This post is a reading note to Bernstein’s Le “centre” de Bernstein, on the part of idempotented algebras (section 1.1 to 1.7, ibid.). Let \(R\) be a ring, an element \(e\) is called an idempotent if \(e^2=e\). Let \(k\) be a field.

Definition 1. A \(k\)-idempotented algebra \((H, I)\) is a \(k\)-algebra \(H\) together with the set idempotents \(I\) satisfying

(*) For any finite family \((x_i)\), there exists an idempotent \(e \in I\) such that \(ex_ie = x_i\) for all \(i\).

The \(k\)-structures of \(H\) is not involved in the discussion of this post, so we will ignore it. Let \((H, I)\) be an idempotented algebra for the rest of the post.

Lemma 2. Let \(e, f \in I\). The following are equivalent.

(1) \(eHe \subset fHf\)
(2) \(e \in fHf\)
(3) \(e = fef\)

Pf. If \(eHe \subset fHf\), then in particular, \(e = eee \in fHf\). If \(e \in fHf\), then \(e = fhf\) for some \(h \in H\), so \[fef = f(fhf)f = fhf = e.\] If \(e = fef\), then \(eHe = fefHfef \subset fHf\). \(\vartriangleleft\)

For \(e, f \in I\), we say \(e \le f\) if the equivalent conditions in Lemma 2 hold. This gives an order on the set \(I\). Definition 1 of idempotented algebra forces \(I\) to be directed under this order: For any \(e, f \in I\), there exists by definition a \(g \in I\) such that \(e = geg\) and \(f = gfg\), i.e., \(e \le g\) and \(f \le g\). Moreover, \(H\) is the union of \(eHe\), for \(e\) runs over \(I\): for any \(h \in H\), there exists an \(e \in I\) such that \(h = ehe \in eHe\).

Definition 3. An \(H\)-module \(M\) is non-degenerate if \(HM=M\).

Below is a very useful characterization of non-degenerate modules we shall use later:

Proposition 4. An \(H\)-module \(M\) is non-degenerate if and only if for any \(m \in M\), there exists an \(e \in I\) such that \(em = m\).

Proof. “if” part is obvious.

“only if” part: For any \(h \in H\), there exists an \(e \in I\) such that \(h = ehe\). Therefore, \(eh = e(ehe) = ehe = h\). Now for any \(m \in M\), there exist \(h \in H\) and \(n \in M\) such that \(hn = m\), since \(HM=M\). We can find an \(e \in I\) with \(eh = h\), then \(em = ehn = hn = m\). \(\square\)

In particular, we can see from the proof that \(H\) itself is non-degenerate as an \(H\)-module. We denote \(\mathcal{M}\) the category of all non-degenerate \(H\)-modules.

The next thing we would like to do is to put a topology on \(H\), then consider the completion \(\hat{H}\) of \(H\) under this topology. It turns out that \(\hat{H}\) is the endomorphism ring of the forgetful functor \(\omega: \mathcal{M} \to \mathcal{A}\), where \(\mathcal{A}\) is the category of abelian groups. We shall see the details of these in the following.

Definition 5 (Topology on \(H\)). The topology \(\tau\) on \(H\) is given by a fundamental system of neighborhoods \(\{H(1-e): e \in I\}\) at \(0 \in H\), where \[H(1-e) = \{h-he: h \in H\},\] is the set of left annihilators of \(e\).

Since \(H\) has a group structure, the topology \(\tau\) above is well-defined. More specifically, for any \(h \in H\), there associates to it a fundamental system of neighborhoods \(\{h+H(1-e): e \in I\}\). A Cauchy sequence in \(H\) is a sequence \(\{x_n\}_{n \in \N} \subset H\) that, for any \(e \in I\), there exits an \(N_e > 0\) such that for any \(n, m \ge N_e\), \(x_n - x_m \in H(1-e)\). Given two Cauchy sequences \(\{x_n\}\) and \(\{y_n\}\), we say that \(\{x_n\} \sim \{y_n\}\) if for any \(e \in I\), there exists an \(N_e > 0\) such that for any \(n \ge N_e\), \(x_n - y_n \in H(1-e)\). One can easily check that \(\sim\) is an equivalent relation for these Cauchy sequences. We define the completion of \(H\) under \(\tau\) to be \[\hat{H} = \{\text{Cauchy sequences in }H\}/\sim.\]

Theorem 6. \(\displaystyle \hat{H} = \varprojlim_{e \in I} He\), where \(\varprojlim\) is the inverse limit, taking over the inverse system \(x \mapsto xe: Hf \to He\) for \(e \le f\), of all the \(He\) with discrete topology.

