Lemma 8

a) The mapping $\varrho : (g, h)\mapsto \varrho (g, h)$ is a unitary representation of $G\times H$ in E ;

b) For every orthonormal basis $(e_i)_{i\in I}$ of $E_1$, the mapping from the Hilbert sum $\bigoplus_{i\in I}E_2$ into E defined by $(y_i)_{i\in I}\mapsto \sum_{i\in I}e_i\otimes y_i$ is an isometric H-isomorphism of the Hilbert sum $\bigoplus_{i\in I}\varrho_2$ onto Res$^{G\times H}_{\{e\}\times H}(\varrho )$;

c) For every orthonormal basis $(f_j)_{j\in J}$ of $E_2$, the mapping from the Hilbert sum $\bigoplus_{j\in J}E_1$ into E defined by $(x_j)_{j\in J}\mapsto \sum_{j\in J}x_j\otimes f_j$ is an isometric G-isomorphism of the Hilbert sum $\bigoplus_{j\in J}\varrho_1$ onto Res$^{G\times H}_{G\times \{e\}}(\varrho )$.

The mapping $(g, h)\mapsto \varrho (g, h)$ is a homomorphism of G into $\mathbf{G}\mathbf{L}(E) ($cf. EVT, V, p. 28, n$^o2)$. Let $(e_i)_{i\in I}$ be an orthonormal basis of $E_1$. By EVT, V, p. 29, prop. 3 and cor. 2, the mapping

is an isometry of the Hilbert direct sum $F_2=\bigoplus_{i\in I}E_2$ onto E. By means of this isometry, the representation $\varrho_H:h\mapsto \varrho (e, h)$ of H in E is identified with the direct-sum representation of the representations $(\varrho_2)_{i\in I}$ in $F_2$. In particular, it is a unitary representation of H in E (lemma 7). Analogously, the homomorphism $\varrho_G:g\mapsto \varrho (g, e)$ is a unitary representation of G in E.

Let $(g, h)\in G\times H$. We have $\varrho (g, h) =\varrho_G(g)\circ \varrho_H(h)$; therefore $\varrho (g, h)$ is unitary; moreover, $\varrho_G(g)$ and $\varrho_H(h)$ commute, hence lemma 1 of V, p. 377 implies that $\varrho$ is a unitary representation of $G\times H$.

Finally, the assertions concerning the restriction of $\varrho$ to the subgroups $\{e\} \times H$ and $G\times \{e\}$ were obtained in the course of the preceding argument.

The representation $(g, h)\mapsto \varrho_1(g)\otimes \varrho_2(h)$ of $G\times H$ is called the external tensor product of the unitary representations $\varrho_1$ and $\varrho_2$, and is denoted by $\varrho_1\boxtimes \varrho_2$.

Let $n\in \mathbf{N}$ and let $(G_i)_{1\leqslant i\leqslant n}$ be a finite family of topological groups. Let $\varrho_i$ be a unitary representation of $G_i$ in a Hilbert space $E_i$ for $1\leqslant i\leqslant n$. One defines analogously a representation

of $G_1\times \cdots \times G_n$ in the Hilbert space $E = E_1\widehat{\otimes}_2\cdots \widehat{\otimes}_2E_n$ (EVT, V, p. 27).

Suppose that $G_i= G$ for $1\leqslant i\leqslant n$. Let $\Delta_n: G\rightarrow G^n$ be the homomorphism defined by $g\mapsto (g, . . . , g)$ for every $g\in G$. One denotes by $\varrho_1\otimes \cdots \otimes \varrho_n$ the unitary representation $(\varrho_1\boxtimes \cdots \boxtimes \varrho_n)\circ \Delta_n$ of G. One says that it is the tensor product of the unitary representations $\varrho_i$.

For every permutation $\sigma$ of $\{1, . . . , n\}$, the canonical isomorphism

(EVT, V, p. 28) is an isometric isomorphism

of representations of G.

If $\varrho_i=\varrho$ for $1\leqslant i\leqslant n$, where $\varrho$ is a unitary representation of G, one also denotes by $\varrho^{\otimes n}$ the tensor-product representation of the $\varrho_i$, and one says that it is the $n^e$ tensor power of $\varrho$.