Unique Pseudo-Expectations for C∗C^*-Inclusions

Given an inclusion D $\subseteq$ C of unital C*-algebras, a unital completely positive linear map $\Phi$ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. The set PsExp(C,D) of all pseudo-expectations is a convex set, and for abelian D, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D. Natural classes of inclusions have a unique pseudo-expectation (e.g., when D is a regular MASA in C). Uniqueness of the pseudo-expectation implies interesting structural properties for the inclusion. For example, when D $\subseteq$ C $\subseteq$ B(H) are W*-algebras, uniqueness of the pseudo-expectation implies that D' $\cap$ C is the center of D; moreover, when H is separable and D is abelian, we characterize which W*-inclusions have the unique pseudo-expectation property. For general inclusions of C*-algebras with D abelian, we characterize the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property. Applications include: a) if an inclusion D $\subseteq$ C has a unique pseudo-expectation $\Phi$ which is also faithful, then the C*-envelope of any operator space X with D $\subseteq$ X $\subseteq$ C is the C*-subalgebra of C generated by X; b) for many interesting classes of C*-inclusions, having a faithful unique pseudo-expectation implies that D norms C. We give examples to illustrate the theory, and conclude with several unresolved questions.Comment: 26 page

On the smoothness of centres of rational Cherednik algebras in positive characteristic

In this article we study rational Cherednik algebras at $t = 1$ in positive characteristic. We study a finite dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series $G(m,d,n)$, we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter

Pseudo-Free Families and Cryptographic Primitives

In this paper, we study the connections between pseudo-free families of computational $\Omega$-algebras (in appropriate varieties of $\Omega$-algebras for suitable finite sets $\Omega$ of finitary operation symbols) and certain standard cryptographic primitives. We restrict ourselves to families $(H_d\,|\,d\in D)$ of computational $\Omega$-algebras (where $D\subseteq\{0,1\}^*$) such that for every $d\in D$, each element of $H_d$ is represented by a unique bit string of length polynomial in the length of $d$. Very loosely speaking, our main results are as follows: (i) pseudo-free families of computational mono-unary algebras with one-to-one fundamental operation (in the variety of all mono-unary algebras) exist if and only if one-way families of permutations exist; (ii) for any $m\ge2$, pseudo-free families of computational $m$-unary algebras with one-to-one fundamental operations (in the variety of all $m$-unary algebras) exist if and only if claw-resistant families of $m$-tuples of permutations exist; (iii) for a certain $\Omega$ and a certain variety $\mathfrak V$ of $\Omega$-algebras, the existence of pseudo-free families of computational $\Omega$-algebras in $\mathfrak V$ implies the existence of families of trapdoor permutations