Multipliers for the convolution algebra of left and rightK-finite compactly supported smooth functions on a semi-simple Lie group

1984 ◽  
Vol 75 (1) ◽  
pp. 9-23 ◽  
Author(s):  
P. Delorme
Keyword(s):  
Lie Group ◽  
Smooth Functions ◽  
Convolution Algebra ◽  
Compactly Supported

Compactly supported analytic indices for Lie groupoids

2009 ◽  
Vol 4 (2) ◽  
pp. 223-262 ◽  
Author(s):  
Paulo Carrillo Rouse
Keyword(s):  
Index Theorem ◽  
Lie Groupoids ◽  
Smooth Functions ◽  
Lie Groupoid ◽  
Convolution Algebra ◽  
Compactly Supported ◽  
Potential Applications ◽  
Analytic Index ◽  
Image Position ◽  
Symbol Class

AbstractFor any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairingwith τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

scholarly journals On the weakly-* dense subsets in $L^{\infty}(\Omega)$

2021 ◽  
Vol 16 ◽  
pp. 149
Author(s):  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Weak Convergence ◽  
Density Property ◽  
Smooth Functions ◽  
Compactly Supported

In this paper we study the density property of the compactly supported smooth functions in the space $L^{\infty}(\Omega)$. We show that this set is dense with respect to the weak-* convergence in variable spaces.

scholarly journals Extension theorems for smooth functions on real analytic spaces and quotients by Lie groups and smooth stability

1977 ◽  
Vol 24 (4) ◽  
pp. 440-457
Author(s):  
G. S. Wells
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Analytic Space ◽  
Main Step ◽  
Compact Lie Group ◽  
Orbit Type ◽  
Smooth Functions ◽  
Extension Theorems ◽  
Real Analytic

AbstractExtension theorems are proved for smooth functions on a coherent real analytic space for which local defining functions exist which are finitely determined in the sense of J. Mather, (1968), and for smooth functions invariant under the action of a compact lie groupG. thus providing the main step in the proof that smooth infinitesimal stability implies smooth stability in the appropriate categories. In addition the remaining step for the category ofCxG-manifolds of finite orbit type is filled in.

Cyclic-parallel Ricci tensors on a class of almost Kenmotsu 3-manifolds

2019 ◽  
Vol 16 (06) ◽  
pp. 1950092 ◽  
Author(s):  
Yaning Wang ◽  
Xinxin Dai
Keyword(s):  
Hyperbolic Space ◽  
Lie Group ◽  
Ricci Tensor ◽  
Lie Derivative ◽  
Smooth Functions ◽  
Cyclic Parallel Ricci Tensor ◽  
Contact Distribution ◽  
Local Characterization ◽  
Parallel Ricci Tensor

In this paper, we give a local characterization for the Ricci tensor of an almost Kenmotsu [Formula: see text]-manifold [Formula: see text] to be cyclic-parallel. As an application, we prove that if [Formula: see text] has cyclic-parallel Ricci tensor and satisfies [Formula: see text], (where [Formula: see text] is the Lie derivative of [Formula: see text] along the Reeb flow and both [Formula: see text] and [Formula: see text] are smooth functions such that [Formula: see text] is invariant along the contact distribution), then [Formula: see text] is locally isometric to either the hyperbolic space [Formula: see text] or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

scholarly journals Invariant Whitney functions

2019 ◽  
Vol 30 (02) ◽  
pp. 1950009
Author(s):  
Hans-Christian Herbig ◽  
Markus J. Pflaum
Keyword(s):  
Lie Group ◽  
Invariant Function ◽  
Compact Lie Group ◽  
Similar Statement ◽  
Real Vector ◽  
Linear Action ◽  
Smooth Functions ◽  
Finite Dimensional ◽  
Subanalytic Set ◽  
Whitney Functions

Theorem 1 of [G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975) 63–68.] says that for a linear action of a compact Lie group [Formula: see text] on a finite dimensional real vector space [Formula: see text], any smooth [Formula: see text]-invariant function on [Formula: see text] can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set [Formula: see text] fulfilling some regularity assumptions. In order to deal with the case when [Formula: see text] is not [Formula: see text]-stable, we use the language of groupoids.

scholarly journals Compactly Supported Curvelet-Type Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Kenneth N. Rasmussen ◽  
Morten Nielsen
Keyword(s):  
Sparse Representation ◽  
Compact Support ◽  
Type Systems ◽  
Linear Combinations ◽  
Smooth Functions ◽  
Constructive Description ◽  
Compactly Supported ◽  
Single Function ◽  
Type Spaces ◽  
Finite Linear

We study a flexible method for constructing curvelet-type frames. These curvelet-type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to give a constructive description of how to construct curvelet-type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet-type spaces. Such a system of curvelet molecules can then be constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay.

scholarly journals A Convenient Notion of Compact Set for Generalized Functions

2017 ◽  
Vol 61 (1) ◽  
pp. 57-92
Author(s):  
Paolo Giordano ◽  
Michael Kunzinger
Keyword(s):  
Function Spaces ◽  
Generalized Functions ◽  
Distribution Theory ◽  
Compact Set ◽  
Compact Sets ◽  
Smooth Functions ◽  
Test Function ◽  
Compactly Supported ◽  
Nonlinear Generalized Functions

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

scholarly journals OPERATOR KERNELS FOR IRREDUCIBLE REPRESENTATIONS OF EXPONENTIAL LIE GROUPS

2008 ◽  
Vol 78 (2) ◽  
pp. 301-316
Author(s):  
DETLEV POGUNTKE
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Kernel Function ◽  
Lie Group ◽  
Linear Form ◽  
Irreducible Representations ◽  
Compactly Supported ◽  
Exponential Lie Group ◽  
Exponential Lie Groups

AbstractA nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

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