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We show that layer potential groupoids for conical domains constructed in an earlier paper (Carvalho-Qiao, Central European J. Math., 2013) are Fredholm groupoids, which enables us to deal with many analysis problems on singular spaces in a unified treatment. As an application, we obtain Fredholm criteria for operators on layer potential groupoids.
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We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $Gamma$-equivariant vector bundles $E_0$ and $E_1$. Let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces by restriction a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $pi_alpha(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $Gamma$ is finite abelian. We prove then that the restriction $pi_alpha(P)$ is Fredholm if, and only if, $P$ is $alpha$-elliptic, a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $alpha$-elliptic, but the converse is not true in general. However, if $Gamma$ acts freely on a dense open subset of $M$, then $P$ is $alpha$-elliptic for the given fixed $alpha$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $psi^{m}(M; E)^Gamma$ of classical, $Gamma$-invariant pseudodifferential operators acting on sections of the vector bundle $E to M$ and of the structure of its restrictions to the isotypical components of $Gamma$. These structures are described in terms of the isotropy groups of the action of the group $Gamma$ on $E to M$.
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.
We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $ C^{ast} $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{ast} $-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence-uniqueness results for KMS states associated to the graphs.
Let $Gamma$ be a compact group acting on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. When $Gamma$ is finite, we explicitly characterize the operators $P$ for which the map $pi_alpha(P)$ is Fredholm in terms of the principal symbol of $P$ and the action of $Gamma$ on the vector bundles $E_i$. When $Gamma = {1}$, that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol $C^*$-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. Whenever our results also hold for non-discrete groups, we prove them in this greater generality. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.
In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.
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