Let A and B be separable C∗-algebras, A unital and B stable. It is shown that there is a natural six-terms exact sequence which relates the group which arises by considering all semi-split extensions of A by B to the group which arises by restricting the attention to unital semi-split extensions of A by B. The six-terms exact sequence is an unpublished result of G. Skandalis. Let A,B be separab...

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of ...

A C-metric algebra consists of a unital C-algebra and a Leibniz Lip-norm on the C-algebra. We show that if the Lip-norms concerned are lower semicontinuous, then any unital -homomorphism from a C-metric algebra to another one is necessarily Lipschitz. It results that the free product of two Lipschitz unital -homomorphisms between C-metric algebras coming from -filtrations is still a Lipschitz u...

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T ) for not necessarily unital C∗-algebras. The stronger one of the two is called “property (T )” and the weaker one is called “property (Te)”. It is shown that all non-unital C*-algebras do not have property (T ) (neither do their unitalizations). Moreover, all non-unital σ-unital C*-algebras do not have prope...

We will use commutators to provide decompositions of 3×3 matrices as sums whose terms satisfy some polynomial identities, and we apply them bounded linear operators endomorphisms free modules infinite rank. In particular it is proved that every operator an infinite-dimensional complex Hilbert space a sum four automorphisms order 3 simple ring obtained quotient the endomorphism vector modulo its...

Generalization, also called anti-unification, is the dual of unification. A generalizer two terms t and $t^{\prime }$ a term \prime which are substitution instances. The most general equational unifiers that least generalizers, i.e., specific anti-instances modulo equations. In previous work, we extended classical untyped generalization algorithm to: (1) an order-sorted typed setting with sorts...

An l-group G is an abelian group equipped with a translation invariant lattice order. Baker and Beynon proved that G is finitely generated projective iff it is finitely presented. A unital l-group is an l-group G with a distinguished order unit, i.e., an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. While every finitely generated projective unital l-...

Abstract Every finite subunital of any generalized hermitian unital is itself a unital; the embedding given by an quadratic field extensions. In particular, with one (i.e., it originates from separable extension).

Let A be a unital separable C∗-algebra and B = C ⊗ K, where C is a unital C∗-algebra. Let τ : A → M(B)/B be a weakly unital full essential extensions of A by B. We show that there is a bijection between a quotient group of K0(B) onto the set of strong unitary equivalence classes of weakly unital full essential extensions σ such that [σ] = [τ ] in KK(A,B). Consequently, when this group is zero, ...

For any nite volume hyperbolic 3-manifold M we use ideal tri-angulation to deene an invariant (M) in the Bloch group B(C). It actually lies in the subgroup of B(C) determined by the invariant trace eld of M. The Chern-Simons invariant of M is determined modulo rationals by (M). This implies rationality and | assuming the Ramakrishnan conjecture | ir-rationality results for Chern Simons invariants.