Schatten class operators form Banach algebra?

Question

I am reading about Schatten $p$-class operators. Denote by $S_p(H)$ the space of all bounded linear operators with finite Schatten $p$-norm. I know that $S_p(H)$ is an ideal of $B(H)$ and is a Banach space with respect to $p$-norm. I was wondering if it is a Banach algebra as well?(I know they form Banach algebra for p=1,2) Also can someone suggest me some references where I can read about these in detail?

Answer 1

Just use the module property $\| a \, b \, c\|_{p} \leq \| a\|_\infty \| b \|_{p} \| c \|_\infty$ and use fact that $\| x \|_{\infty} \leq \| x \|_{p}$.