TY - JOUR

T1 - Approximately multiplicative maps from weighted semilattice algebras

AU - Choi, Yemon

N1 - http://journals.cambridge.org/action/displayJournal?jid=AJZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 95 (1), pp 36-67 2013, © 2013 Cambridge University Press.

PY - 2013/8

Y1 - 2013/8

N2 - We investigate which weighted convolution algebras ℓ1ω(S), where S is a semilattice, are AMNM in the sense of Johnson [‘Approximately multiplicative functionals’, J. Lond. Math. Soc. (2) 34(3) (1986), 489–510]. We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all ℓ1ω(S) where S has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein [‘Strong Ditkin algebras without bounded relative units’, Int. J. Math. Math. Sci. 22(2) (1999), 437–443]. We also investigate when (ℓ1ω(S),M2) is an AMNM pair in the sense of Johnson [‘Approximately multiplicative maps between Banach algebras’, J. Lond. Math. Soc. (2) 37(2) (1988), 294–316], where M2 denotes the algebra of 2×2 complex matrices. In particular, we obtain the following two contrasting results: (i) for many nontrivial weights on the totally ordered semilattice Nmin, the pair (ℓ1ω(Nmin),M2) is not AMNM; (ii) for any semilattice S, the pair (ℓ1(S),M2) is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent 2×2 matrices.

AB - We investigate which weighted convolution algebras ℓ1ω(S), where S is a semilattice, are AMNM in the sense of Johnson [‘Approximately multiplicative functionals’, J. Lond. Math. Soc. (2) 34(3) (1986), 489–510]. We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all ℓ1ω(S) where S has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein [‘Strong Ditkin algebras without bounded relative units’, Int. J. Math. Math. Sci. 22(2) (1999), 437–443]. We also investigate when (ℓ1ω(S),M2) is an AMNM pair in the sense of Johnson [‘Approximately multiplicative maps between Banach algebras’, J. Lond. Math. Soc. (2) 37(2) (1988), 294–316], where M2 denotes the algebra of 2×2 complex matrices. In particular, we obtain the following two contrasting results: (i) for many nontrivial weights on the totally ordered semilattice Nmin, the pair (ℓ1ω(Nmin),M2) is not AMNM; (ii) for any semilattice S, the pair (ℓ1(S),M2) is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent 2×2 matrices.

KW - AMNM

KW - approximate homomorphism

KW - Feinstein algebra

KW - Semilattice

KW - weighted convolution algebra

U2 - 10.1017/S1446788713000189

DO - 10.1017/S1446788713000189

M3 - Journal article

VL - 95

SP - 36

EP - 67

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

SN - 1446-7887

IS - 1

ER -