Arens regularity of Banach algebras

Question

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.

Answer 1

For the question about trace-class operators, you could look at my New York Journal article There is a survey of sorts; see Theorem 5.39 for your question. For the trace-class operators in particular, I cite Dales's book, Theorem 2.6.23, where the result is attributed to Ulger. (See the notes at the end of Section 2.6 in Dales's book. There he references Palmer's work and Ulger's work: you could chase these references to see if there is a definitive first source.)

For the projective tensor product question, you could look at Ulger's paper. I must admit to being a little wary of this paper, because of the errata. Assuming there is no mistake, you could combine Theorems 3.4 and 4.5 to, possibly, show that $S_{p_1} \otimes_{\gamma} S_{p_2}$ is Arens regular. To do this, you'd need to know that certain maps from $S_{p_1}$ to $S_{p_2'}$ were compact. I'm not an expert here in the Banach space geometry (the analogous result for $\ell^p$ spaces is true, and is "Pitt's Theorem").