A local global principle for regular operators in Hilbert C*-modules

Hilbert C*-modules are the analogues of Hilbert spaces where a C*-algebra plays the role of the scalar field. With the advent of Kasparov's celebrated KK-theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert C*-modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this paper we present a new criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. We discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results.Comment: 31 pages, 2 figures; v3: revised version containing an erratum to v2 clarifying the contributions by Fran\c{c}ois Pierro

Closed quantum subgroups of locally compact quantum groups

We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions - one due to S.Vaes and one due to S.L.Woronowicz - are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podle\'s from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.Comment: 24 pages, v3 adds another reference. The paper will appear in Advances in Mathematic

C*-Segal algebras with order unit

We introduce the notion of a (noncommutative) C*-Segal algebra as a Banach algebra which is a dense ideal in a C*-algebra. Several basic properties are investigated and, with the aid of the theory of multiplier modules, the structure of C*-Segal algebras with order unit is determined.Comment: 23 pages; slightly enlarged and amended versio

Rental Housing Assistance for the 21st Century

Current rental housing assistance programs are not designed to provide a safety net for people whose lives are volatile, or to encourage poor people to live in good locations. These failings can be corrected. HUD should establish a program of rental insurance-like mortgage insurance, but for renters. Low income housing assistance formulas should be revised to reward good neighborhood features, and punish bad