Let $ A$ and $ B$ be Banach algebras. Consider a right Banach $ A$ -module, $ E$ , and a right Banach $ B$ -module, $ F$ , as well as a Banach algebra morphism $ \pi\colon A\to\mathcal L_B(F)$ into the bounded $ B$ -linear operators on $ F$ . Then $ \pi$ makes $ F$ into a left Banach $ A$ -module, and the actions of $ A$ and $ B$ commute. Thus, the (completed) tensor product $ E\otimes_AF$ has the structure of a right Banach $ B$ -module.

**Question:** Is the tensor product of two compact operators again compact?

To be more precise, let $ F\in\mathcal K_A(E)$ and $ G\in\mathcal K_B(F)$ . Is $ F\otimes_AG\in\mathcal K_B(E\otimes_AF)$ ? As for Hilbert C*-modules, the compact operators on a Banach module are the closure of the linear span of the finite rank operators. Thus, it suffices to consider the case where $ F=F_1F_2$ with $ F_1\in\mathcal L_A(A,E)$ , $ F_2\in\mathcal L_A(E,A)$ , and where $ G=G_1G_2$ with $ G_1\in\mathcal L_B(B,F)$ , $ G_2\in\mathcal L_B(F,B)$ . However, the tensor product $ F_i\otimes_AG_i$ is not defined since $ B$ itself does not carry an $ A$ -module structure.

**Motivation:** In Lafforgue’s $ KK^{\mathrm{ban}}$ -theory, there is a natural map $ \Psi\colon KK^{\mathrm{ban}}(A,B)\to\mathrm{Hom}(K_0(A),K_0(B))$ which may be described as follows: Elements of $ KK^{\mathrm{ban}}(A,B)$ are represented by Banach $ (A,B)$ -bimodules, i.e. triples $ (F,\pi,T)$ with $ F$ a (graded) right Banach $ B$ -module, $ \pi\colon A\to\mathcal L_B(F)$ an even contractive Banach algebra morphism, and $ T\in\mathcal L_B(F)$ an odd operator such that $ \pi(a)(T^2-1)$ and $ [\pi(a),T]$ are compact for all $ a\in A$ . Now if $ p\in M_n(A)$ is an idempotent, one may pushforward $ p$ via $ \pi$ to obtain an idempotent $ \pi_*p\in M_n(\mathcal L_B(F))\cong\mathcal L_B(F^n)$ . Then $ \Psi[F,\pi,T]([p])=[\pi_*pF^n,(\pi_*p)T^n(\pi_*p)]\in KK^{\mathrm{ban}}(\mathbb C,B)\cong K_0(B)$ .

Now I wondered if one could find a similar description when an element of $ K_0(A)$ is represented not by an idempotent over $ A$ , but rather as an element of $ KK^{\mathrm{ban}}(\mathbb C,A)\cong K_0(A)$ , i.e. by a *Fredholm module* $ (E,S)$ where $ E$ is a graded right Banach $ A$ -module, and $ S\in\mathcal L_A(E)$ is an odd operator satisfying $ S^2-1\in\mathcal K_A(E)$ . A first try would be to put $ \Psi[F,\pi,T]([E,S])=[E\otimes_AF,T\otimes_AS]$ , where the tensor product is built using $ \pi$ . However, one would have to prove that $ (T\otimes_AS)^2-1=(T^2-1)\otimes_A(S^2-1)$ is compact, which boils down to the question above.