Topologizing $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$

Question

Given a separable Hilbert space $\mathcal{H}$ I would like to know how one could topologize the quotient algebra $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$? Here $\mathcal{L}(\mathcal{H})$ denotes the bounded operators on $\mathcal{H}$ and $\mathcal{S}^p(\mathcal{H})$ are the Schatten class operators of order $p \geq 2$. The latter is an ideal in the $C^{\ast}$-algebra of bounded operators. But the quotient has no obvious interpretation as a Banach algebra. I strongly suspect any non-trivial complete algebra topology will turn out to be non-normable (since the quotient is certainly infinite-dimensional). My first attempt is to order the singular numbers for a given $S \in \mathcal{S}^p$ and define the semi-norms for $T$ in the quotient: $$ p_i(T + \mathcal{S}_p) := \inf_{S \in \mathcal{S}^p} \mathrm{max}\{\|T + S\|, \sigma_i(S)\} $$ or something like that... Can one recommend books/papers where this is spelled out? A first search did not yield anything for me.