I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $*$-algebras". There he have discussed the Arens regularity of common Banach algebras like $L^1(G)$, $C^*$-algebras,$M(G)$, $K(H)$ etc. Some other primary Banach algebras that comes to my mind are Schatten p-class operators and their tensor products. So my questions are-

1) Is $S_1(H)$ (algebra of trace class operators on Hilbert space) Arens regular?

2) What about Arens regularity of projective tensor products $S_{p_1}(H)\otimes_\gamma S_{p_2}(H)$? ($1\leq p_1,p_2<\infty$).

These seems to be the very first objects people might have investigated for Arens regularity. Please suggest a reference(book,papers etc) where these things have been discussed or provide some hints.