Wendel Theorem for center of group algebra

Question

Let $G$ be a locally compact SIN-group. Then $ZL^{1}(G)$ has a bounded approximate identity. I want to prove that the multiplier algebra of $ZL^{1}(G)$ is equal to $ZM(G)$ (center of measure algebra). If $(e_{\alpha})$ is a central approximate identity for $ZL^{1}(G)$ and T is a multiplier of $ZL^{1}(G)$ then $T\in B(ZL^{1}(G))$. I want to know whether $T(e_{\alpha})$ must be an element of $ZM(G)$.

Answer 1

This is a proof for $G$ a compact group.

Define $ZC(G)=\{ f\in C_b(G): f(xyx^{-1})=f(y) \ \forall x,y \in G\}$. To prove the fact that $ZM(G)$ is the multiplier algebra of $ZL^1(G)$, first we need the following lemma.

Lemma 1. Let $G$ be a compact group. Then $ZM(G)$ is the dual of $ZC(G)$.

Proof. Let us define a norm decreasing map $P: C(G) \rightarrow ZC(G)$ by $$ P(f)(x)=\int_{G} f(yxy^{-1}) dy. $$ Hence, $P^*:ZC(G)^* \rightarrow M(G)$, the adjoint of $P$, is a norm decreasing map.

Let $\iota: ZC(G) \rightarrow C(G)$ be the canonical embedding which takes every function to itself. Then $\iota^*: M(G) \rightarrow ZC(G)^*$. Let us define $j:=\iota^*|_{ZM(G)}$ which is still a norm decreasing map. We claim that $P^*\circ j$ is the identity map. To prove this, first note that for each $f\in C(G)$, $\iota\circ P(f)=P(f)$. Therefore, for each $\mu \in ZM(G)$, we get \begin{eqnarray*} \langle P^*\circ j(\mu), f\rangle &=& \langle j(\mu), P(f)\rangle \\ &=& \langle \mu, \iota\circ P(f)\rangle\\ &=& \langle \mu, P(f)\rangle\\ &=& \int_G P(f)(x) d\mu(x)\\ &=& \int_{G\times G} f(yxy^{-1}) dy d\mu(x)\\ &=& \int_{G} \int_{G} f(y) d(\delta_x*\mu * \delta_{x^{-1}})(y) dx\\ &=& \int_G f(y) d\mu(y). \end{eqnarray*} $\Box$

Now we can prove the main result:

Proof of the result. First note that for each $\mu \in ZM(G)$, $\mu \in M(ZL^1(G))$ because on one hand $\mu *f \in ZM(G)$ for every $f\in ZL^1(G)$ and on the other hand, $L^1(G)$ is an ideal in $M(G)$.

Conversely, let $(e_\alpha)$ be a central bounded approximate identity of $L^1(G)$. For each $T\in M(ZL^1(G))$, $(T(e_\alpha))$ is a bounded net in $ZM(G)$. So weakly$^*$ it approaches some $\mu \in ZM(G)$. Hence, for each $f\in ZL^1(G)$, $$ T(f)= w^*-\lim_\alpha T(e_\alpha * f)= w^*-\lim_\alpha T(e_\alpha)* f = \mu*f. $$

$\Box$

Note. For a general locally compact group $G$, $ZL^1(G)$ always has a bounded approximate identity (by J. LIUKKONEN and A. MOSAK, HARMONIC ANALYSIS AND CENTERS OF GROUP ALGEBRAS, Trans. Amer. Math. Soc, 1974).