We use essential ideals in a C * -algebra A to extend the inner product on a Hilbert C * -module M to a greater C * -submodule of the dual Banach module M ′ . This extension may go beyond the second dual module M ′′ .

Let A and B be C * -algebras, A separable, and B σ-unital and stable. It is shown that there are natural isomorphismsis the generalized Calkin algebra and K denotes the C * -algebra of compact operators of an infinite dimensional separable Hilbert space.

Let A be a separable C Ã -algebra and B a stable C Ã -algebra containing a strictly positive element. We show that the group Ext À1=2 ðSA; BÞ of unitary equivalence classes of extensions of SA by B; modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between Ext À1=2 ðSA; BÞ and the E-theory group EðA; BÞ of homotopy classes of asymptotic homomorphisms from S 2 A to B: r 2004 Elsevier Inc. All rights reserved.

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D = 4, N = 4 super-Yang-Mills coupled to supergravity. The reduction cannot be done on the action itself, but must be done either on the field equations or on a duality invariant form of the action, such as one in the doubled formalism in which potentials are introduced for both electric and magnetic fields. The resulting theory in odd-dimensions has massive form fields satisfying a self-duality condition dA ∼ m * A. We construct such theories in D = 3, 5, 7.

Induced conditional expectations of finite index on crossed product C*algebras are considered which are non-algebraically of finite index. The characteristics of actions of (amenable) topological groups on compact Hausdorif spaces X are investigated, ensuring the appearance of a well-defined induced conditional expectation on the corresponding commutative C*a1gebra C(X) and its property to be of finite index. For this purpose Hilbert C* module and topological techniques are used. Special emphasis is placed on discrete group actions.