Linear orthogonality preservers of Hilbert bundles

Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $\theta$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if \left =0 whenever \left =0. We prove that if $\theta$ is an orthogonality preserving map from a full Hilbert $C_0(\Omega)$-module $E$ into another Hilbert $C_0(\Omega)$-module $F$ that satisfies a weaker notion of $C_0(\Omega)$-linearity (known as "localness"), then $\theta$ is bounded and there exists $\phi\in C_b(\Omega)_+$ such that \left\ =\ \phi\cdot\left, \quad \forall x,y \in E. On the other hand, if $F$ is a full Hilbert $C^*$-module over another commutative $C^*$-algebra $C_0(\Delta)$, we show that a "bi-orthogonality preserving" bijective map $\theta$ with some "local-type property" will be bounded and satisfy \left\ =\ \phi\cdot\left\circ\sigma, \quad \forall x,y \in E where $\phi\in C_b(\Omega)_+$ and $\sigma: \Delta \rightarrow \Omega$ is a homeomorphism

Linear orthogonality preservers of Hilbert C∗C^*-modules over general C∗C^*-algebras

As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $\Phi : E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$\langle \Phi(x),\Phi(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0,$$ then there exists a unique central positive multiplier $u\in M(I_E)$ such that $$\langle \Phi(x), \Phi(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad (x,y\in E).$$ As a consequence, $\Phi$ is automatically bounded, the induced map $\Phi_0: E\to \overline{\Phi(E)}$ is adjointable, and $\overline{Eu^{1/2}}$ is isomorphic to $\overline{\Phi(E)}$ as Hilbert $A$-modules. If, in addition, $\Phi$ is bijective, then $E$ is isomorphic to $F$.Comment: 15 page

The Rate of Convergence to Perfect Competition of a Simple Matching and Bargaining Mechanism

We study the steady-state of a market with inflowing cohorts of buyers and sellers who are randomly matched pairwise and bargain under private information. Two bargaining protocols are considered: take-it-or-leave-it offering and the double auction. There are frictions due to costly search and time discounting, parameterized by a single number t > 0 proportional to the waiting time until the next meeting. We study the efficiency of these mechanisms as the frictions are removed, i.e. t 0. We find that all equilibria of the take-it-or-leave-it offering mechanism converge to the Walrasian limit, at the fastest possible rate O(t) among all bargaining mechanisms. For the double auction mechanism, we find that there are equilibria that converge at the linear rate, those that converge at a slower rate or even not converge at all.Matching and Bargaining, Search, Double Auctions, Foundations for Perfect Competition, Rate of Convergence

Identification in First-Price and Dutch Auctions when the Number of Potential Bidders is Unobservable

Within the IPV paradigm, we show nonparametric identification of model primitives for first-price and Dutch auctions with a binding reserve price and auction-specific, unobservable sets of potential bidders.auctions, identification

Estimation of the Autoregressive Order in the Presence of Measurement Errors

Most of the existing autoregressive models presume that the observations are perfectly measured. In empirical studies, the variable of interest is unavoidably measured with various kinds of errors. Thus, misleading conclusions may be yielded due to the inconsistency of the parameter estimates caused by the measurement errors. Thus far, no theoretical result on the direction of bias of the lag order estimate is available in the literature. In this note, we will discuss the estimation an AR model in the presence of measurement errors. It is shown that the inclusion of measurement errors will drastically increase the complexity of the problem. We show that the lag lengths selected by the AIC and BIC are increasing with the sample size at a logarithmic rate.Autoregressive Process Measurement Error Akaike Information Criterion Bayesian Information Criterion

Bilateral Matching and Bargaining with Private Information

We explore the role of private information in bilateral matching and bargaining. Our model is a replica of Mortensen and Wright (2002), but with private information. A simple necessary and sufficient condition on the parameters of the model for existence of equilibrium with entry is obtained. As in Mortensen and Wright (2002), we find that equilibrium is unique and has the property that every meeting results in trade when the discount rate is sufficiently small. There are also equilibria in which not every meeting results in trade. All equilibria converge to perfect competition as the frictions of search costs and discounting are removed. We find that private information may deter entry. Because of matching externalities, this entry-deterring effect of private information may be welfare-enhancing.Matching and Bargaining, Search, Foundations for Perfect Competi- tion, Two-sided Incomplete Information