A band is a semigroup consisting of idempotents. It is proved that for any field K and any band S with finitely many components, the semigroup algebra K[S] can be embedded in upper triangular matrices over a commutative K-algebra. The proof of a theorem of Malcev [4, Theorem 10] on embeddability of algebras into matrix algebras over a field is corrected and it is proved that if S = F ∪ E is a b...

Finite-dimensional square-freeK-algebras have been completely characterized by Anderson and D’Ambrosia as certain semigroup algebras A ∼= KξS over a square-free semigroup S twisted by some ξ ∈ Z (S,K), a two-dimensional cocycle of S with coefficients in the group of units K∗ of K. D’Ambrosia extended the definition of square-free to artinian rings with unity and showed every square-free ring ha...

In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a Zorder with hyperbolic unit group. In this paper we complete this classification by handling the case in which the semi...

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S = k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.

Let A be a simple, unital, finite, and exact C-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases — Z-stab...

For every uncountable cardinal κ there are 2 nonisomorphic simple AF algebras of density character κ and 2 nonisomorphic hyperfinite II1 factors of density character κ. These estimates are maximal possible. All C*-algebras that we construct have the same Elliott invariant and Cuntz semigroup as the CAR algebra.

The recently developed theory of partial actions of discrete groups on C-algebras is extended. A related concept of actions of inverse semigroups on C-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.

Abstract We provide an abstract characterization for the Cuntz semigroup of unital commutative AI-algebras, as well a semigroups form $\operatorname {\mathrm {Lsc}} (X,\overline {\mathbb {N}})$ some $T_1$ -space X . In our investigations, we also uncover new properties that all AI-algebras satisfies.

We introduce a concept and develop a theory of Galois subalgebras in skew semigroup rings. Proposed approach has a strong impact on the representation theory, first of all the theory of Harish-Chandra modules, of many infinite dimensional algebras including the Generalized Weyl algebras, the universal enveloping algebras of reductive Lie algebras, their quantizations, Yangians etc. In particula...

We say that a nonselfadjoint operator algebra is partly free if it contains a free semigroup algebra. Motivation for such algebras occurs in the setting of what we call free semigroupoid algebras. These are the weak operator topology closed algebras generated by the left regular representations of semigroupoids associated with finite or countable directed graphs. We expand our analysis of partl...