We prove that Krivine’s Function Calculus is compatible with integration. Let $(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ be a finite measure space, $X$ a Banach lattice, $\mathbf{x}\in X^{n}$, and $f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ a function such that $f(\cdot ,\unicode[STIX]{x1D714})$ is continuous and positively homogeneous for every $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$, and $f(\mathbf{s},\cdot )$ is integrable for every $\mathbf{s}\in \mathbb{R}^{n}$. Put $F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$ and define $F(\mathbf{x})$ and $f(\mathbf{x},\unicode[STIX]{x1D714})$ via Krivine’s Function Calculus. We prove that under certain natural assumptions $F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$, where the right hand side is a Bochner integral.

Let A be a commutative Banach algebra with identity of norm 1, X a Banach A-module and G a locally compact abeian group with Haar measure. Then the multipliers from an A -valued function algebra into an X-valued function space is studied. We characterize the multiplier spaces as the following isometrically isomorphic relations under some appropriate conditions: