On some norm equalities in pre-Hilbert C*-modules

We characterize when the norm of the sum of two elements in a pre-Hilbert C*-module equals the sum of their norms. We also give the necessary and sufficient conditions for two orthogonal elements of a pre-Hilbert C*-module to satisfy Pythagoras’ equality.     

On K-theoretic invariants of semigroup C*-algebras attached to number fields

We show that semigroup C*-algebras attached to ax+b$ax+b$-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Symmetric topological*-algebras

V. Pták's inequality is valid for every hermitian completeQ locallym-convex (:l.m.c.) algebra. Every algebra of the last kind is, in particular, symmetric. Besides, a (Hausdorff) locallyC ^*-algebra (being always symmetric) with the propertyQ is, within a topological algebraic isomorphism, aC ^*-algebra. Furthermore, a type of Raikov's criterion for symmetry is also valid for non-normed topological^*-algebras. Concerning topological tensor products, one gets that symmetry of the-completed tensor product of two unital Fréchet l.m.c.^*-algebrasE, F ( denotes the projective tensorial topology) is always passed toE, F, while the converse occurs when moreover either ofE, F is commutative.     

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ^?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.     

On the asymptotic tensor C*-norm

We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C^*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and discuss its relation to semi-invertibility of C^*-extensions. Received: 23 September 2004; revised: 30 May 2005     

On the geometry of the set of square roots of elements inC *-algebras

This research was partially supported by the CONICET and the University of Buenos Aires.     

C*-Bundles for Certain Liminal C*-Algebras

It is shown that certain liminal C*-algebras whose limit sets in their primitive ideal space are discrete can be described as algebras of continuous sections of a C*-bundle associated with them. Their multiplier algebras are also described in a similar manner. The class of C*-algebras under discussion includes all the liminal C*-algebras with Hausdorff primitive ideal spaces but also many other liminal algebras. A large sub-class of examples is examined in detail.      

On a class of stably finite nuclear C*-algebras

In this note we show that a separable C^*-algebra is nuclear and has a quasidiagonal extension by (the ideal of compact operators on an infinite-dimensional separable Hilbert space) if and only if it is anuclear finite algebra (NF-algebra) in the sense of Blackadar and Kirchberg, and deduce that every nuclear C^*-subalgebra of aNF-algebra isNF. We show that strongNF-algebras satisfy a Følner type condition.     

Elementary operators on antiliminal C^*-algebras

We establish the equivalence of the following three properties of a -algebra A. (a) Every positive elementary operator on A is completely positive. (b) The norm and the cb-norm coincide for every elementary operator on A. (c) A is an extension of an antiliminal -algebra by an abelian one. Received: 15 July 1998 / in revised form: 22 September 1998     

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.     

Unitary equivalence of automorphisms of separable C*-algebras

We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures.