Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C * -algebra is the stationary inductive limit of a C * -stable Fell algebra that has compact spectrum and trivial Dixmier-Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to in principle compute the K-theory of the stable C * -algebra. A specific onedimensional Smale space (the aab/ab-solenoid) is considered as an illustrative running example throughout. 1 2 ROBIN J. DEELEY AND ALLAN YASHINSKIunstably equivalent to some point in P. On the set X u (P), we consider the stable equivalence relation ∼ s , viewed as a groupoidThe groupoid G s (P) has anétale topology, and the stable C * -algebra of (X, ϕ) is the groupoid C * -algebra C * (G s (P)).In the case where X is a Wieler solenoid, we use the inverse limit structure to define a subrelation ∼ 0 of ∼ s . There is a corresponding subgroupoid (P), and therefore G 0 (P) isétale. Building on this, we use the fact that the inverse limit is stationary to prove the following result.Theorem 0.1. There is a nested sequence ofétale subgroupoidsThis allows one to reduce the study of G s (P) to G 0 (P), which is easier to understand. To see why it is easier, first note that the space X u (P) has a natural topology, which coincides with the topology of the diagonal subspace of G s (P). Note we never consider the subspace topology of X u (P) it inherits from X, as X u (P) is dense as a subset of X. Likewise the topology defined on G s (P) is not the same as the subspace topology it inherits as a subspace of X u (P) × X u (P). However, the topology of G 0 (P) does coincide with the subspace topology from
The main result of the present paper is that the stable and unstable C * -algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C * -algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.
These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (http://www.math.psu.edu/mass/).The lectures are meant to be accessible to advanced undergraduate and early graduate students in mathematics. We have placed a great emphasis on clarity and exposition, and we have included many exercises. Hints and solutions for most of the exercises are provided in the end.The lectures discuss piecewise distance preserving maps from a 2-dimensional polyhedral space into the plane. Roughly speaking, a polyhedral space is a space that is glued together out of triangles, for example the surface of a polyhedron. If one imagines such a polyhedral space as a paper model, then a piecewise distance preserving map into the plane is essentially a way to fold the model so that it lays flat on a table. We have five lectures on the following topics:⋄ Zalgaller's folding theorem, which guarantees the existence of a piecewise distance preserving map from a 2-dimensional polyhedral space into the plane. In other words, it is always possible to fold the paper model onto the table. ⋄ Brehm's extension theorem, which allows one to build piecewise distance preserving maps from a convex polygon into the plane with prescribed images on a finite subset of the polygon. ⋄ Akopyan's approximation theorem, which allows one to approximate maps from a 2-dimensional polyhedral space into the plane by piecewise distance preserving maps. ⋄ Gromov's rumpling theorem, which shows the existence of a length-preserving map from the sphere into the plane; i.e., a map that preserves the lengths of all curves. ⋄ An entertaining problem of Arnold on paper folding, which asks if it is possible to fold a square in the plane so that the perimeter increases.3 4We only consider the 2-dimensional case to keep things easy to visualize. However, most of the results admit generalizations to higher dimensions. These results are discussed in the Final Remarks, where proper credit and references are given.Acknowledgments. We would like to thank Arseniy Akopyan, Robert Lang, Alexei Tarasov for their help. Also we would like to thank all the students in our class for their participation and true interest.
Given a smooth deformation of topological algebras, we define Getzler's Gauss-Manin connection on the periodic cyclic homology of the corresponding smooth field of algebras. Basic properties are investigated including the interaction with the Chern-Connes pairing with K-theory. We use the Gauss-Manin connection to prove a rigidity result for periodic cyclic cohomology of Banach algebras with finite weak bidimension. Then we illustrate the Gauss-Manin connection for the deformation of noncommutative tori. We use the Gauss-Manin connection to identify the periodic cyclic homology of a noncommutative torus with that of the commutative torus via a parallel translation isomorphism. We explicitly calculate the parallel translation maps and use them to describe the behavior of the Chern-Connes pairing under this deformation. * This research was partially supported under NSF grant DMS-1101382. Preliminaries 2.1 Locally convex algebras and modulesSee [34] for background in the theory of locally convex topological vector spaces. More details concerning topological tensor products can be found in [14,15,34].We shall work in the category LCTVS of complete, Hausdorff locally convex topological vector spaces over C and continuous linear maps. All bilinear/multilinear maps we work with will be assumed to be jointly continuous. For example, by a locally convex algebra, we mean a space A ∈ LCTVS with a jointly continuous associative multiplication. Similarly, a locally convex module over a locally convex algebra has a jointly continuous module action.Dealing with joint continuity leads naturally to projective tensor products. If R is a unital commutative locally convex algebra and M and N are locally convex R-modules, then M ⊗ R N denotes the completed projective tensor product over R, which is universal for jointly continuous R-bilinear maps from M × N into locally convex R-modules, see [15, Chapter II]. Given X, Y ∈ LCTVS, we shall simply write X ⊗Y for X ⊗ C Y .Given X, Y ∈ LCTVS, we equip the space Hom(X, Y ) of continuous linear maps from X to Y with the topology of uniform convergence on bounded subsets of X. It is a Hausdorff locally convex topological vector space, but it may not be complete. However, it is for many nice cases, for example if X is a Fréchet space or an LF -space. The strong dual of X is X * = Hom(X, C). We remark that the strong dual of a Banach space is a Banach space, but the strong dual of a Fréchet space is never a Fréchet space, unless the original space is actually a Banach space.Given two locally convex R-modules M and N , we topologize Hom R (M, N ) as a subspace of Hom C (M, N ). When N = R, we obtain the topological R-linear dual M := Hom R (M, R).
Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous trace C * -algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arise naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper.
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