Proof. Let \(\tilde{H} = \varprojlim_{e \in I} He\). \(\tilde{H}\) is a topological space, with initial topology induced from the projections \(\{\pi_e: \tilde{H} \to He, e \in I\}\). \(H\) is embedded into \(\tilde{H}\) via diagonal embedding, i.e., \(h \mapsto (he)_{e \in I}\). We view \(H\) is a subgroup in \(\tilde{H}\) under this embedding. The proof contains two steps: (i) the subspace topology of \(H\) in \(\tilde{H}\) is the same of the one define in Definition 5; (ii) \(H\) is dense in \(\tilde{H}\).

Since \(\tilde{H}\) is also a group, we just need to study the topology at \(0\). The initial topology of \(\tilde{H}\) at \(0\) is generated by \(\{\pi_e^{-1}(0)\}\). Since \(I\) is directed, \(\{\pi_e^{-1}(0)\}\) is actually a fundamental system of neighborhoods. But \(\pi_e^{-1}(0) \cap H = H(1-e)\), so (i) is done.

Let \((h_e)_{e \in I} \in \tilde{H}\). We need to show that \(((h_e)+\pi_f^{-1}(0)) \cap H\) is not empty, for any \(f \in I\). We take \(x = h_f \in H\), then \[\pi_f((h_e) - (xe)) = h_f - xf = h_f - h_f f=0.\] Therefore, \(x \in ((h_e)+\pi_f^{-1}(0)) \cap H\). This proves (ii). \(\square\)

Let \(M \in \mathcal{M}\) be a non-degenerate \(H\)-module. If we endow \(M\) with discrete topology, the action \(H \times M \to M\) is continuous: for any \(x \in M\), let \(U = \{(h, m): hm = x\}\), we need to show \(U\) is open in \(H \times M\). For any \((h, m) \in U\), since \(M\) is non-degenerate, there exists an \(e \in I\) such that \(em = m\). Consider \(V = (h + H(1-e)) \times \{m\}\). Then \(V\) is open in \(H \times M\), and a simple calculation shows \(V \subset U\). Thus, \(U\) is indeed open. By continuity, we can extend the action of \(H\) on \(M\) to that of \(\hat{H}\). By Theorem 6, we can write elements in \(\hat{H}\) as \((h_e)_{e \in I}\). For any \(m \in M\), there exists \(e \in I\) such that \(em = m\). Then \((h_e) \cdot m\) is defined to be \(h_e \cdot m\). It’s easy to check that the action of \(\hat{H}\) on \(M\) is well-defined.

Recall that \(\omega: \mathcal{M} \to \mathcal{A}\) is the forgetful functor, from the category of non-degenerate \(H\)-modules to the category of abelian groups. Here is another characterization of \(\hat{H}\).

Theorem 7. \(\displaystyle \hat{H} \cong \mathrm{End}(\omega)\).

Before the proof, let’s recall the notion of endomorphisms of a functor. Let \(F: \mathcal{C} \to \mathcal{D}\) be a functor. Then an endomorphism \(\varphi\) of \(F\) is a collection of morphisms \(\{\varphi_C \in \mathrm{Hom}_\mathcal{D}(FC, FC): C \in \mathcal{C}\}\), such that, for any morphism \(f: C_1 \to C_2\) in \(\mathcal{C}\), we have the following commutative diagram: \[\begin{equation*} \require{AMScd} \begin{CD} FC_1 @>{\varphi_{C_1}}>> FC_1 \\ @V{Ff}VV @VV{Ff}V \\ FC_2 @>>{\varphi_{C_2}}> FC_2 \end{CD} \end{equation*}\]

Proof. [Proof of Theorem 7] Since every non-degenerate \(H\)-module \(M\) is a quotient of \(H^{\oplus S}\) for some \(S\), it’s enough to show that \(\hat{H}\) is exactly those morphisms in \(\mathrm{End}_{\Z}(H)\) that commute with \(\mathrm{End}_H(H)\). Let’s set \[A = \{\varphi \in \mathrm{End}_{\Z}(H): \varphi(f(x)) = f(\varphi(x)), f \in \mathrm{End}_H(H), x \in H\}.\]

