TY - JOUR
T1 - Approximate amenability of tensor products of Banach algebras
AU - Ghahramani, F.
AU - Loy, R. J.
N1 - Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/10/15
Y1 - 2017/10/15
N2 - Examples constructed by the first author and Charles Read make it clear that many of the hereditary properties of amenability no longer hold for approximate amenability. These and earlier results of the authors also show that the presence of a bounded approximate identity often entails positive results. Here we show that the tensor product of approximately amenable algebras need not be approximately amenable, and investigate conditions under which A and B being approximately amenable implies, or is implied by, A⊗ˆB or A#⊗ˆB# being approximately amenable. Once again, the rôle of having a bounded approximate identity comes to the fore. Our methods also enable us to prove that if A⊗ˆB is amenable, then so are A and B, a result proved by Barry Johnson in 1996 under an additional assumption.
AB - Examples constructed by the first author and Charles Read make it clear that many of the hereditary properties of amenability no longer hold for approximate amenability. These and earlier results of the authors also show that the presence of a bounded approximate identity often entails positive results. Here we show that the tensor product of approximately amenable algebras need not be approximately amenable, and investigate conditions under which A and B being approximately amenable implies, or is implied by, A⊗ˆB or A#⊗ˆB# being approximately amenable. Once again, the rôle of having a bounded approximate identity comes to the fore. Our methods also enable us to prove that if A⊗ˆB is amenable, then so are A and B, a result proved by Barry Johnson in 1996 under an additional assumption.
KW - Amenable Banach algebra
KW - Approximate identity
KW - Approximately amenable Banach algebra
KW - Tensor product
UR - http://www.scopus.com/inward/record.url?scp=85019592230&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2017.05.018
DO - 10.1016/j.jmaa.2017.05.018
M3 - Article
SN - 0022-247X
VL - 454
SP - 746
EP - 758
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -