Compact support, $C_c(X)$, and $C_0(X)$

Depends on

• Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space
• Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
• Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right

Used by

• Compact and discrete transforms are the two extreme Plancherel cases Corollary
• Complex Haar Lᵖ spaces and compactly supported functions Definition
• Left Haar integral and left Haar measure Definition
• Positive linear functionals on C_c(X) Definition
• The compactly supported cutoff relation f≺ U Definition
• The transformation (covariance) algebra C_c(G× X) Definition
• C zero of a locally compact space Example
• Convolution on a discrete group Example
• Multiplicity-two diagonal representation Example
• Unitization corresponds to one point compactification Example
• Weak-star compactness of probability measures Example
• C_c(X) is dense in Lⁱnfinity(mu) for every Radon measure False statement
• A bounded real C₀(X) functional is a difference of positive functionals Lemma
• A finite compactly supported partition of unity near a compact set Lemma
• A system of imprimitivity integrates to a nondegenerate representation of the transformation algebra Lemma
• Bounded density and finite-vector transitivity for C*-representations Lemma
• Characters of continuous functions are evaluations Lemma
• Compact-open neighbourhoods on the dual give a neighbourhood basis on the group Lemma
• Compactly supported kernels admit commuting radon integrals Lemma
• Compactly supported nonnegative transform bumps on the dual Lemma
• Completeness of the complex Haar L1 and L2 spaces and density of Cc Lemma
• Continuous characters separate points of an LCA group Lemma
• Convolution preserves compact support and is associative Lemma
• Fourier-Stieltjes transforms determine finite Radon measures Lemma
• Haar measure on an abelian group is invariant under inversion Lemma
• L1 convolution smooths bounded functions into UCB Lemma
• L1 of a second-countable locally compact group is separable Lemma
• LCH Urysohn cutoff Lemma
• Nonzero multiplicative functionals on L¹ of an LCA group are Fourier evaluations Lemma
• Normal relative property (T) controls the distance to the invariant subspace Lemma
• Positive C₀(X) functionals have finite regular representing measures Lemma
• Positive convolution squares form a dense inversion core Lemma
• Positive definite functions give positive bounded functionals on the transform core Lemma
• Restriction of the regular representation to a closed subgroup Lemma
• The Bochner functional extends and has a Radon representing measure Lemma
• The character topology on L¹ of an LCA group is the compact-open topology Lemma
• The Plancherel transform range is dense in L² of the dual Lemma
• Translation continuity and normalised local approximate identities on an LCA group Lemma
• The L1 group algebra has a unit exactly when the group is discrete Proposition
• Bochner's theorem for LCA groups Theorem

…and 12 more results.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Donald L. Cohn, Measure Theory, 2nd ed., §7.1 (standard reference, not scraped)