We introduce a new invariant for C?-algebras of stable rank one that merges the Cuntz semigroup information together with K1-group information. This semigroup, termed Cu1-semigroup, is constructed as equivalence classes pairs consisting positive element in stabilization given C?-algebra unitary unitization hereditary subalgebra generated by element. show Cu1-semigroup well-defined continuous fu...

Abstract We unify various étale groupoid reconstruction theorems such as the following: • Kumjian and Renault’s from a C*-algebra; Exel’s an ample inverse semigroup; Steinberg’s ring; Choi, Gardella Thiel’s Lp {L^{p}} -algebra. do this by working with certain bumpy semigroups S of functions ...

Guided by the construction of the unital *-algebra of closed operators which are ‘affiliated’ with a given unital *-algebra, we introduce the notion of the ‘monotone closure’ for certain increasing sequences of unital *-algebras. Then we study an example of a unital *-algebra F0(A) constructed from a countable number of copies of a unital *-algebra A. We endow F0(A) with a quantum semigroup str...

Let M denote the multiplicative semigroup of all n-by-n matrices over a finite field F and K a commutative ring with an identity element in which the characteristic of F is a unit. It is proved here that the semigroup algebra K[M] is the direct sum of n + 1 algebras, namely, of one full matrix algebra over each of the group algebras K[GL(r, F)] with r = 0, 1, ..., n . The degree of the relevant...

The semigroup of binary relations on {1, . . . , n} with the relative product is isomorphic to the semigroup Bn of n×n matrices over {0, 1} with the Boolean matrix product. Over any field F , we prove that there is an ideal Kn in FBn of dimension (2n−1)2, and we construct an explicit isomorphism of Kn with the matrix algebra M2n−1(F ). Let Zn = {1, . . . , n} (n ≥ 1); then P (Z n), the power se...

We show that every continuous product system of correspondences over a unital C∗–algebra occurs as the product system of a strictly continuous E0–semigroup.

The paper presents a construction of the crossed product of a C *-algebra by a semigroup of endomorphisms generated by partial isometries.

Let n, r ∈ N. The affine Schur algebra S̃(n, r) (of type A) over a field K is defined to be the endomorphism algebra of certain tensor space over the extended affine Weyl group of type Ar−1. By the affine Schur–Weyl duality it is isomorphic to the image of the representation map of the U(ĝl n ) action on the tensor space when K is the field of complex numbers. We show that S̃(n, r) can be defined...