Are the algebras of compact operators $K(\ell_p)$ and $K(\ell_q)$ isomorphic as Banach algebras for $1\leq p<q<\infty$?
2026-03-29 05:50:11.1774763411
Algebra of compact operators on $\ell_p$
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No, they are not isomorphic as Banach algebras. By the proof of Eidelheit's theorem, if $A_1$ and $A_2$ are subalgebras, respectively, of $B(X)$ and $B(Y)$ that contain all finite-rank operators, ($X$ and $Y$ are Banach spaces) and are Banach-algebra isomorphic, then $X$ and $Y$ are Banach-space isomorphic.
This was rediscovered in this paper which surprisingly does not mention Eidelheit's result.