Let’s show \(\hat{H} \subseteq A\) first. Let \(\hat{h} = (h_e)_{e \in I} \in \hat{H}\). \(H\) is itself a non-degenerate \(H\)-module, so for any \(x \in H\), there exists \(e \in I\) such that \(ex = x\). Then \(\hat{h} \cdot x := h_e x\). It’s easy to check that this action is well-defined and respects abelian structures, so \(\hat{h} \in \mathrm{End}_{\Z}(H)\). Moreover, if \(\hat{h_1} \ne \hat{h_2}\), then they maps differently into \(\mathrm{End}_{\Z}(H)\): there must be an \(e \in I\) so that \(h_{1, e} \ne h_{2, e}\), so \(\hat{h_1} \cdot e \ne \hat{h_2} \cdot e\).

Take any \(f \in \mathrm{End}_H(H)\), and any \(x \in H\). Then there exists by Definition 1 an \(e \in I\) such that \(ex = x\) and \(ef(x) = f(x)\). Hence, \[f(\hat{h} \cdot x) = f(h_e x) = h_ef(x) = \hat{h} \cdot f(x).\] This proves \(\hat{h} \in A\), so \(\hat{H} \subseteq A\).

Next we would like to show \(A \subseteq \hat{H}\). Let \(\varphi \in A\). Define \(\hat{h} = (h_e)\) by setting \(h_e = \varphi(e)\). The claim is that \(\varphi(x) = \hat{h} \cdot x\) for \(x \in H\). For any element \(y \in H\), right multiplication by \(y\) is an element in \(\mathrm{End}_H(H)\), so \(\varphi\) commutes with right multiplication in \(H\). For any \(x \in H\), let \(e \in I\) be such that \(ex = x\). Then, \[\varphi(x) = \varphi(ex) = \varphi(e)x = h_e x = \hat{h} \cdot x.\] Thus, \(A \subseteq \hat{H}\). \(\square\)

Remark. In the original proof, Bernstein actually considered also the topology on \(H\) and \(\hat{H}\). The topology of \(H\) (see Definition 5) induces a topology of pointwise convergence on \(\mathrm{End}_{\Z}(H)\). Under these topologies, \(\hat{H}\) is embedded in \(\mathrm{End}_{\Z}(H)\) as a closure of \(H\), where \(H\) is viewed a subset in \(\mathrm{End}_{\Z}(H)\) via left multiplication. Lastly, we would like to discuss the center of \(\mathcal{M}\). Let \(\mathcal{C}\) be an abelian category, and \(\mathrm{Id}_{\mathcal{C}}: \mathcal{C} \to \mathcal{C}\) be the identity functor. The center of \(\mathcal{C}\) is defined to be the endomorphism ring \(\mathrm{End}(\mathrm{Id}_{\mathcal{C}})\).

Theorem 8. \(\displaystyle Z(\hat{H}) = \mathrm{End}(\mathrm{Id}_{\mathcal{M}}) = \varprojlim_{e \in I} Z(eHe)\), where \(Z(R)\) is the center of a ring \(R\).

Proof. By Theorem 7, we know already that \(\hat{H} = \mathrm{End}(\omega)\). For \(\hat{h} \in \hat{H}\), \(\hat{h} \in \mathrm{End}(\mathrm{Id}_{\mathcal{M}})\) if and only if \(\hat{h}\) also commutes with left multiplication by elements in \(H\). \(\hat{h}\) commutes with left multiplication by elements in \(H\), is exactly the same as \(\hat{h} \in Z(\hat{H})\). Therefore, we have \(Z(\hat{H}) = \mathrm{End}(\mathrm{Id}_{\mathcal{M}})\).

Let \(C = \varprojlim_{e \in I} Z(eHe)\). Let’s see that \(C\) is actually well-defined, i.e., \(\{Z(eHe): e \in I\}\) forms an inverse system. For \(f \ge e\), let \(fzf \in Z(fHf)\), we need to show \(eze \in Z(eHe)\). Let \(exe \in eHe\), we note that \(ef = fe = e\) and \(ez = ze = eze\), then \[exe \cdot eze = efxfze = ezfxfe = eze \cdot exe.\] Therefore, \(eze \in Z(eHe)\), so \(C\) is well-defined.

