Timeline for Two $E_\infty$ structures on infinite matrices

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Dec 5, 2023 at 14:03	comment	added	guest313131		So, by Dunn additivity (which says an $E_n$-algebra in $E_1$-algebras is an $E_{n+1}$-algebra) and the Eckmann-Hilton argument (which says the $n+1$ multiplications of an $E_{n+1}$ algebra given by Dunn additivity are all equivalent) we can conclude that O (with matrix multiplication) is equivalent to O (with direct sum of block matrices) as an $E_1$-algebra; and in fact, as an $E_\infty$-algebra. Is that correct?
Dec 5, 2023 at 12:04	comment	added	John Rognes		The block sum of matrices extends to make $O$ an $E_\infty$-algebra in topological groups, and forgetting this operad action leaves you with the group $O$ under matrix multiplication. Reformulated after applying the bar construction: The Whitney sum of vector bundles extends to make $BO$ an $E_\infty$-algebra in spaces, and forgetting this operad action leaves you with the space $BO$. The Eckmann-Hilton argument corresponds to the block matrix with entries A, 0; 0; B being conjugate to the matrix with entries AB, 0; 0; I.
Dec 5, 2023 at 9:55	answer	added	Neil Strickland		timeline score: 6
S Dec 5, 2023 at 9:04	review	First questions
Dec 5, 2023 at 9:50
S Dec 5, 2023 at 9:04	history	asked	guest313131	CC BY-SA 4.0