123620301543), (10.1016/j.jfa.2020.108611))
AU - Brown, Jonathan H.
AU - Gillaspy, Elizabeth
N1 - Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2022/3/15
Y1 - 2022/3/15
N2 - There is a flaw in the proof of [2, Proposition 4.1]. Namely, we assume that if [Formula presented] , where [Formula presented] , then [Formula presented]. This is false in general. However, the proof can be salvaged, as follows. This correction has been discussed with all of the authors of [2]. Proof Suppose [Formula presented] where each [Formula presented] is a finite linear combination of elements from N, and [Formula presented]. To begin, we will assume [Formula presented]. Fix [Formula presented] , so that [Formula presented]. Define [Formula presented] and choose Q so that if [Formula presented] we have [Formula presented]. Since [Formula presented] by hypothesis, and the conditional expectation [Formula presented] is linear and contractive (hence continuous), [Formula presented] Observing that for [Formula presented] we have [Formula presented] it follows that [Formula presented]. As we assumed [Formula presented] , there must exist at least one i for which [Formula presented]. In particular, as [Formula presented] , there is an open neighborhood U of x such that [Formula presented]. Therefore, by Lemma 4.4, we have [Formula presented] and [Formula presented] for all [Formula presented]. Since [Formula presented] and [Formula presented] , we have now shown that every element [Formula presented] of the Weyl groupoid G (with [Formula presented]) can be written as [Formula presented] for some [Formula presented]. Furthermore, the fact that [Formula presented] for all [Formula presented] and [Formula presented] means that our restriction [Formula presented] does not actually restrict which groupoid elements we can apply the above logic to. This completes the revised proof of [2, Proposition 4.1]. □ We also record here the observation that all of the results [2, Section 4], with the exception of Lemma 4.3, still hold if B is not assumed maximal abelian, as long as Φ is a faithful conditional expectation. In particular, even in the more general setting of [1], every element of the Weyl groupoid can be described using only elements of the dense spanning set N of normalizers.
AB - There is a flaw in the proof of [2, Proposition 4.1]. Namely, we assume that if [Formula presented] , where [Formula presented] , then [Formula presented]. This is false in general. However, the proof can be salvaged, as follows. This correction has been discussed with all of the authors of [2]. Proof Suppose [Formula presented] where each [Formula presented] is a finite linear combination of elements from N, and [Formula presented]. To begin, we will assume [Formula presented]. Fix [Formula presented] , so that [Formula presented]. Define [Formula presented] and choose Q so that if [Formula presented] we have [Formula presented]. Since [Formula presented] by hypothesis, and the conditional expectation [Formula presented] is linear and contractive (hence continuous), [Formula presented] Observing that for [Formula presented] we have [Formula presented] it follows that [Formula presented]. As we assumed [Formula presented] , there must exist at least one i for which [Formula presented]. In particular, as [Formula presented] , there is an open neighborhood U of x such that [Formula presented]. Therefore, by Lemma 4.4, we have [Formula presented] and [Formula presented] for all [Formula presented]. Since [Formula presented] and [Formula presented] , we have now shown that every element [Formula presented] of the Weyl groupoid G (with [Formula presented]) can be written as [Formula presented] for some [Formula presented]. Furthermore, the fact that [Formula presented] for all [Formula presented] and [Formula presented] means that our restriction [Formula presented] does not actually restrict which groupoid elements we can apply the above logic to. This completes the revised proof of [2, Proposition 4.1]. □ We also record here the observation that all of the results [2, Section 4], with the exception of Lemma 4.3, still hold if B is not assumed maximal abelian, as long as Φ is a faithful conditional expectation. In particular, even in the more general setting of [1], every element of the Weyl groupoid can be described using only elements of the dense spanning set N of normalizers.
UR - http://www.scopus.com/inward/record.url?scp=85122540620&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2021.109354
DO - 10.1016/j.jfa.2021.109354
M3 - Comment/debate
AN - SCOPUS:85122540620
SN - 0022-1236
VL - 282
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 6
M1 - 109354
ER -