Math 2.0

Abstract harmonic analysis Definition

Measure algebra M(G)

Builds on Locally compact groupComplex measureTotal variation of a measureRadon measureSpace of functions vanishing at infinityGroup algebra of a locally compact groupBanach algebra

Definition

Let GG be a locally compact group. Let M(G)M(G) be the space of complex measures μ\mu on the Borel sets of GG whose total variation ∣μ∣\abs{\mu} is a Radon measure. The measure algebra of GG is M(G)M(G) with the following structure.

  1. The norm is the total variation ∥μ∥=∣μ∣(G)\norm{\mu} = \abs{\mu}(G).
  2. The product is the convolution μ∗ν∈M(G)\mu * \nu \in M(G), determined by ∫Gf d(μ∗ν)=∫G∫Gf(xy) dμ(x) dν(y)\int_G f \, d(\mu * \nu) = \int_G \int_G f(xy) \, d\mu(x) \, d\nu(y) for every ff in the space C0(G)C_0(G) of continuous functions vanishing at infinity.
  3. The involution is μ∗(A)=μ(A−1)‾\mu^*(A) = \overline{\mu(A^{-1})} for Borel sets AA, where A−1={x−1:x∈A}A^{-1} = \set{x^{-1} : x \in A}.

Remarks

By the Riesz representation theorem, M(G)M(G) is the dual of the Banach space C0(G)C_0(G), and the norm in clause 1 is the dual norm. The convolution is well defined since the right side of clause 2 is a bounded linear functional of ff, of norm at most ∥μ∥∥ν∥\norm{\mu} \norm{\nu}. With these operations M(G)M(G) is a Banach algebra with isometric involution, with unit the point mass δe\delta_e at the identity ee. It is commutative exactly when GG is abelian. For a left Haar measure mm, the map f↦f dmf \mapsto f \, dm embeds the group algebra L1(G)L^1(G) isometrically into M(G)M(G) as a closed two-sided ideal.

Example: point masses. For x,y∈Gx, y \in G, clause 2 gives ∫f d(δx∗δy)=f(xy)\int f \, d(\delta_x * \delta_y) = f(xy), so δx∗δy=δxy\delta_x * \delta_y = \delta_{xy}. Thus x↦δxx \mapsto \delta_x is an injective homomorphism of GG into the group of invertible elements of M(G)M(G).

Non-example: Lebesgue measure λ\lambda on ℝ\R is a Radon measure but not an element of M(ℝ)M(\R), since λ(ℝ)=∞\lambda(\R) = \infty and so its total variation norm is infinite.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Banach algebra, concept, no evidence levelBanach algebraconceptComplex measure, concept, no evidence levelComplex measureconceptGroup algebra of a locally compact group, concept, no evidence levelGroup algebra of a lo…conceptLocally compact group, concept, no evidence levelLocally compact groupconceptRadon measure, concept, no evidence levelRadon measureconceptSpace of functions vanishing at infinity, concept, no evidence levelSpace of functions va…conceptTotal variation of a measure, concept, no evidence levelTotal variation of a …conceptMeasure algebra M(G), definition, no evidence levelMeasure algebra M(G)definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Banach algebra concept
  2. Complex measure concept
  3. Group algebra of a locally compact group concept
  4. Locally compact group concept
  5. Radon measure concept
  6. Space of functions vanishing at infinity concept
  7. Total variation of a measure concept
  8. Measure algebra M(G) definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_algebra_of_a_locally_compact_group
About this page
TagG6YK28C0Z6
URIm2://G6YK28C0Z6
Kinddefinition
WritingW0 stub
MSC43A10, 22D15
DefinesMeasure algebra M(G)
UsesLocally compact group, Complex measure, Total variation of a measure, Radon measure, Space of functions vanishing at infinity, Group algebra of a locally compact group, Banach algebra
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.