Characterizations of compact and discrete quantum groups through second duals
نویسنده
چکیده
A locally compact group G is compact if and only if L(G) is an ideal in L(G), and the Fourier algebra A(G) of G is an ideal in A(G)∗∗ if and only if G is discrete. On the other hand, G is discrete if and only if C0(G) is an ideal in C0(G)∗∗. We show that these assertions are special cases of results on locally compact quantum groups in the sense of J. Kustermans and S. Vaes. In particular, a von Neumann algebraic quantum group (M,Γ) is compact if and only if M∗ is an ideal in M ∗, and a (reduced) C∗-algebraic quantum group (A,Γ) is discrete if and only if A is an ideal in A∗∗.
منابع مشابه
A characterization of compact quantum groups through second duals
A locally compact group G is compact if and only if L(G) is an ideal in L(G). On the other hand, the Fourier algebra A(G) of G is an ideal in A(G)∗∗ if and only if G is discrete. We show that both results are special cases of one general theorem on locally compact quantum groups in the sense of J. Kustermans and S. Vaes: a von Neumann algebraic quantum group (M,Γ) is compact if and only if M∗ i...
متن کاملFusion Rules for Representations of Compact Quantum Groups
The compact quantum groups are objects which generalise at the same time the compact groups, the duals of discrete groups and the q−deformations (with q > 0) of classical compact Lie groups. A compact quantum group is an abstract object which may be described by (is by definition the dual of) the algebra of “continuous functions on it”, which is a Hopf C-algebra. A system of axioms for Hopf C-a...
متن کاملCompare and contrast between duals of fusion and discrete frames
Fusion frames are valuable generalizations of discrete frames. Most concepts of fusion frames are shared by discrete frames. However, the dual setting is so complicated. In particular, unlike discrete frames, two fusion frames are not dual of each other in general. In this paper, we investigate the structure of the duals of fusion frames and discuss the relation between the duals of fusion fram...
متن کاملLacunary Fourier Series for Compact Quantum Groups
This paper is devoted to the study of Sidon sets, Λ(p)-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, Λ(p)-sets and lacunarities for LFourier multipliers, generali...
متن کاملAround Property (T) for Quantum Groups
WestudyProperty (T) for locally compact quantumgroups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with l...
متن کامل