Hecke algebra of a locally compact group

Remarks

Each double coset $KgK$ is compact and open, and a compact set meets only finitely many of them. So the indicator functions $\mathbf{1}_{KgK}$ form a basis of $\mathcal{H}(G, K)$ as a vector space. The function $\mu(K)^{-1} \mathbf{1}_K$ is the unit element.

Example: $G$ discrete and $K = \set{e}$, with $\mu$ the counting measure. Both clauses say only that $f$ is finitely supported, and $\mathbf{1}_{\set{a}} * \mathbf{1}_{\set{b}} = \mathbf{1}_{\set{ab}}$. So $\mathcal{H}(G, \set{e})$ is the group algebra $\C[G]$.

Example: $G = \mathrm{GL}_2(\Q_p)$ and $K = \mathrm{GL}_2(\Z_p)$. Then $\mathcal{H}(G, K)$ is commutative. It is generated by the indicators of $K \operatorname{diag}(p, 1) K$ and $K \operatorname{diag}(p, p) K$ and the inverse of the latter.

Non-example: for $G = \Q_p$ and $K = \Z_p$, the indicator of $p \Z_p$ fails clause 1, since it is not constant on the coset $\Z_p$.

When $G$ is totally disconnected, the union of the $\mathcal{H}(G, K)$ over all compact open $K$ is the full Hecke algebra $\mathcal{H}(G)$ of locally constant compactly supported functions.