Frames and bases in tensor products of Hilbert spaces and Hilbert C*-modules

In this article, we study tensor product of Hilbert $C^*$-modules and Hilbert spaces. We show that if $E$ is a Hilbert $A$-module and $F$ is a Hilbert $B$-module, then tensor product of frames (orthonormal bases) for $E$ and $F$ produce frames (orthonormal bases) for Hilbert $A\otimes B$-module $E\otimes F$, and we get more results. For Hilbert spaces $H$ and $K$, we study tensor product of frames of subspaces for $H$ and $K$, tensor product of resolutions of the identities of $H$ and $K$, and tensor product of frame representations for $H$ and $K$.Comment: 12 page

Serre's Tensor Construction and Moduli of Abelian Schemes

Given a polarized abelian scheme with action by a ring, and a projective finitely presented module over that ring, Serre's tensor construction produces a new abelian scheme. We show that to equip these abelian schemes with polarizations it's enough to equip the projective modules with non-degenerate positive-definite hermitian forms. As an application, we relate certain moduli spaces of principally polarized abelian schemes with action by the ring of integers of a CM field. More specifically, we consider integral models of zero-dimensional Shimura varieties associated to compact unitary groups. We show that all abelian schemes in such moduli spaces are, \'etale locally over their base schemes, Serre constructions of CM abelian schemes with positive-definite hermitian modules. We also describe the morphisms between such objects in terms of morphisms between the constituent data, and formulate these relations as an isomorphism of algebraic stacks.Comment: 36 page

Internet and Users. Who is the Reader?

Internet has turned into a fundamental component of everyday life, as it plays a major role in advancing the globalization process. Globalization was fostered by the idea of creating equalaccess opportunities for all and facilitating communication worldwide. Using internet as the core platform, billions of people try to access and benefit from this opportunity through search engines, service providers, websites and social media. However, given the profound difference between internet and user’s languages, users end up on relying on search engines and tools to translate their ideas into a computer-readable language and derive information from them. In order to provide the best possible services, search engines and social media need to accumulate comprehensive data on each user’s identity. The challenge is that once they are fed with convenient information on each user, they tend to personalize the idea they grasp of him or her based on their given regulations and policies, which in the mid- and long-term results in managing users’ access to information.. By applying the reader-response theory, this paper seeks to focus on the challenges stemming from the adoption of users’ personalized profiles by Google, Facebook and Amazon as the most common part of users’ performance in internet. It also explores how the reading differences of the users and the tools result not only in personalized versions of users, but also engender an unrecognized virtual in-betweenness of users’ own perception of themselves and the tools’ perception of users