Active lattices determine AW*-algebras

We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure.     

AW*-algebras Which are Enveloping C*-algebras of JC-algebras

In the given article, enveloping C*-algebras of AJW-algebras are considered. Conditions are given, when the enveloping C*-algebra of an AJW-algebra is an AW*-algebra, and corresponding theorems are proved. In particular, we proved that if $\mathcal{A}$ is a real AW*-algebra, $\mathcal{A}_{sa}$ is the JC-algebra of all self-adjoint elements of $\mathcal{A}$ , $\mathcal{A}+i\mathcal{A}$ is an AW*-algebra and $\mathcal{A}\cap i\mathcal{A} = \{0\}$ then the enveloping C*-algebra $C^*(\mathcal{A}_{sa})$ of the JC-algebra $\mathcal{A}_{sa}$ is an AW*-algebra. Moreover, if $\mathcal{A}+i\mathcal{A}$ does not have nonzero direct summands of type I₂, then $C^*(\mathcal{A}_{sa})$ coincides with the algebra $\mathcal{A}+i\mathcal{A}$ , i.e. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

Translation invariant topologies on commutative *-algebras

Let A be a commutative *-algebra. Under certain conditions on the involution, we construct a topology makingA a HausdorffQ-ring with continuous involution and inversion; this topology is induced by anA-valued norm.Applying the previous considerations to the algebraC(X) of continuous complex-valued functions over a topological spaceX, we obtain a new characterization of Weierstrass spaces.Furthermore, we provide every projective finitely generated moduleM over a topological ring R with a unique topology, under whichM is a topologicalR-module and every R-linear mapf :M N is continuous, for any topologicalR-moduleN. In caseR =A, we prove that this topology is induced by anA-norm. Mathematics subject classification numbers, 1980/85. Primary 46K05, Secondary 16A80.     

Inner ideals in C*-algebras

Research supported by a grant from the Schweizerische Nationalfonds/Fonds national suisse     

Approximating derivations on ideals ofC *-algebras

For each^*-derivation of a separableC ^*-algebraA and each >0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that (–ad(ix))|I< and x.     

The category of commutative HopfC *-algebras

In this paper, we first continue our study of group duality, and prove that the duality we established earlier is natural. Then we use this naturality to study the category of commutative, cocommutative HopfC ^*-algebras, and show that the category of compact Abelian semigroups and the category of commutative, cocommutative HopfC ^*-algebras with units are isomorphic. By using this result, we show that the category of commutative, cocommutative quantum groups is Abelian. This is a generalization of a result of Grothendieck about the catrgory of finite-dimensional commutative, cocommutative Hopf algebras with antipodes.     

Primitive ideals of locally C*-algebras

This paper is concerned with the connection between the structure space of a locally C*-algebra and the set of its continuous topologically irreducible *-representations. Properties of primitive ideals in such algebras are further investigated, for instance, closed ideals are expressed as intersections of primitive ideals, by using that the Jacobson radical reduces to 0.     

On the closed Lie ideals of certainC *-algebras

In this article we determine the closed Lie ideals of a uniformly hyperfiniteC ^*-algebra, and of the tensor product of such an algebra withC(X), the space of continuous functions on a compact, Hausdorff space. This is done by localizing the Lie ideals in algebras of the form , where is an algebra over a field of characteristic not equal to 2.This research is partially supported by NSERC (Canada)     

Derivations and nil ideals

IfR is a torsion free ring, then the nil radical ofR is stable relative to any derivation ofR. In this note, we show the same is true for the Levitzki radical ofR, which is the largest locally nilpotent ideal ofR.     

Derivations on Lie ideals in semiprime rings

We prove that ifR is a semiprime 2-torsion free ring with a derivationd andU a Lie ideal ofR such thatd ² (U)=0, thend(U)?Z(R), the center ofR.