We compute the generator rank of a subhomogeneous $C^*\!$-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every $\mathcal {Z}$-stable approximately subhomogeneous algebra has generator rank one, which means that a generic element in such an algebra is a generator.

This leads to a strong solution of the generator problem for classifiable, simple, nuclear $C^*\!$-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear $C^*\!$-algebras.

Let M be a closed simply connected 2n-dimensional manifold. The paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M.

We study the E-theory group E[0, 1](A, B) for a class of C*-algebras over the unit interval with finitely many singular points, called elementary C[0, 1]-algebras. We use results on E-theory over non-Hausdorff spaces to describe E[0, 1](A, B) where A is a sky-scraper algebra. Then we compute E[0, 1](A, B) for two elementary C[0, 1]-algebras in the case where the fibers A(x) and B(y) of A and B are such that E1 (A(x), B(y)) = 0 for all x, y ∈ [0, 1]. This result applies whenever the fibers satisfy the UCT, their K0-groups are free of finite rank and their K1-groups are zero. In that case we show that E[0, 1](A, B) is isomorphic to Hom(0(A), 0(B)), the group of morphisms of the K-theory sheaves of A and B. As an application, we give a streamlined partially new proof of a classification result due to the first author and Elliott.

The most basic structure of chiral conformal field theory (CFT) is the Verlinde ring. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFT's associated to loop groups, as twisted equivariant K-theory. We build on their work to express K-theoretically the structures of full CFT. In particular, the modular invariant partition functions (which essentially parametrise the possible full CFTs) have a rich interpretation within von Neumann algebras (subfactors), which has led to the developments of structures of full CFT such as the full system (fusion ring of defect lines), nimrep (cylindrical partition function), alpha-induction etc. Modular categorical interpretations for these have followed. For the generic families of modular invariants (i.e. those associated to Dynkin diagram symmetries), we provide a K-theoretic framework for these other CFT structures, and show how they relate to D-brane charges and charge-groups. We also study conformal embeddings and the modular invariant of SU(2), as well as some families of finite group doubles. This new K-theoretic framework allows us to simplify and extend the less transparent, more ad hoc descriptions of these structures obtained previously within CFT.

We define the filtrated $\text{K}$-theory of a ${{\text{C}}^{*}}$-algebra over a finite topological space $X$ and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over $X$ in terms of filtrated $\text{K}$-theory.

For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification.

We also exhibit an example where filtrated $\text{K}$-theory is not yet a complete invariant. We describe two ${{\text{C}}^{*}}$-algebras over a space $X$ with four points that have isomorphic filtrated $\text{K}$-theory without being $\text{KK}\left( X \right)$-equivalent. For this space $X$, we enrich filtrated $\text{K}$-theory by another $\text{K}$-theory functor to a complete invariant up to $\text{KK}\left( X \right)$-equivalence that satisfies a Universal Coefficient Theorem.

In this paper, we investigate the higher simplicial cohomology groups of the convolution algebra ${{\ell }^{1}}\left( S \right)$ for various semigroups $S$. The classes of semigroups considered are semilattices, Clifford semigroups, regular Rees semigroups and the additive semigroups of integers greater than $a$ for some integer $a$. Our results are of two types: in some cases, we show that some cohomology groups are 0, while in some other cases, we show that some cohomology groups are Banach spaces.

It is proved that the measure algebra M(G) of a locally compact group G is Connes-amenable if and only if G is amenable.

For a well-behaved measure μ, on a locally compact totally ordered set X, with continuous part μc, we make Lp (X, μc) into a commutative Banach bimodule over the totally ordered semigroup algebra Lp (X, μ), in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from Lp (X, μ) into its dual module and in particular shows that if μc is not identically zero then Lp (X, μ) is not weakly amenable. We show that all bounded derivations from L1 (X, μ) into its dual module arise in this way and also describe all bounded derivations from Lp(X, μ) into its dual for 1 < p < ∞ the case that X is compact and μ continuous.

We study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt} is a path of such operators, we can associate to {Bt} an integer, sf({Bt}), called the spectral flow of the path. This notion, due to M. Atiyah and G. Lusztig, assigns to the path {Bt} the net number of eigenvalues (counted with multiplicities) which pass through 0 in the positive direction. There are difficulties in making this precise — the usual argument involves looking at the graph of the spectrum of the family (after a suitable perturbation) and then counting intersection numbers with y = 0.

We present a completely different approach using the functional calculus to obtain continuous paths of eigenprojections (at least locally) of the form . The spectral flow is then defined as the dimension of the nonnegative eigenspace at the end of this path minus the dimension of the nonnegative eigenspace at the beginning. This leads to an easy proof that spectral flow is a well-defined homomorphism from the homotopy groupoid of onto Z. For the sake of completeness we also outline the seldom-mentioned proof that the restriction of spectral flow to is an isomorphism onto Z.

A new proof is given for the weak amenability of the group algebras of locally compact groups.

We are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space.

In § 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted ℳ, and the set of unital (real) algebra homomorphisms from B into C, denoted by ΦB. This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex.

Finally, in § 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras “almost complex C*- algebras” and provide a natural classification scheme.