Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

Continuous Bundles of C*-Algebras with Discontinuous Tensor Products

For each non-exact C*-algebra A and infinite compact Hausdorffspace X there exists a continuous bundle B of C*-algebras onX such that the minimal tensor product bundle AB is discontinuous.The bundle B can be chosen to be unital with constant simplefibre. When X is metrizable, B can also be chosen to be separable.As a corollary, a C*-algebra A is exact if and only if A Bis continuous for all unital continuous C*-bundles B on a giveninfinite compact Hausdorff base space. The key to proving theseresults is showing that for a non-exact C*-algebra A there existsa separable unital continuous C*-bundle B on [0,1] such thatA B is continuous on [0,1] and discontinuous at 1, a counter-intuitiveresult. For a non-exact C*-algebra A and separable C*-bundleB on [0,1], the set of points of discontinuity of A B in [0,1]can be of positive Lebesgue measure, and even of measure 1.2000 Mathematics Subject Classification 46L06 (primary), 46L35(secondary).     

Relatively Weakly Open Sets in Closed Balls of C*-Algebras

Let A be an infinite-dimensional C*-algebra. It is proved thatevery nonempty relatively weakly open subset of the closed unitball B_A of A has diameter equal to 2. This implies that B_A isnot dentable, and that there is not any point of continuityfor the identity mapping (B_A,weak)(B_A,norm).     

Continuity of Lie Isomorphisms of Banach Algebras

We prove that if A and B are semisimple Banach algebras, thenthe separating subspace of every Lie isomorphism from A ontoB is contained in the centre of B. 1991 Mathematics SubjectClassification 17B40, 17B60, 46H40.     

The A-Complexity of a Space

Suppose that A is a pointed CW-complex. The paper looks at howdifficult it is to construct an A-cellular space B from copiesof A by repeatedly taking homotopy colimits; this is determinedby an ordinal number called the complexity of B. Studying thecomplexity leads to an iterative technique, based on resolutions,for constructing the A-cellular approximation CW_A(X) of an arbitraryspace X.     

Morita Equivalence of Operator Algebras and Their C*-Envelopes

If two operator algebras A and B are strongly Morita equivalent(in the sense of [5]), then their C*-envelopes C*(A) and C*(B)are strongly Morita equivalent (in the usual C*-algebraic sensedue to Rieffel). Moreover, if Y is an equivalence bimodule fora (strong) Morita equivalence of A and B, then the operation,Y_hA–, of tensoring with Y, gives a bijection between theboundary representations of C*(A) for A and the boundary representationsof C*(B) for B. Thus the ‘noncommutative Choquet boundaries’of Morita equivalent A and B are the same. Other important objectsassociated with an operator algebra are also shown to be preservedby Morita equivalence, such as boundary ideals, the Shilov boundaryideal, Arveson's property of admissability, and the latticeof C*-algebras generated by an operator algebra. 1991 MathematicsSubject Classification 47D25, 46L05, 46M99, 16D90.     

Uniform Embeddings of Free Products into Uniformly Convex Banach Spaces

Let A and B be countable discrete groups, and let = A * B betheir free product. We show that if A and B are uniformly embeddableinto a uniformly convex Banach space, then so is . 2000 MathematicsSubject Classification 46L89, 20F65.     

Constructions of stable equivalences of Morita type for finite-dimensional algebras III

In this paper, we provide a new method to produce stable equivalencesof Morita type. Our main results can be stated as follows. LetA and B be two finite-dimensional k-algebras over a field k.Suppose that two bimodules _AM_B and _BN_A define a stable equivalenceof Morita type between A and B and that R is a generator forA-modules. Then there is a stable equivalence of Morita typedefined by X and Y between the endomorphism algebra End_A(R)of the module R and the endomorphism algebra End_B(N_AR) of themodule N_AR. If M and N satisfy the property that both (N_A–,M_B–) and (M_B–, N_A–) are adjoint pairs of functors,then so do the modules X and Y. Moreover, we show that the self-injectivedimension and the Gorenstein property are invariant under stableequivalences of Morita type with the above-mentioned adjointproperty.     

Derived Subgroups of Products of an Abelian and a Cyclic Subgroup

Let G be a finite group and suppose that G = AB, where A andB are abelian subgroups. By a theorem of Ito, the derived subgroupG' is known to be abelian. If either of the subgroups A or Bis cyclic, then more can be said. The paper shows, for example,that G'/(G'A) is isomorphic to a subgroup of B in this case.     

Normal Subgroups of Groups Which Split Over The Infinite Cyclic Group

Let G be either a free product with amalgamation A*_CB or anHNN group A*_C, where all normal subgroups of C are finitelygenerated. Suppose that both A and B have no non-trivial finitelygenerated normal subgroups of infinite indices. We show thatif G contains a finitely generated normal subgroup N which intersectsA or B non-trivially but is not contained in C, then the indexof N in G is finite. 1991 Mathematics Subject Classification20E06.