Proposition 5

Tag 0394, permanent. Cite it as [Bourbaki, Tag 0394].

Théories spectrales, chapter V, § 1. Représentations unitaires

Let G be a topological group. The sets Θ(G) and Υ(G) are unital involutive subalgebras of $\mathscr{C}_b(G)$ .

The constant function 1 is a matrix coefficient of the trivial representation of G on $\mathbf{C}$ . Let $\varrho$ be a unitary representation of G and let $x$ and $y$ be vectors of the space of $\varrho$ . For every $\lambda \in \mathbf{C}$ and every $g\in G$ , we have $\lambda \langle x|\varrho (g)y\rangle =\langle x|\varrho (g)(\lambda y)\rangle$ . Moreover, we have

\langle x|\varrho (g)y\rangle =\langle \varrho (g^{-1})x|y\rangle =\overline{\langle y|\varrho(g^{-1})x\rangle}

for every $g\in G$ . Consequently, the sets Θ(G) and Υ(G) are stable under multiplication by scalars and under conjugation.

Let $\varrho_1$ and $\varrho_2$ be unitary representations of G; let $(x_1, y_1)$ be vectors of the space of $\varrho_1$ and $(x_2, y_2)$ vectors of the space of $\varrho_2$ . For every $g\in G$ , we then have

\langle x_1|\varrho_1(g)y_1\rangle +\langle x_2|\varrho_2(g)y_2\rangle =\langle (x_1, x_2)|(\varrho_1\oplus \varrho_2)(g)(y_1, y_2)\rangle

\langle x_1|\varrho_1(g)y_1\rangle \langle x_2|\varrho_2(g)y_2\rangle =\langle x_1\otimes x_2|(\varrho_1\otimes \varrho_2)(g)(y_1\otimes y_2)\rangle

which proves that Θ(G) and Υ(G) are stable under addition and multiplication. The proposition follows.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.