Positive definite ∗*-spherical functions, property (T), and C∗C^*-completions of Gelfand pairs

The study of existence of a universal $C^*$-completion of the $^*$-algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (\operatorname{SL}_2(\Qp), \operatorname{SL}_2(\Zp)) does not admit a universal $C^*$-completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell-Rieffel equivalence, and highlighted the role of other $C^*$-completions. In the case of the pair (\operatorname{SL}_n(\Qp), \operatorname{SL}_n(\Zp)) for $n\geq 3$ we show, invoking property (T) of \operatorname{SL}_n(\Qp), that the $C^*$-completion of the $L^1$-Banach algebra and the corner of C^*(\operatorname{SL}_n(\Qp)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least $2$ over a $\mathfrak{p}$-adic field with a good choice of a maximal compact open subgroup.Comment: 15 page

Representations of Hecke algebras and dilations of semigroup crossed products

We consider a family of Hecke C*-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.Comment: 16 page

How Would the President's Fiscal Commission's Social Security Proposals Affect Future Beneficiaries?

Estimates how proposed changes to Social Security, including raising the cap on earnings subject to the payroll tax, would affect benefits over time and by income and lifetime payroll tax contributions

Nica-Toeplitz algebras associated with product systems over right LCM semigroups

We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of $C^*$-correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for $C^*$-precategories. Our results provide an interpretation of conditions identified in work of Fowler and Fowler-Raeburn, and apply also to their crossed product twisted by a product system, in the new context of right LCM semigroups, as well as to a new, Doplicher-Roberts type $C^*$-algebra associated to the Nica-Toeplitz algebra. As a derived construction we develop Nica-Toeplitz crossed products by actions with completely positive maps. This provides a unified framework for Nica-Toeplitz semigroup crossed products by endomorphisms and by transfer operators. We illustrate these two classes of examples with semigroup $C^*$-algebras of right and left semidirect products.Comment: Title changed from "Nica-Toeplitz algebras associated with right tensor C*-precategories over right LCM semigroups: part II examples". The manuscript accepted in J. Math. Anal. App

On C*-algebras associated to right LCM semigroups

We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.Comment: 31 page