$G$-invariant operators on $L^2(G)$

Question

Let $G$ be a Lie group equipped with a left-invariant Haar measure. Then elements of $G$ act as bounded operators on $L^2(G)$, with action given by translation:

$$(g\cdot f)(h):=f(g^{-1}h),$$

for any $f\in L^2(G)$, $g,h\in G$.

Let us say that an operator $T\in\mathcal{B}(L^2(G))$ is $G$-invariant if

$$g\circ T\circ g^{-1} = T$$

for all $g\in G$. One can check that, for example, convolution by a function in $C_c(G)$ is $G$-invariant.

Question: does every $G$-invariant operator in $\mathcal{B}(L^2(G))$ arise as the limit of a sequence of elements of $C_c(G)$ acting by convolution?

In other words, are the $G$-invariant operators in $\mathcal{B}(L^2(G))$ precisely the elements of the reduced group $C^*$-algebra $C^*_\lambda(G)$?