International Mathematics Research Notices (10737928)(10)
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 : H ? G we introduce quantum groups H/ker and im corresponding to the classical quotient by kernel and closure of image. We show that if the action of H on G associated to is integrable then H/ker~= im and characterize such . As a particular case, we consider an injective continuous homomorphism : H ? G between locally compact groups H and G. Then yields an integrable action of H on L8(G) if and only if its image is closed and is a homeomorphism of H onto im 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. © The Author(s) 2017. Published by Oxford University Press. All rights reserved.
Quarterly Journal of Mathematics (00335606)(2)
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 ΨG-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 Podlé condition, normality and regularity, and we generalize a number of results known before to hold under the coamenability assumption. © 2017. Published by Oxford University Press.
Amini M.,
Asadi, Mohammad B.,
Elliott, George A.,
Khosravi, F. Glasgow Mathematical Journal (00170895)(1)
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. © Glasgow Mathematical Journal Trust 2016.
