We introduce a new concept of \(\sigma\)-biflatness for Banach algebras. We show difference between this new concept and the classical one. A characterization of \(\sigma\)-biflatness is also given. As we shall see, \(\sigma\)-biflatness is closely related to the notions of \(\sigma\)-amenability and \(\sigma\)-biprojectivity. We compare our definition with the existing notion of \(\sigma\)-biflat Banach algebras in the sense of Ghorbani and Bami (Bull Iran Math Soc 39(3):507–515, 2013). In addition, \(\sigma\)-biflatness of tensor product of Banach algebras is discussed.
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We would like to thank the anonymous reviewer(s) for much helpful advice that improved the presentation of the paper, especially for directing us to the reference Mirzavaziri and Moslehian (2011) and pointing out Theorems 4.7 and 5.2.