Let \(\hat{c} = (c_e) \in C\), then \(\hat{c} \in \hat{H}\). Let \(x \in H\), for any \(y \in H\), then there exists an \(e \in I\) such that \(exe = x\) and \(eye = y\). Thus, \(exye = xy\) and \[x\hat{c}(y) = xc_ey = c_e xy = \hat{c}(xy).\] This proves that \(\hat{c}\) commutes with left multiplication in \(H\), so \(\hat{c} \in Z(\hat{H})\).

Now let \(\hat{h} = (h_e) \in Z(\hat{H})\). Then \(\hat{h}\) commutes with left multiplication in \(H\). In particular, \(e\hat{h}(e) = \hat{h}(e)\), i.e., \(eh_ee = h_ee = h_e \in eHe\) (last equality uses the fact that \(h_e \in He\)). Let \(exe \in eHe\), we use again the fact that \(\hat{h}\) commutes with left multiplication: \[exeh_e = exe \hat{h}(e) = \hat{exe} = h_e exe.\] This shows that \(h_e \in Z(eHe)\), so \(\hat{h} \in C\). \(\square\)

An example of idempotented algebras

Let \(G\) be a locally compact, totally disconnected group. Let \(H(G)\) be the set of locally constant functions from \(G\) to \(\C\) with compact open support. The addition in \(H(G)\) is just pointwise addition: for \(f, g\in H(G)\), \((f+g)(s) = f(s) + g(s)\), for \(s \in G\). The multiplication in \(H(G)\) is given by convolution: for \(f, g \in H(G)\), \[f \ast g(t) = \int_G f(s)g(s^{-1}t)d\mu(s),\] where \(\mu\) is a left Haar measure on \(G\). The above addition and multiplication give a ring structure on \(H(G)\). There are natural idempotents in \(H(G)\). Let \(K\) be an open compact subgroup in \(G\), and let \(e_k\) be the normalized characteristic function on \(K\), i.e., \[\begin{equation*} e_K(s) = \left\{ \begin{array}{ll} \frac{1}{\mu(K)}, & s \in K; \\ 0, & s \not\in K. \end{array} \right. \end{equation*}\] Here, \(\mu(K) = \int_Kd\mu(s)\) is the measure of \(K\). In a simpler form, we can write \(e_K = \frac{1}{\mu(K)} \mathbf{1}_K\), where \(\mathbf{1}_K\) is the indicator function on \(K\). A direct computation shows \(e_K \ast e_K = e_K\), so \(e_K\) is an idempotent. Let \(I\) be a set of idempotents in \(H(G)\). We will check that \((H(G), I)\) is an idempotented algebra.

Let \(K \subset H\) be two open compact supgroups in \(G\), then \[\begin{equation*} \begin{split} e_K \ast e_H(t) &= \int_G e_K(s)e_H(s^{-1}t) d\mu(s) \\ &= \frac{1}{\mu(K)}\int_K e_H(s^{-1}t)d\mu(s) \\ &= \frac{1}{\mu(K)\mu(H)} \mathbf{1}_H(t) \int_K d\mu(s) \\ &= \frac{1}{\mu(H)} \mathbf{1}_H(t) = e_H. \end{split} \end{equation*}\] Similarly, \(e_H \ast e_K = e_H\). Therefore, \(e_K \ast e_H \ast e_K = e_H\), i.e., \(e_K \ge e_H\). Now consider \(\varphi = \mathbf{1}_C\) for some compact open set \(C\) in \(G\). We can then cover \(C\) with open sets \[\{gK \subset C: g \in G, K \subset G \text{ is a compact open subgroup}\}.\] This can be done because \(\{K \subset G \text{ is a compact open subgroup}\}\) is a basis for the topology at the identity of \(G\) and \(C\) is open. Since \(C\) is compact, we can then select finite many \(g_iK_i\) that cover \(C\). Since \(K_i\) are also open, we can further make \(g_iK_i\) disjoint, so we can rewrite \(\varphi\) as \[\varphi = \sum_i \mathbf{1}_{g_iK_i}.\] For simplicity, let’s consider for now \(\psi = \mathbf{1}_{gK}\), for some \(g \in G\) and \(K \subset G\) some compact open subgroup. Let \(H = K \cap gKg^{-1}\). It’s easy to see that \(\psi\) is right invariant under multiplication by elements in \(H\). In fact, \(\psi(xk) = \psi(x)\) for any \(x \in G\) and \(k \in K\). Moreover, \(\psi\) is left invariant under multiplication by elements in \(H\), i.e., \(\psi(hx) = \psi(x)\), for any \(x \in G\) and \(h \in H\): write \(h = gkg^{-1}\) for some \(k \in K\), then \[\begin{equation*} \begin{split} \psi(hx) &= \mathbf{1}_{gK} = (gkg^{-1}x) = \mathbf{1}_{K}(kg^{-1}x) \\ &= \mathbf{1}_{K}(g^{-1}x) = \mathbf{1}_{gK}(x) = \psi(x). \end{split} \end{equation*}\]

