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In this short note, we show that the generalized group algebra $L^1(G,A)$ isnot strongly Arens irregular for any infinite discrete group $G$ and any Banachalgebra $A$ with an orthonormal Schauder basis equivalent to the canonicalSchauder basis of $\ell^1(\mathbb N)$. This is interestingly in contrast to thecelebrated result by Losert and Lau that the group algebra $L^1(G)$ is stronglyArens irregular for any locally compact group $G$.

Failure of strong Arens irregularity in generalized group alegebras