I saw this particular question on stackexchange. Since there has been zero answers and since I've been interested in this question myself I want to ask it here. Given the $C^{\ast}$-algebra of bounded operators $\mathcal{L}(\mathcal{H})$ on some separable Hilbert space. Then the Schatten class operators of order $p \geq 2$ denoted $\mathcal{S}^p(\mathcal{H})$ is an ideal and the quotient $\mathcal{L} / \mathcal{S}^p$ is an associative algebra again. What is the "appropriate" topology on this quotient? It does not seem to be easily completely normable in a non-trivial way. The reason why I am interested in this is the connection to singular integral operators and index theory. If we have an exact sequence and can view this quotient as some kind of locally convex algebra we may apply K-theory, obtain six-term exact sequence etc.. (K-theory after all also "works" for locally convex algebras as shown by Cuntz). Sorry, if this question turns out to be elementary and unsuitable for MO.
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asked Aug 1, 2014 at 18:27
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2$\begingroup$ One small comment: one should probably try to understand the "commutative version" $\ell_\infty/\ell_p$ first $\endgroup$– Yemon ChoiCommented Aug 2, 2014 at 2:27
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