Translation Invariant Asymptotic Homomorphisms and Extensions of C*-Algebras

2005 ◽  
Vol 39 (3) ◽  
pp. 236-239 ◽  
Author(s):  
V. M. Manuilov ◽  
K. Thomsen
Keyword(s):  
Translation Invariant ◽  
C Algebras

scholarly journals Extensions of $C^*$-algebras and translation invariant asymptotic homomorphisms

2007 ◽  
Vol 100 (1) ◽  
pp. 131
Author(s):  
V. Manuilov ◽  
K. Thomsen
Keyword(s):  
The Other ◽  
Translation Invariance ◽  
Homotopy Classes ◽  
Translation Invariant ◽  
C Algebras

Let $A$, $B$ be $C^*$-algebras; $A$ separable, $B$ $\sigma$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathsf{R})\otimes A$ to $B$ and show that the Connes-Higson construction applied to any extension of $A$ by $B$ is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of $A$ by $B$ out of such a translation invariant asymptotic homomorphism. This leads to our main result; that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.

An Introduction to K-Theory for C*-Algebras

2000 ◽  
Author(s):  
M. Rørdam ◽  
F. Larsen ◽  
N. Laustsen
Keyword(s):  
C Algebras ◽  
K Theory

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

scholarly journals Optimal transport and unitary orbits in C⁎-algebras

2021 ◽  
Vol 281 (5) ◽  
pp. 109068
Author(s):  
Bhishan Jacelon ◽  
Karen R. Strung ◽  
Alessandro Vignati
Keyword(s):  
Optimal Transport ◽  
C Algebras

Non‐associative C *‐algebras

2021 ◽  
pp. 111-153
Author(s):  
Ángel Rodríguez Palacios ◽  
Miguel Cabrera García
Keyword(s):  
C Algebras

High-Performance Image Filters via Sparse Approximations

2020 ◽  
Vol 3 (2) ◽  
pp. 1-19
Author(s):  
Kersten Schuster ◽  
Philip Trettner ◽  
Leif Kobbelt
Keyword(s):  
High Performance ◽  
Hardware Acceleration ◽  
Optimization Method ◽  
Translation Invariant ◽  
Approximation Quality ◽  
Trade Offs ◽  
Sparse Approximations ◽  
Image Filters ◽  
Good Trade ◽  
And Performance

We present a numerical optimization method to find highly efficient (sparse) approximations for convolutional image filters. Using a modified parallel tempering approach, we solve a constrained optimization that maximizes approximation quality while strictly staying within a user-prescribed performance budget. The results are multi-pass filters where each pass computes a weighted sum of bilinearly interpolated sparse image samples, exploiting hardware acceleration on the GPU. We systematically decompose the target filter into a series of sparse convolutions, trying to find good trade-offs between approximation quality and performance. Since our sparse filters are linear and translation-invariant, they do not exhibit the aliasing and temporal coherence issues that often appear in filters working on image pyramids. We show several applications, ranging from simple Gaussian or box blurs to the emulation of sophisticated Bokeh effects with user-provided masks. Our filters achieve high performance as well as high quality, often providing significant speed-up at acceptable quality even for separable filters. The optimized filters can be baked into shaders and used as a drop-in replacement for filtering tasks in image processing or rendering pipelines.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

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