Lemma 9. Under the above notations, we have \(e_H \ast \psi = \psi \ast e_H = \psi\). In particular, \(e_H \ast \psi \ast e_H = \psi\).

Pf. We need to use the fact that \(\psi\) is left and right invariant under multiplication by elements of \(H\). Here, only calculation for \(e_H \ast \psi = \psi\) is given. \(\psi \ast e_H = \psi\) can be shown using a similar calculation.

\[\begin{equation*} \begin{split} e_H \ast \psi(t) &= \int_G e_H(s)\psi(s^{-1}t) d\mu(s) \\ &= \frac{1}{\mu(H)} \int_H \psi(s^{-1}t) d\mu(s) \\ &= \frac{\psi(t)}{\mu(H)}\int_H d\mu(s) = \psi(t). \end{split} \end{equation*}\] Note that we need the fact that \(\psi\) is left invariant under multiplication by elements of \(H\) in the above calculation. \(\vartriangleleft\)

Come back to our \(\varphi = \sum_i \mathbf{1}_{g_iK_i}\). Then for each \(\mathbf{1}_{g_iK_i}\), we can find by Lemma 9 a compact open subgroup \(H_i\) such that \[\begin{equation} e_{H_i} \ast \mathbf{1}_{g_iK_i} \ast e_{H_i} = \mathbf{1}_{g_iK_i}. \label{20161004-1} \end{equation}\]

Let \(H = \cap_i H_i\), then \(H\) is also a compact open subgroup. \(H_i \subset H\), so \(e_H \ge e_{H_i}\), so \[\begin{equation} e_{H} \ast e_{H_i} \ast e_{H} = e_{H_i}, \label{20161004-2} \end{equation}\]

for each \(i\). Combining \(\eqref{20161004-1}\) and \(\eqref{20161004-2}\), we have \[e_{H} \ast \mathbf{1}_{g_iK_i} \ast e_{H} = \mathbf{1}_{g_iK_i},\] for all \(i\). This implies that \(e_H \ast \varphi \ast e_H = \varphi\). To summarize, we have the following theorem:

Theorem 10. Let \(C\) be a compact open subset in \(G\) and let \(\varphi = \mathbf{1}_C\), then there exist a compact open subgroup \(K\) such that \[e_K \ast \varphi \ast e_K = \varphi.\]

Basically, we have already shown that \((H(G), I)\) is an idempotented algebra by Theorem 10. Given a finite set of elements \(\{\varphi_i\}\) in \(H(G)\), we can assume \(\varphi_i = \mathbf{1}_{C_i}\) for some compact open subset. The assumption is harmless, since elements in \(H(G)\) are of form \[\sum_i \alpha_i \mathbf{1}_{C_i},\] where \(\alpha_i\) are constants in \(\C\) and \(C_i\) are compact open subsets in \(G\). Then by Theorem 10, there exists a compact open \(K_i\) so that \[e_{K_i} \ast \varphi_i \ast e_{K_i} = \varphi_i,\] for each \(i\). Now take \(K = \cap_i K_i\), then \(e_K \ast \varphi_i \ast e_K = \varphi_i\), for all \(i\). Therefore, we conclude that \((H(G), I)\) is an idempotent algebra.

Remark. Let \(I_C = \{e_K: K \text{ is a compact open subgroup in }G\} \subset I\). Then the above argument actually shows that we can find for each finite set \(\{\varphi_i\} \subset H(G)\) an idempotent \(e \in I_C\) such that \(e \ast \varphi_i \ast e = \varphi_i\). Therefore, \(I_C\) is cofinal in \(I\). Instead of taking inverse limit over \(I\), we can take it over \(I_C\): \[\widehat{H(G)} = \varprojlim_{e \in I} H(G) \ast e = \varprojlim_{e \in I_C} H(G) \ast e.\]