!newop ext
\renewcommand{\H}{\mathcal{H}}
\newcommand{\K}{\mathcal{K}}
\newcommand{\Pos}{\mathcal{P}}
\newcommand{\Posn}{\mathcal{P}_1}
\newcommand{\extPG}{\ext(\mathcal{P}_1(G))}
\newcommand{\normise}[1]{\widehat{#1}}
\newcommand{\abar}{\overline{a}}
\newcommand{\bbar}{\overline{b}}
\newcommand{\cbar}{\overline{c}}
\newcommand{\orth}{\perp}

## GNS constructions, and functions of positive type

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

# Cyclic representations
Definition:
    A unitary representation $(\pi,\H)$ is _cyclic_ if there exists
    $\xi (- \H$ s.t.\ the span of the orbit of $\xi$ is dense. Call such $\xi$
    a _cyclic vector_.

Remark:
    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 $==>$ cyclic (any $\xi != 0$ 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.)

# Functions of positive type

Definition:
    **(i) For $X$ a topological space, a _kernel of positive type_ is a
	continuous function $\Phi : X^2 --> \C$ such that for any finite tuple
	$(x_1,...,x_n)$ from $X$ and any $(a_1,...,a_n) \in \C^n$,
	\[ \sum_{i,j} a_i \Phi(x_i,x_j) \abar_j \geq 0 .\]
    **(ii) For $G$ a topological group, a _function of positive type_ is a
	continuous function $\phi : G --> \C$ such that $(g,h) |->
	\phi(h^{-1}g)$ is a kernel of positive type.
    *ee*

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

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

Lemma:
    **(i) If $\Phi : X^2 --> \C$ is of +ve type,
	**(a) $\overline{\Phi(x,y)} = \Phi(y,x)$ (so $\Phi(x,x) \in \R$)
	**(b) $|\Phi(x,y)|^2 \leq \Phi(x,x)\Phi(y,y)$
	*ee*
    **(ii) If $\phi : G --> \C$ is of +ve type,
	**(a) $\overline{\phi(g)} = \phi(g^{-1})$ (so $\phi(e) \in \R$)
	**(b) $|\phi(g)| \leq \phi(e)$
	*ee*
    *ee*
Proof:
    **(i) 
	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.

    **(ii) Follows, by considering $\Phi(g,e)$.
    *ee*

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

Theorem ["GNS construction"]:
    $G$ topological group, $\phi : G --> \C$ 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 $\H_\phi -~=-> \H'; \xi_\phi |->
    \xi'$.

Remark:
    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.

# Cone of functions of +ve type

$\Pos(G) := \{ \phi : G --> \C \;\text{of +ve type} \}$.
$\Pos(G)$ 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 $||\xi|| = 1$.

Definition:
  A _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 $t \in \{0,1\}$.

Theorem:
    $\phi \in \Posn(G)$ is pure iff the GNS representation
    $\pi_\phi$ is irreducible.
Proof:
    We have $\phi = \phi_{\pi_\phi,\xi_\phi}$, with $\xi_\phi$ cyclic.
    Suppose $\pi_\phi$ is reducible.
    Say $\H_\phi = \K (+) \K^\orth$ 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 $s != 0 != t$ since $\xi$ is cyclic.
    So $\phi$ is not pure.
    
    Conversely, suppose $\phi$ is not pure,
    say $\phi = s^2 \phi_1 + t^2 \phi_2$, $s^2+t^2=1$, $s != 0 != t$, $\phi_i\in\Posn(G)$,
    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 $\pi_1 (+) \pi_2$ of $G$ on $\H_1 (+) \H_2$,
    and let $\xi' := s\xi_1 + t\xi_2$.
    Then $\phi_{\pi_1 (+) \pi_2,\xi'}(g)
	= <{ s g\xi_1 + t g\xi_2, s \xi_1 + t \xi_2 }>
	= s^2 \phi_1(g) + t^2 \phi_2(g)
	= \phi(g)$.

    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 $T : \H' --> \H_1; \xi' |-> \xi_1$ 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

Fact:
    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
    $G$.

# 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 $G$.
$<{ g, h }> := \phi(h^{-1}g)$ extends to a conjugate-symmetric sesquilinear form on $W$,
$<{ \sum_i a_i g_i, \sum_i b_i h_i }> = \sum_{ij} a_i\phi(h_i^{-1}g_i)\bbar_j$.
Now $<{ \sum_i a_i h_i, \sum_i a_i h_i }> = \sum_{ij} a_i \phi(h_i^{-1}h_j) \abar_j >= 0$
by definition of a positive function, so $<{ , }>$ 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 $<{ , }>$ on $V$.

Let $\H$ be the Hilbert space completion of $V$ with respect to $<{ , }>$;
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 $g |-> gf$ 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
$\phi_{\pi,\xi}(h) = <{ \phi_h, \phi_e }> = \phi(h)$.

It remains to show the uniqueness.

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

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

Define an isomorphism $T : \H -~=-> \H'$ 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

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