Helton-Howe trace formula gives an explicit description of the traces of antisymmetric sums of Toeplitz operators on the Bergman space. On a submodule of the Bergman module with relatively nice zero locus, previous results show that the cross commutators between the generalized Toeplitz operators with coordinate symbols belong to some Schatten class. A natural question is whether the same Helton-Howe trace formula remain true for such submodules. We answer this question in affirmative, using finite rank approximation and dilation techniques.
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Given a metric space $X$, the Roe algebra of $X$, denoted by $\mathrm{C}^*(X)$, is a $ \mathrm{C} ^*$-algebra which encodes many of $X$'s large scale geometric properties. In this talk, I will discuss some uniform approximability results for maps between Roe algebras (we call those "coarse-like properties"). I will then talk about applications of these uniform approximability results to isomorphisms between Roe algebras. In particular, given a uniformly locally finite metric space $X$, we obtain that the canonical map from the group of coarse equivalences of $X$ modulo the relation of closeness to the group of outer automorphisms of $ \mathrm{C} ^*(X)$ is surjective if $X$ has property A. This is a joint work with Alessandro Vignati.
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A completely positive (cp) map is called order zero when it preserves orthogonality. Such maps enjoy a rich structure, which has made them a key component of completely positive approximations of nuclear C*-algebras. Motivated by generalized inductive limits arising from such cp approximations, we consider the structure of the image of a cp order zero map. It turns out that this is captured by a few key properties that one can ask of a self-adjoint subspace of a C*-algebra. We will discuss these properties and the implications for generalized inductive sequences. This is joint work with Wilhelm Winter.
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We present results in index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a ``long neck principle'' for a compact Riemannian spin n-manifold with boundary X, stating that if scal(X) ≥ n(n-1) and there is a nonzero degree map f into the n-sphere which is area decreasing, then the distance between the support of the differential of f and the boundary of X is at most π/n. This answers, in the spin setting, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing a small n-ball from a closed spin n-manifold Y. We show that if scal(X) ≥ σ >0 and Y satisfies a certain condition expressed in terms of higher index theory, then the width of a geodesic collar neighborhood Is bounded from above from a constant depending on σ and n. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to N x [-1,1], with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if scal(V) ≥ n(n-1), then the distance between the boundary components of V is at most 2π/n. This last constant is sharp by an argument due to Gromov.
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Higher general for a G-proper manifold without boundary can be defined in analogy with Galois coverings and they are, by definition, geometric objects. To understand their stability properties we need to connect them to higher C^*-indices of suitable Dirac operators. This is possible but under additional assumptions on the group G, for example G semisimple and connected and more generally G satisfying the Rapid Decay condition and G/K of nonpositive sectional curvature. I will begin my talk by explaining these results. I will then move to manifolds with boundary and explain how it is possible to define higher genera in this more complicated situation. Crucial to the analysis is a higher C^*-index theorem of Atiyah-Patodi-Singer type. All these results, the last very recent, are in collaboration with Hessel Posthuma.
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Recently Rufus Willett and I showed that all bounded derivations on Uniform Roe Algebras associated to a bounded geometry metric space X are inner in our paper “Bounded Derivations on Uniform Roe Algebras”. This is equivalent to the first Hochschild cohomology group $H^1(C^*_u(X), C^*_u(X))$ vanishing. It is then natural to ask if all the higher groups $H^n(C^*_u(X), C^*_u(X))$ vanish. To investigate the continuous cohomology of a Uniform Roe Algebra we employ the technique of “reduction of cocycles” where we modify a given cocycle by a coboundary to obtain certain properties. I will discuss this procedure and give examples of calculating the higher cohomology groups.
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In the early days of Banach algebra K-theory, Novodvorskii proved that the Gelfand transform for any commutative Banach induces an isomorphism in Banach algebra K-theory. This is a version of the Oka principle in several complex variables, which identifies equivalence classes of structures, including vector bundles, in the holomorphic and continuous categories in a variety of contexts. Since the Oka principle has long been proposed as a mechanism to understand and indeed prove the Baum-Connes conjecture, Novodvorskii’s theorem continues to be of interest in noncommutative geometry. I shall give a more or less self-contained proof of Novodvorskii’s theorem, along with a rough sketch of possible future extensions into the noncommutative realm. This is joint work with Jacob Bradd.
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