Corrigendum to ‘A Stinespring type theorem for completely positive multilinear maps on Hilbert C*-modules’ [Linear Multilinear Algebra 67 (2019), 121–140]

Abstract

Throughout Section 5, ϕ is an invariant symmetric completely positive and completely bounded multilinear map and (Formula presented.) is its symmetric representation associated, which is a multilinear map. We affirm that ‘If, in addition, (Formula presented.) is commutative, we see that (Formula presented.) for all (Formula presented.).’ (page 136, lines −7 and −8), which is not true because (Formula presented.) is not linear and so the equality (Formula presented.) does not hold (page 136, lines −4 and −3). Also, we affirm that ‘Then (Formula presented.) extends to a representation of (Formula presented.) ’ (page 137, line 5), and ‘By the universal property of the crossed product (Formula presented.), the map (Formula presented.) extends to a representation of (Formula presented.) into (Formula presented.), still denoted by (Formula presented.) ’, (page 137, lines 14–15), which are not true because (Formula presented.) defined in (7) (page 136) is not linear. Unfortunately, in the proof of Theorem 5.2 we use the linearity of the map ϕ (page 139, line 4), and we do not know how to eliminate this gap. For the accuracy of the exposure, we use this opportunity to make the following corrections: On the page 125, line 9 (Remark 2.7 (Formula presented.)),‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.On the page 127, line −2, the page 128, line 13 and the page 129, line 6,‘the linear span of (Formula presented.) (Formula presented.) ’ should be replaced by ‘the linear span of (Formula presented.) ’.On the page 128, lines 1, 4, 10 and 11,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.On the page 128, lines 6, 7, 8, 9 and 10,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.On the page 128, line 11,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.On the page 129, lines 1 and 5,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.On the page 129, line 12,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.on the page 132, lines 32,‘ (Formula presented.) ’ must replaced by ‘ (Formula presented.) ’.On the page 133, line 5,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.Throughout the proof of Theorem 4.4,‘ (Formula presented.) ’ must replaced by ‘ (Formula presented.) ’.On the page 134, line 22,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.Throughout Section 5 ‘ (Formula presented.) ’, ‘ (Formula presented.) ’ and ‘ (Formula presented.) ’ should be replaced by ‘dt’, ‘ds’ respectively ‘dr’.On the page 137, line (Formula presented.) and on the page 138, line 7,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.On the page 137, line (Formula presented.) and on the page 138, line 16,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.On the page 138, line 5,‘ (Formula presented.) ’ should be replaced by ‘ (Formula presented.) ’.