  In this paper we discuss the $L^p$-$L^q$ boundedness of both spectral and
Fourier multipliers on general locally compact separable unimodular groups $G$
for the range $1<p\leq q<\infty$. We prove a Lizorkin type multiplier theorem
for $1<p\leq q<\infty$, and then refine it as a H\"ormander type multiplier
theorem for $1<p\leq 2\leq q<\infty$. In the process, we establish versions of
Paley and Hausdorff-Young-Paley inequalities on general locally compact
separable unimodular groups. As a consequence of the H\"ormander type
multiplier theorem we derive a spectral multiplier theorem on general locally
compact separable unimodular groups. We then apply it to obtain embedding
theorems as well as time-asymptotics for the $L^p$-$L^q$ norms of the heat
kernels for general positive unbounded invariant operators on $G$. We
illustrate the obtained results for sub-Laplacians on compact Lie groups and on
the Heisenberg group. We show that our results imply the known results for
$L^p$-$L^q$ multipliers such as H\"ormander's Fourier multiplier theorem on
$\mathbb{R}^{n}$ or known results for Fourier multipliers on compact Lie
groups. The new approach developed in this paper relies on the analysis in the
group von Neumann algebra for the derivation of the desired multiplier
theorems.
