Dilating Semigroup Representations to the Boundary Quotient

Abstract

We prove that a representation of a group embeddable or right LCM cancellative semigroup may be dilated to a representation of its reduced boundary quotient $C^\star$-algebra if and only if it extends to a completely contractive representation of the reduced semigroup operator algebra. We show that the latter property is satisfied not only by all constructible representations of amenable semigroups, but also a very large class of other representations, which encompasses several classical dilation theorems and the corresponding matricial von Neumann inequalities. In fact, our criterion of completely contractive extension to the operator algebra turns out to be an appropriate generalisation of the matricial von Neumann inequality to semigroups more general than $\mathbb{N}^k$, and dilation to the boundary quotient turns out to be an apt generalisation of unitary dilations. Thus, our theorem is in spirit and in practice a generalisation of Sz.-Nagy and Ando's dilation theorems for general semigroups. In addition, this also demonstrates that any completely contractive representation of the operator algebra dilates to a representation with additional relations among its generators, the new relations coming from the boundary quotient. In particular, for Ore semigroups, this completely characterises which representations admit unitary dilations.