2022/10/29 by M. J. Mehdipour, Mehdipour, M. J., Ali Rejali +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2210.16596
openalex publication_date 2022/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate homological properties of Banach algebras. We show that retractions Banach algebras preserve biprojectivity, contractibility and biflatness. We also prove that contractibility of second dual of a Banach algebra implies contractibility of the Banach algebra. For a Banach algebra A with Δ(A)≠∅, let \frakF(X, A) be one of the Banach algebras Cb(X, A), C0(X, A), \hboxLipα(X, A) or \hboxlipα(X, A). In the following, we study homological properties of Banach algebra \frakF(X, A), especially contractibility of it. We prove that contractibility of \frakF(X, A) is equivalent to finiteness of X and contractibility of A. In the case where, A is commutative, we show that \frakF(X, A) is contractible if and only if A is a C^*-algebra and both X and Δ(A) are finite. In particular, \hboxlipα0(X, A) is contractible if and only if X is finite. We also investigate contractibility of L1(G, A) and establish L1(G, A) is contractible if and only if G finite and A is contractible. Finally, we show that biprojectivity of the Beurling algebra L1(G, ω) is equivalent to compactness of G, however, biprojectivity of the Banach algebras L1(G, ω)** is equivalent to finiteness of G. This result holds for the Banach algebra M(G, ω)** instead of L1(G, ω)**.