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We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space$X$is compactly locally reflexive if and only if $\mathcal{K}(Y,X^{**})\subseteq\mathcal{K}(Y,X)^{**}$ for all reflexive Banach spaces$Y$. We show that$X$^{*}has the approximation property if and only if$X$has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If$X$is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation property with conjugate operators for dual spaces.

We give complete detail of the description of the GNS representation of the quantum plane A and its dual  as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary W is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes' construction of the multiplicative unitary Wm, we give the GNS description of the quantum double D(A) which is equivalent to GLq+(2, ℝ). Furthermore, we study the fundamental corepresentation Tλ, t and its matrix coefficients, and show that it can be expressed by the b-hypergeometric function. We also study the regular corepresentation and representation induced by Wm and prove that the space of L2 functions on the quantum double decomposes into the continuous series representation of Uqq̃(gl(2, ℝ)) with the quantum dilogarithm {pipe}Sb(Q+2iα){pipe}2 as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.