# Arens regularity of Banach algebras

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.

• My guess is that answers to both of your questions might be found in the reference book of Dales, "Banach Algebras and Automatic Continuity", but I am not completely sure. – Yemon Choi Oct 16 '19 at 23:50
• I think @MatthewDaws is an expert in this stuff; let's see if he'll give an answer. – Nik Weaver Oct 17 '19 at 1:15
• jstor.org/stable/2044028#metadata_info_tab_contents answers my first question. – NewB Oct 17 '19 at 7:05
• @LavKumar That's a good reference: better than the ones I mention. A direct link for those without JSTOR access. – Matthew Daws Oct 17 '19 at 7:23

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").