We study homomorphisms of locally compact quantum groups from the point of view of integrability of the associated action. For a given homomorphism of quantum groups Π : H → G we introduce quantum groups H/ker Π and im Π corresponding to the classical quotient by kernel and closure of image. We show that if the action of H on G associated to Π is integrable then H/ker Π ∼ = im Π and characterize such Π. As a particular case we consider an injective continuous homomorphism Π : H → G between locally compact groups H and G. Then Π yields an integrable action of H on L ∞ (G) if and only if its image is closed and Π is a homeomorphism of H onto im Π.We also give characterizations of open quantum subgroups and of compact quantum subgroups in terms of integrability and show that a closed quantum subgroup always gives rise to an integrable action. Moreover we prove that quantum subgroups closed in the sense of Woronowicz whose associated homomorphism of quantum groups yields an integrable action are closed in the sense of Vaes.2010 Mathematics Subject Classification. Primary: 46L89 Secondary: 46L85, 46L52. Key words and phrases. Locally compact quantum group, quantum subgroup, homomorphism of quantum groups, integrable action.

We establish a one to one correspondence between idempotent states on a locally compact quantum group G and integrable coideals in the von Neumann algebra L ∞ (G) that are preserved by the scaling group. In particular we show that there is a one to one correspondence between idempotent states on G and ψ G -expected left-invariant von Neumann subalgebras of L ∞ (G). We characterize idempotent states of Haar type as those corresponding to integrable normal coideals preserved by the scaling group. We also establish a one to one correspondence between open subgroups of G and central idempotent states on the dual G. Finally we characterize coideals corresponding to open quantum subgroups of G as those that are normal and admit an atom. As a byproduct of this study we get a number of universal lifting results for Podleś condition, normality and regularity and we generalize a number of results known before to hold under the coamenability assumption.2010 Mathematics Subject Classification. Primary: 46L65 Secondary: 43A05, 46L30, 60B15.

We show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.

For a given graph G, a function f : V(G) → {0, 1, …, diamG} such that for every vertex v of G, f(v) ≤ e(v) (where e(v) is the eccentricity of the vertex v and diamG is the diameter of G), is a broadcast on G. A broadcast f is a dominating broadcast if Nf[V+] = V(G), where V+ = {u | f(u) > 0}, Nf[V+] = ⋃u∈V+Nf[u], and Nf[u] = {v | d(u, v) ≤ f(u)}. In this paper, inspired by Dunbar et al. [3], we study limited dominating broadcast in graphs and obtain some properties, bounds, and characterizations for limited broadcast domination number in graphs.

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete Kac quantum groups, we show that the existence of unique stationary SAT states entails rigidity results concerning amenable extensions of quantum group von Neumann algebras.