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Math 992 - Locally Compact Groups

Notes—based on Folland's A Course in Abstract Harmonic Analysis but including supplementary content on Radon measures and semidirect products—are available in the sidebar. The course began with Chapter 2 of this textbook, opting to return to Chapter 1 as needed. Below are optional exercises based on this material, many of which appear in the notes.

Section 2.1 Topological Groups
  • 1. Show that the continuity of G×G(x,y)x1yG is equivalent to the continuity of both multiplication and inversion.
  • 2. For a compact subset K of a locally compact Hausdorff space X, show that there exists KX satisfying K(K).
  • 3. Let K be a compact subset of a locally compact group G and let U be a neighborhood of K. Show that there exists a symmetric neighborhood V of 1G so that xK,KxU for all xV.
  • 4. Show that Adx(y):=xyx1 defines a continuous action G for a topological group G.
  • 5. Given a continuous action H\overset{\alpha}{\curvearrowright} N of topological groups, show that H_\alpha \ltimes N:=H\times N equipped with the operations (x,a)(y,b):=(xy, \alpha_y^{-1}(a)b) \qquad \text{ and } \qquad (x,a)^{-1} = (x^{-1}, \alpha_x(y^{-1}) is a topological group.
  • 6. Show that \begin{align*} G _{\text{Ad}}\ltimes G & \to G\\ (x,y) & \mapsto xy \end{align*} is a continuous surjective group homomorphism with kernel \{(x,x^{-1}\colon x\in G\}.
  • 7. Let G be a topological group and let \text{Aut}(G) denote the set of homeomorphic automorphisms. Equip \text{Aut}(G) with the topology determined by the neighborhood basis for the identity isomorphism consisting of sets of the following form \mathcal{U}(K,V):=\{ \alpha\in \text{Aut}(G)\colon \alpha(t), \alpha^{-1}(t)\in Vt \text{ for all } t\in K\}, where K\subset G is compact and V is an neighborhood of 1\in G.
    • (a) Show that \text{Aut}(G) is a topological group when equipped with this topology.
    • (b) For \beta\in \text{Aut}(G), show that \mathcal{U}(\beta,K,V):=\{\alpha \in \text{Aut}(G)\colon \alpha(t)\in V\beta(t) \text{ and }\alpha^{-1}(t)\in V \beta^{-1}(t) \text{ for all }t\in K\} forms a neighborhood basis for \beta, where K\subset G is compact and V is a neighborhood of 1\in G. [Hint: Show that \mathcal{U}(\beta, K\cup \beta^{-1}(K), V\cap \beta^{-1}(V)) \subset \mathcal{U}(K,V)\beta.]
    For the remaining parts, assume G is a locally compact group.
    • (c) Show that the evaluation map \begin{align*} \text{Aut}(G) \times G &\to G\\ (\alpha, x) & \mapsto \alpha(x) \end{align*} is continuous.
    • (d) Suppose G is a locally compact group. Show that if a net (\alpha_i)_{i\in I}\subset \text{Aut}(G) converges to \alpha\in \text{Aut}(G) in this topology then \lim_{i\to\infty} \| f\circ \alpha_i - f\circ \alpha\|_\infty =0 for all f\in C_c(G). Does the converse hold?
    • (e) Let \mu be a left Haar measure on G. Show that \text{Aut}(G)\ni \alpha \mapsto \frac{d(\mu\circ \alpha)}{d\mu} \in (0,\infty) is a continuous group homomorphism.
Section 2.2 Haar Measure
  • 1. Show that a Radon measure \mu on a locally compact Hausdorff space X is inner regular on all \mu-semifinite sets.
  • 2. Let X be a locally compact Hausdorff space.
    • (a) Show that f\colon X\to (-\infty, +\infty] is lower semicontinuous if and only if f(x_0)\leq \liminf_{x\to x_0} f(x) for all x_0\in X.
    • (b) Show that g\colon X\to [-\infty, +\infty) is upper semicontinuous if and only if g(x_0)\geq \limsup_{x\to x_0} g(x) for all x_0\in X.
  • 3. Let X be a locally compact Hausdorff space. For a lower semicontinuous function f\colon X\to [0,+\infty], show that f(x) = \sup\{ g(x)\colon g\in C_c(X),\ 0\leq g\leq f\} for all x\in X.