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

Cauchy-Davenport type theorems for semigroups

Let $\mathbb{A} = (A, +)$ be a (possibly non-commutative) semigroup. For $Z \subseteq A$ we define $Z^\times := Z \cap \mathbb A^\times$, where $\mathbb A^\times$ is the set of the units of $\mathbb{A}$, and $$\gamma(Z) := \sup_{z_0 \in Z^\times} \inf_{z_0 \ne z \in Z} {\rm ord}(z - z_0).$$ The paper investigates some properties of $\gamma(\cdot)$ and shows the following extension of the Cauchy-Davenport theorem: If $\mathbb A$ is cancellative and $X, Y \subseteq A$, then $$|X+Y| \ge \min(\gamma(X+Y),|X| + |Y| - 1).$$ This implies a generalization of Kemperman's inequality for torsion-free groups and strengthens another extension of the Cauchy-Davenport theorem, where $\mathbb{A}$ is a group and $\gamma(X+Y)$ in the above is replaced by the infimum of $|S|$ as $S$ ranges over the non-trivial subgroups of $\mathbb{A}$ (Hamidoune-K\'arolyi theorem).Comment: To appear in Mathematika (12 pages, no figures; the paper is a sequel of arXiv:1210.4203v4; shortened comments and proofs in Sections 3 and 4; refined the statement of Conjecture 6 and added a note in proof at the end of Section 6 to mention that the conjecture is true at least in another non-trivial case