L1(G) has an approximate identity
Let G be a locally compact topological group. In general, the Banach *-algebra L1(G) (parent entry (http://planetmath.org/L1GIsABanachAlgebra)) does not have an identity element. In fact:
- L1(G) has an identity element if and only if G is discrete.
When G is discrete the identity element of L1(G) is just the Dirac delta, i.e. the function that takes the value 1 on the identity element of G and vanishes everywhere else.
Nevertheless, L1(G) has always an approximate identity.
Theorem - L1(G) has an approximate identity (eλ)λ∈Λ. Moreover the approximate identity (eλ)λ∈Λ can be chosen to the following :
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eλ is self-adjoint (http://planetmath.org/InvolutaryRing),
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∥eλ∥1=1,
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eλ∈Cc(G)
where Cc(G) stands for the space of continuous functions G⟶ℂ with compact support.
Title | L1(G) has an approximate identity |
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Canonical name | L1GHasAnApproximateIdentity |
Date of creation | 2013-03-22 17:42:40 |
Last modified on | 2013-03-22 17:42:40 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 9 |
Author | asteroid (17536) |
Entry type | Theorem |
Classification | msc 46K05 |
Classification | msc 43A20 |
Classification | msc 22D05 |
Classification | msc 22A10 |
Defines | L1(G) has an identity element iff G is discrete |