Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$ be a continuous representation of $A$ on a separable Hilbert space $H$ with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of $\pi$ with respect to an orthonormal basis and suppose that for each $1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m} \|\pi_{ij}\|_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$ left $\pi$-invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all $a\in A$. In this paper we prove a link between the existence of left $\pi$-invariant elements and the vanishing of certain Hochschild cohomology groups of $A$. Our results extend an earlier result by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pym and the second named author in the special case that $\pi \colon A \longrightarrow \mathbf C$ is a non-zero character on $A$.