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Some Cohomological Notions in Banach Algebras Based on Maximal Ideal Space

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Abstract

In this paper, we investigate the homological notion of left \(\phi \)-biprojectivity for certain Banach algebras, where \(\phi \) is a nonzero multiplicative linear functional. Our initial result states that this notion is equivalent to left \(\phi \)-contractibility, provided that \(\mathcal {A}\) has a left approximate identity. As an application, we study left \(\phi \)-biprojectivity of Banach algebras related to locally compact groups. For instance, we show that for a locally compact group \(\mathcal {G}\), the Segal algebra \(S(\mathcal {G})\) is left \(\phi \)-biprojective if and only if \(\mathcal {G}\) is compact and the Fourier algebra \(A(\mathcal {G})\) is left \(\phi \)-biprojective if and only if \(\mathcal {G}\) is discrete. Finally, we give some examples which show the differences between our new notion and the classical ones.

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Acknowledgements

The authors wish to thank the anonymous referee for his/her careful reading of the manuscript and his/her useful suggestions and comments. The first author is thankful to Ilam university, for its support.

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Correspondence to Mehdi Rostami.

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Sahami, A., Rostami, M. Some Cohomological Notions in Banach Algebras Based on Maximal Ideal Space. Iran J Sci Technol Trans Sci 46, 173–179 (2022). https://doi.org/10.1007/s40995-021-01224-y

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