I have a question on a paper in the Corvallis proceedings on automorphic forms. Background: Let $G$ be a topological group of td type. This means that $G$ is Hausdorff, and every neighborhood of the identity of $G$ contains a compact open subgroup. The Hecke algebra $H(G)$ is the vector space of all locally constant functions of compact support $G \rightarrow \mathbb{C}$, with multiplication given by convolution:
$$f_1 \ast f_2(g) = \int\limits_G f_1(x)f_2(x^{-1}g)dx$$
$H(G)$ is in general not commutative or unital, but for each open compact subgroup $K$ of $G$, $H(G,K)$, the subalgebra of $H(G)$ consisting of all bi $K$-invariant functions, has an identity element $e_K = \frac{1}{\mu(K)} \chi_K$.
A representation $(\pi,V)$ of $G$ is called smooth if the stabilizer of each vector is an open subgroup of $G$. Every smooth representation is associated to a canonical nondegenerate $H(G)$-module structure given by a vector valued integral
$$\pi(f)v = \int\limits_G f(g)\pi(g)v \space dg$$
$V$ is irreducible if and only if it is simple as an $H(G)$-module, and a $G$-map is the same thing as an $H(G)$-linear map. Moreover, $V$ is irreducible if and only if $V^K = \{ v \in V : \pi(k).v = v \textrm{ for all } k \in K \}$ is simple as an $H(G,K)$-module for a fundamental system of open compact subgroups $K$.
A smooth representation is called admissible if $V^K$ is finite dimensional for all open compact $K$.
The problem: let $G_1, G_2$ be groups of td-type, and let $G = G_1 \times G_2$. The Hecke algebra of $G$ can be identified with $H(G_1) \otimes H(G_2)$. I'm trying to understand why every irreducible admissible representation $(\pi,V)$ of $G$ decomposes as a tensor product of irreducible admissible representations $V_1 \otimes V_2$ for $G_1$ and $G_2$. My reference is the article Decomposition of Representations into Tensor Products by D. Flath:
The reference [2] is a result in Bourbaki, Algebra, Chap VIII, which says that if $A, B$ are unital algebras over an algebraically closed field $k$, and $P$ is a simple unitary $A \otimes_k B$-module, then $P$ is isomorphic to $M \otimes_k N$ for simple modules $M, N$ over $A, B$.
What I'm confused on is where the maps $b_1, b_2$ are coming from, why they can be chosen to be injections, and why they can be chosen to form a directed system over the fundamental system of open compact neighborhoods $K_1 \times K_2$ of $G_1 \times G_2$. I would appreciate any reference or a hint.
