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- W3025528407 abstract "The main result in the above-mentioned paper, namely the existence and uniqueness theorem for host algebras, is wrong. The existence and uniqueness theorem for host algebras, as given in [2, Theorem 3.4], is wrong for the following reasons. In Definition 3.1 we wanted to generalise the concept of an open projection away from universal enveloping Von Neumann algebras, so that it makes sense for any Von Neumann algebra N . We defined a projection P ∈ N as open, if NP is the intersection of left kernels of normal states of N . Unfortunately, every projection P ∈ N will satisfy this criterion, by [3, Theorem 3.6.11 and its proof]. In particular, if N = A′′ for a C*-algebra A, then this definition of an open projection does not define the usual open projections (cf. [3, Proposition 3.11.9]). Hence the proof of [2, Theorem 3.4], which rests on this, fails. The proof of [2, Theorem 3.4] cannot be repaired, because there is no intrinsic way to define the open projections in N = A′′ without the knowledge of the subalgebra A ⊂ N . In fact, the knowledge of the open projections in N will uniquely determine the subalgebra A ⊂ N by the subsequent argument in the proof of [2, Theorem 3.4], using the characterisation of M(A)sa as q-continuous elements of N in [1]. Since one can find a different C*-subalgebra B ⊂ A′′ such that B′′ = A′′ and B = A (for example, by unitary conjugation of A in A′′), we see that the open projections with respect to the two subalgebras A, B must differ, but as they produce the same Von Neumann algebra N = B′′ = A′′, their open projections cannot be defined purely from N itself, without the knowledge of how the subalgebras are included in N . This mistake affects most of Section 3, as well as the isomorphism claim in line 2, p. 745. The rest of the paper is not affected by it nor, as far as the author knows, has it compromised any other published results. Acknowledgement. I wish to thank Professor Karl-Hermann Neeb for pointing out this error." @default.
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- W3025528407 date "2008-01-01" @default.
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- W3025528407 title "Corrigendum: generalising group algebras" @default.
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