C*-algebras over topological spaces: the bootstrap class

2001

These notes cover the contents of three survey lectures held at the ICTP Trieste Summer school on High dimensional manifold theory 2001. They introduce techniques coming from the theory of operator algebras. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. An central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. It implies the Novikov conjecture. In the first talk, the Baum-Connes conjecture will be explained and put into our context. One application of the Baum-Connes conjecture is to the positive scalar curvature question. It implies the so called ``stable Gromov-Lawson-Rosenberg conjecture''. The unstable version of this conjecture said that, given a closed spin manifolds M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterex...

Journal of Noncommutative Geometry

Via Gelfand duality, a unital C*-algebra A induces a functor from compact Hausdorff spaces to sets, CHaus → Set. We show how this functor encodes standard functional calculus in A as well as its multivariate generalization. Certain sheaf conditions satisfied by this functor provide a further generalization of functional calculus. Considering such sheaves CHaus → Set abstractly, we prove that the piecewise C*-algebras of van den Berg and Heunen are equivalent to a full subcategory of the category of sheaves, where a simple additional constraint characterizes the objects in the subcategory. It is open whether this additional constraint holds automatically, in which case piecewise C*-algebras would be the same as sheaves CHaus → Set. Intuitively, these structures capture the commutative aspects of C*-algebra theory. In order to find a complete reaxiomatization of unital C*-algebras within this language, we introduce almost C*-algebras as piecewise C*-algebras equipped with a notion of inner automorphisms in terms of a self-action. We provide some evidence for the conjecture that the forgetful functor from unital C*-algebras to almost C*-algebras is fully faithful, and ask whether it is an equivalence of categories. We also develop an analogous notion of almost group, and prove that the forgetful functor from groups to almost groups is not full. In terms of quantum physics, our work can be seen as an attempt at a reconstruction of quantum theory from physically meaningful axioms, as realized by Hardy and others in a different framework. Our ideas are inspired by and also provide new input for the topos-theoretic approach to quantum theory. Contents 2010 Mathematics Subject Classification. Primary: 46L05 (General theory of C*-algebras), 46L60 (Applications of selfadjoint operator algebras to physics). Secondary: 18F20 (Presheaves and sheaves), 20A05 (Group theory, axiomatics and elementary properties).

Annals of the New York Academy of Sciences, 1996

Journal of Mathematical Analysis and Applications, 2007

A considerable number of non-normed topological *-algebras admit a C *-enveloping algebra. Various characterizations of such algebras are given in terms of diverse properties of decisive features of a given topological *-algebra, like the sets of its continuous *-representations, its continuous positive functionals, the hermitian spectrum, etc. Interconnections of these sorts of algebras with hermiticity and C *-spectrality are also discussed. An analogy in structure, between (not necessarily involutive) commutative Q-algebras and topological *-algebras having a C *-enveloping algebra, leads to the study of some automatic continuity results. Furthermore, different examples are presented concerning the question: When does a dense Fréchet *-subalgebra of a C *-algebra have a C *-enveloping algebra? Some questions and conjectures are stated that arise from the whole study.

2017

In this paper, two sufficient and necessary conditions are given. The first one considers the boundary path groupoid of a topological graph without singular vertices, and it characterizes when the interior of its isotropy group bundle is closed. The second one concerns the path groupoid of a row-finite k-graph without sources, and it demonstrates when the interior of its isotropy is closed. It follows that the associated topological graph algebra and the associated k-graph C*-algebra have Cartan subalgebras due to a result of Brown–Nagy–Reznikoff–Sims–Williams.

2004

Let M be a smooth Fredholm manifold modeled on a separable infinite-dimensional Euclidean space E with Riemannian metric g. Given an (augmented) Fredholm filtration F of M by finite-dimensional submanifolds (M_n), we associate to the triple (M, g, F) a non-commutative direct limit C*-algebra A(M, g, F) = lim A(M_n) that can play the role of the algebra of functions vanishing at infinity on the non-locally compact space M. The C*-algebra A(E), as constructed by Higson-Kasparov-Trout for their Bott periodicity theorem for infinite dimensional Euclidean spaces, is isomorphic to our construction when M = E. If M has an oriented Spin_q-structure (1 <= q <=\infty), then the K-theory of this C*-algebra is the same (with dimension shift) as the topological K-theory of M defined by Mukherjea. Furthermore, there is a Poincare' duality isomorphism of this K-theory of M with the compactly supported K-homology of M, just as in the finite-dimensional spin setting.

Transactions of the American Mathematical Society, 1975

If G G is a group of automorphisms of a C ∗ {C^ \ast } -algebra A A with identity, then G G acts in a natural way as a transformation group on the state space S ( A ) S(A) of A A . Moreover, this action is uniformly almost periodic if and only if G G has compact pointwise closure in the space of all maps of A A into A A . Consideration of the enveloping semigroup of ( S ( A ) , G ) (S(A),G) shows that, in this case, this pointwise closure G ¯ \bar G is a compact topological group consisting of automorphisms of A A . The Haar measure on G ¯ \bar G is used to define an analogue of the canonical center-valued trace on a finite von Neumann algebra. If A A possesses a sufficiently large group G 0 {G_0} of inner automorphisms such that ( S ( A ) , G 0 ) (S(A),{G_0}) is uniformly almost periodic, then A A is a central C ∗ {C^ \ast } -algebra. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional C ∗ {C^ \ast ...

2016

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra A is a *-homomorphism A → M that factors through the canonical inclusion C(X) ⊆ℓ^∞(X) when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where M is a C*-algebra. Any subhomogenous C*-algebra admits an injective functorial discretization, where M is a W*-algebra. However, any functorial discretization, where M is an AW*-algebra, must trivialize A = B(H) for any infinite-dimensional Hilbert space H.

2009

The approach we present is a modification of the Morse theory for unital C*-algebras. We provide tools for the geometric interpretation of noncommutative CW complexes. These objects were introduced and studied in [2], [7] and . Some examples to illustrate these geometric information in practice are given. A classification of unital C*-algebras by noncommutative CW complexes and the modified Morse functions on them is the main object of this work.

Proceedings of the Estonian Academy of Sciences, 2011

Several analogies of Ford lemma for topological algebras (in particular, for topological *-algebras) are proved (without using projective limits). Topological *-algebras, in which a self-adjoint element a with sp A (a) ⊂ (0, ∞) has a self-adjoint square root b with sp A (a) ⊂ (0, ∞) and sp A (h 1 +. .. + h n) ⊂ [0, ∞), if sp A (h k) ⊂ [0, ∞) where h k are self-adjoint elements for each k ∈ {1,. .. , n}, are described.