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\begin{document}
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\section*{GNS constructions, and functions of positive type}

Based on Appendix C of "Kazhdan's Property T" by Bekka, Harpe, and Valeppe.

\subsection*{Cyclic representations}
\paragraph{Definition:}
\mbox{}\\    A unitary representation $(\pi,\H)$ is \underline{cyclic} if there exists
    $\xi \in \H$ s.t.\ the span of the orbit of $\xi$ is dense. Call such $\xi$
    a \underline{cyclic vector}.

\paragraph{Remark:}
\mbox{}\\    If $(\pi,\H)$ is a representation and $\xi\in\H$, then the closure of the
    span of the orbit of $\xi$, $\overline{<G\xi>}$, is an invariant closed
    subspace, so yields a cyclic subrepresentation.
\\    So irreducible $\Rightarrow $ cyclic (any $\xi \neq  0$ is cyclic),
\\    and any unitary representation $(\pi,\H)$ can be decomposed as an
    orthogonal direct sum of cyclic subrepresentations.
\\	(Proof: take a maximal such direct sum; if not $\H$, consider closure
	of orbit of an element of the orthogonal complement for a contradiction.)

\subsection*{Functions of positive type}

\paragraph{Definition:}
\begin{enumerate}[(i)]\item For $X$ a topological space, a \underline{kernel of positive type} is a
	continuous function $\Phi : X^2 \rightarrow  \C$ such that for any finite tuple
	$(x_1,...,x_n)$ from $X$ and any $(a_1,...,a_n) \in \C^n$,
\\	\[ \sum_{i,j} a_i \Phi(x_i,x_j) \abar_j \geq 0 .\]
\item For $G$ a topological group, a \underline{function of positive type} is a
	continuous function $\phi : G \rightarrow  \C$ such that $(g,h) \mapsto 
	\phi(h^{-1}g)$ is a kernel of positive type.
\end{enumerate}

\paragraph{Remark:}
\mbox{}\\    $\Phi$ is of positive type iff for any finite subset $\{x_1,...,x_n\} \subseteq 
    X$, the matrix $(a_{ij})_{ij}$ with $a_{ij} = \Phi(x_i,x_j)$ is positive
    semidefinite.

\paragraph{Example:}
\mbox{}\\    $G := \left<{\R; +}\right>$, $\phi(z) := e^{iz}$.

\paragraph{Lemma:}
\begin{enumerate}[(i)]\item If $\Phi : X^2 \rightarrow  \C$ is of +ve type,
\begin{enumerate}[(a)]\item $\overline{\Phi(x,y)} = \Phi(y,x)$ (so $\Phi(x,x) \in \R$)
\item $|\Phi(x,y)|^2 \leq \Phi(x,x)\Phi(y,y)$
\end{enumerate}
\item If $\phi : G \rightarrow  \C$ is of +ve type,
\begin{enumerate}[(a)]\item $\overline{\phi(g)} = \phi(g^{-1})$ (so $\phi(e) \in \R$)
\item $|\phi(g)| \leq \phi(e)$
\end{enumerate}
\end{enumerate}
\paragraph{Proof:}
\begin{enumerate}[(i)]\item 
	The matrix
    \[ \begin{bmatrix}
		\Phi(x,x) & \Phi(x,y)  \\
		\Phi(y,x) & \Phi(y,y)
    \end{bmatrix} \]
    is positive semidefinite, so is Hermitian with non-negative determinant.

\item Follows, by considering $\Phi(g,e)$.
\end{enumerate}

\paragraph{Remark:}
\mbox{}\\    Let $(\pi,\H)$ be a unitary representation of $G$, let $\xi \in \H$.
\\    Define $\phi_{\pi,\xi} : g \mapsto  \left<{g\xi,\xi}\right>$.
\\    Then $\phi_{\pi,\xi}$ is of +ve type.
\\    Indeed, $(g,h) \mapsto  \left<{h^{-1}g\xi,\xi}\right> = \left<{g\xi, h\xi}\right>$ is of +ve type,
\\    since $\sum_{ij} a_i \left<{g_i\xi, g_j\xi}\right> \abar_j
    = \sum_{ij} \left<{a_i g_i\xi, a_j g_j\xi}\right>
    = \left<{\sum_i a_i g_i\xi, \sum_i a_i g_i\xi}\right> \geq  0$.

\paragraph{Theorem ["GNS construction"]:}
\mbox{}\\    $G$ topological group, $\phi : G \rightarrow  \C$ of +ve type.
\\    Then $\phi = \phi_{\pi_\phi,\xi_\phi}$ for some
    unitary representation $(\pi_\phi,\H_\phi)$ and cyclic vector
    $\xi_\phi\in\H_\phi$.

    Moreover, the pair $(\pi_\phi,\xi_\phi)$ associated to $\phi$ is unique up
    to isomorphism, in the sense that if $\phi = \phi_{\pi',\xi'}$ where
    $\xi'$ is a cyclic vector in another unitary representation $(\pi',\H')$,
\\    then there is an intertwining isomorphism $\H_\phi \xrightarrow{\cong} \H'; \xi_\phi \mapsto 
    \xi'$.

\paragraph{Remark:}
\mbox{}\\    So $(\pi,\xi) \mapsto \phi_{\pi,\xi}$ 
    induces a bijective correspondence between +ve functions and cyclic
    vectors in unitary representations up to isomorphism.

    The zero function corresponds to the trivial representation.

\subsection*{Cone of functions of +ve type}

$\Pos(G) := \{ \phi : G \rightarrow  \C \;\text{of +ve type} \}$.
\\$\Pos(G)$ is a cone (i.e.\ closed under taking positive linear combinations).

$\Posn(G) := \{ \phi \in \Pos(G) \;|\; \phi(e)=1 \}$.
\\So $\Posn(G)$ is a convex subset of $\Pos(G)$.

Note $\phi_{\pi,\xi} \in \Posn(G)$ iff $\|\xi\| = 1$.

\paragraph{Definition:}
\mbox{}\\  A \underline{pure} function of +ve type is an extreme point of $\Posn(G)$;
\\  i.e.\ $\phi\in\Posn(G)$ such that whenever $\phi = t\psi_1 + (1-t)\psi_2$
  with $t\in [0,1]$ and $\psi_i \in \Posn(G)$, actually $t \in \{0,1\}$.

\paragraph{Theorem:}
\mbox{}\\    $\phi \in \Posn(G)$ is pure iff the GNS representation
    $\pi_\phi$ is irreducible.
\paragraph{Proof:}
\mbox{}\\    We have $\phi = \phi_{\pi_\phi,\xi_\phi}$, with $\xi_\phi$ cyclic.
\\    Suppose $\pi_\phi$ is reducible.
\\    Say $\H_\phi = \K \oplus \K^\orth$ is a non-trivial invariant orthogonal decomposition;
\\    decompose $\xi = \xi_\K + \xi_{\K^\orth}$.
\\    Set $s := \| \xi_{\K} \|, t := \| \xi_{\K^\orth} \|$.
\\    So $s^2+t^2 = \| \xi_\phi \|^2 = \phi(e) = 1$.

    Now one may calculate that $\phi = s^2\phi_{\pi_\phi,\normise{\xi_\K}} + t^2\phi_{\pi_\phi,\normise{\xi_{\K^\orth}}}$,
\\    but $s \neq  0 \neq  t$ since $\xi$ is cyclic.
\\    So $\phi$ is not pure.

    Conversely, suppose $\phi$ is not pure,
\\    say $\phi = s^2 \phi_1 + t^2 \phi_2$, $s^2+t^2=1$, $s \neq  0 \neq  t$, $\phi_i\in\Posn(G)$,
\\    and suppose $\pi_\phi$ is irreducible.

    By GNS, say $\phi_1 = \phi_{\pi_1,\xi_1}$ and $\phi_2 = \phi_{\pi_2,\xi_2}$;
\\    consider the direct sum representation $\pi_1 \oplus \pi_2$ of $G$ on $\H_1 \oplus \H_2$,
\\    and let $\xi' := s\xi_1 + t\xi_2$.
\\    Then $\phi_{\pi_1 \oplus \pi_2,\xi'}(g)
	= \left<{s g\xi_1 + t g\xi_2, s \xi_1 + t \xi_2}\right>
	= s^2 \phi_1(g) + t^2 \phi_2(g)
	= \phi(g)$.

    Consider the subrepresentation $\pi_0$ on $\H' := \overline{<G\xi'>}$.
\\    By the uniqueness of GNS, this is equivalent to $\pi_\phi$.
\\    Since $\xi_1$ is cyclic,
\\    the projection $T : \H' \rightarrow  \H_1; \xi' \mapsto  \xi_1$ has dense image,
\\    so by irreducibility of $\pi_\phi$, $T$ is an isomorphism
    \\(lest $(\ker T)^\orth$ be a proper subrepresentation).

    So $\phi = \phi_{\pi_0,\xi'} = \phi_{\pi_1,\xi_1} = \phi_1$, contradicting
    impurity.
\\    \qed

\paragraph{Fact:}
\mbox{}\\    If $G$ is locally compact, the convex hull of $\extPG$ is dense in
    $\Posn(G)$ for the topology of uniform convergence on compact subsets of
    $G$.

\subsection*{Proof of GNS}

Let $\phi$ be a non-zero function of +ve type on $G$.
\\Consider the translates of $\phi$, $\phi_h(g) := \phi(h^{-1}g)$.
\\Let $V$ be the subspace of $C(G)$ they span.
\\Let $W$ be the vector space freely generated by $G$.
\\$\left<{g, h}\right> := \phi(h^{-1}g)$ extends to a conjugate-symmetric sesquilinear form on $W$,
\\$\left<{\sum_i a_i g_i, \sum_i b_i h_i}\right> = \sum_{ij} a_i\phi(h_i^{-1}g_i)\bbar_j$.
\\Now $\left<{\sum_i a_i h_i, \sum_i a_i h_i}\right> = \sum_{ij} a_i \phi(h_i^{-1}h_j) \abar_j \geq  0$
by definition of a positive function, so $\left<{,}\right>$ is a semi-inner product on $W$.

The kernel is $\{ \sum_i b_i h_i \;|\; \sum_i \phi_{h_i} = 0 \}$,
\\so it induces a well-defined inner product $\left<{,}\right>$ on $V$.

Let $\H$ be the Hilbert space completion of $V$ with respect to $\left<{,}\right>$;
\\so $\H$ is the space of functions on $G$ which are pointwise limits of Cauchy
sequences in $V$ with respect to the norm induced by ${< , >}$.

Let $\pi$ be the representation of $G$ acting by translation on $\H$,
\\$\pi(h)f(g) := f(h^{-1}g)$,
\\and let $\xi := \phi \in \H$.
\\This is unitary, since it preserves $<,>$ on the $\phi_g$,
\\and is strongly continuous since $\phi$ is continuous
\\(and so then $g \mapsto  gf$ is continuous for $f=\phi$, hence for any
$f \in V$, and hence for $f\in\H$ since a uniform limit of continuous
functions is continuous),
\\and $\xi$ is cyclic, and
$\phi_{\pi,\xi}(h) = \left<{\phi_h, \phi_e}\right> = \phi(h)$.

It remains to show the uniqueness.

Suppose $\phi_{\pi',\xi'} = \phi_{\pi,\xi}$,
\\with $\xi'$ cyclic for $\pi' : G \rightarrow  \H'$.

(*) $\| \sum_i a_ig_i\xi \|^2
    = \sum_{ij} a_i \left<{g_i\xi, g_j\xi}\right> \abar_j
    = \sum_{ij} a_i \left<{g_i\xi', g_j\xi'}\right> \abar_j 
    = \| \sum_i a_ig_i\xi' \|^2$

Define an isomorphism $T : \H \xrightarrow{\cong} \H'$ by $T(g\xi) := g\xi'$ and extending
linearly and continuously; this is well-defined by (*).
\\$T$ intertwines $\pi$ and $\pi'$, and $T(\xi) = \xi'$.
\\\qed

\paragraph{Example:}
\mbox{}\\    $G := \left<{\R; +}\right>$, $\phi(z) := e^{iz}$.
\\    $\phi_a(z) = e^{i(z-a)} = e^{-ia}\phi(z)$,
\\    so $\H$ is 1-dimensional,
\\    and the representation is $a \mapsto  (e^{-ia}\cdot)$.
\end{document}
