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Remarks on the quantum Bohr compactification. (English) Zbl 1305.43006
Summary: The category of locally compact quantum groups can be described as either Hopf \(\ast\)-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how Sołtan’s quantum Bohr compactification can be used to construct a “compactification” in this category. Depending on the viewpoint, different \(C^{\ast}\)-algebraic compact quantum groups are produced, but the underlying Hopf \(\ast\)-algebras are always, canonically, the same. We show that a complicated range of behaviours, with \(C^{\ast}\)-completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of Sołtan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in \(L^{\infty}(\mathbb{G})\), involving the antipode, which allows one to compute the Hopf \(\ast\)-algebra of the compactification of \(\mathbb{G} \); we later study when the antipode assumption can be dropped. In the cocommutative case, we make a detailed study of Runde’s notion of a completely almost periodic functional – with a slight strengthening, we show that for [SIN] groups this does recover the Bohr compactification of \(\hat{G}\).

MSC:
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A95 Categorical methods for abstract harmonic analysis
47L25 Operator spaces (= matricially normed spaces)
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