Let $G$ be a topological group of totally disconnected (td) type. This means that the identity of $G$ has a fundamental system of neighborhoods consisting of open compact subgroups. Then $G$ is locally compact, and has a Haar measure. We give $C_c^{\infty}(G)$, the vector space of locally constant functions of compact support into $\mathbb{C}$, the structure of an (associative, not necessarily unital) algebra over $\mathbb{C}$ by setting

$$f_1 \ast f_2(g) = \int\limits_G f_1(x)f_2(x^{-1}g) dx$$

As an algebra, we write $H(G)$ instead of $C_c^{\infty}(G)$.

Let $G_1, G_2$ be groups of td type. In the article Decompositions of Representations into Tensor Products (D. Flath, Corvallis proceedings), it is written that

$$H(G_1 \times G_2) \simeq H(G_1) \otimes_{\mathbb{C}} H(G_2)$$

I'm trying to understand why this is true. I have tried to show that $H(G_1 \times G_2)$ satisfies the same universal property as $H(G_1) \otimes_{\mathbb{C}} H(G_2)$ in the category of algebras. To do this, I need to define algebra homomorphisms $\phi_i: H(G_i) \rightarrow H(G_1 \times G_2)$ and show that for any algebra $A$, $\delta \mapsto (\delta \circ \phi_1, \delta \circ \phi_2)$ gives a bijection from $\textrm{Hom}_{\mathbb{C}-\textrm{alg}}(H(G_1 \times G_2),A)$ onto

$$\{ (\psi_1,\psi_2) \in \textrm{Hom}_{\mathbb{C}-\textrm{alg}}(H(G_1),A) \times \textrm{Hom}_{\mathbb{C}-\textrm{alg}}(H(G_2),A) : \psi_1(f_1), \psi_2(f_2) \textrm{ commute for all } f_i \in H(G_i)\}$$

At first, I thought that I should define $\phi_1$ by

$$\phi_1(f)(g_1,g_2) = f(g_1)$$

which is locally constant, but obviously not of compact support, in general. Something more clever is required, but I cannot yet see what. I would appreciate any suggestions or hints.

  • $\begingroup$ What are the compact open subset of $G_1\times G_2$? $\endgroup$ – AdLibitum Jul 13 '17 at 22:19
  • $\begingroup$ They include the products $K_1 \times K_2$ $\endgroup$ – D_S Jul 13 '17 at 22:32

For any t.d. group $G$, $H(G)$ is spanned by the characteristic functions of the (cosets of) compact open subgroups of $G$. So it suffices to define what a map $H(G_1\times G_2)\rightarrow H(G_1)\otimes H(G_2)$ does to each characteristic function. Since we're only working with characteristic functions, we only care about characteristic functions for a basis of the topology on $G_1\times G_2$. So, in particular, we can assume wlog that our function is of the form $\chi_{U\times V}$, where $U\subset G_1$ and $V\subset G_2$ are (cosets of) compact opens. There's an obvious element of $H(G_1)\otimes H(G_2)$ to map this to, namely $\chi_U\otimes\chi_V$. It's not very hard to check that this gives an isomorphism. (What's the inverse?)

  • $\begingroup$ You need the closets of the compact open subgroups, not just the subgroups. $\endgroup$ – Mariano Suárez-Álvarez Jul 14 '17 at 23:00
  • $\begingroup$ Yes, that's right; thanks! (typo: cosets) $\endgroup$ – PL. Jul 14 '17 at 23:02

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