Induced Isometric Representations

Let $\sigma$ be an isometric representation of $\mathbb{N}^d$ on a Hilbert space $\mathcal{H}$. We induce $\sigma$ to an isometric representation $V$ of $\mathbb{R}_{+}^{d}$ on another Hilbert space $\mathcal{K}$. We show that the map $\sigma \to V$, restricted to strongly pure isometric representations, preserves index and irreducibility. As an application, we show that, for $k \in \{0, 1,2,\cdots\} \cup \{\infty\}$, there is a continuum of prime multiparameter CCR flows (i.e, not a tensor product of two non-trivial $E_0$-semigroups) with index $k$.