Let G be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in L ∞ (G) preserved by the scaling group and idempotent states on the dual quantum group G. As a byproduct we give a simple proof that normal integrable coideals in L ∞ (G) which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of G.

We focus on a question raised by M. Daws [Bull. London Math. Soc. 36 (2004), 493-503] concerning the Arens regularity of B(X), the algebra of operators on a Banach space X. In this respect, among other things, we show that B(X) is Arens regular if and only if X is ultra-reflexive.2010 Mathematics Subject Classification. 47L10; 47L50; 46H25 .

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By using the Principle of Local Reflexivity (PLR), we prove that for every two Banach spaces E and X there exists a suitable ultrafilter U such that F (E, X) * , the dual space of the finite rank operators, can be isomorphically identified with certain quotient of the ultrapower space (E ⊗X * ) U , of the projective tensor product space E ⊗X * . This generalizes the identity B(E, X * * ) ∼ = B(E, X) * * , where E is finite-dimensional. We then serve our main result to improve some results on the reflexivity of B(E, X), the space of all bounded linear operators, by showing that: if B(E, X) is reflexive, then B(E, X) = A(E, X), the space of all approximable operators. This particularly implies that, B(E) is reflexive if and only if E is finite-dimensional. Finally, as more by-products of the PLR, some generalizations of the classical Goldstine weak * -density theorem are also included. 2020 Mathematics Subject Classification. 46B07; 46B10; 47L10.

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