$\displaystyle\pi\circ{\star}$ $\displaystyle={\star}\circ\pi$

These properties hold for the representations $\pi$ and $\theta$ by
assumption. Consider the equivalent properties for the map
$(\pi\otimes\theta)\circ\Delta$. The first property is verified by:

$\displaystyle(\pi\otimes\theta)\circ\Delta\circ\eta$
$\displaystyle=(\pi\otimes\theta)\circ(\eta\otimes\eta)$ (3.23)
$\displaystyle=(\pi\circ\eta)\otimes(\theta\circ\eta)$
$\displaystyle=\eta\otimes\eta$

The second property is verified by:

$\displaystyle(\pi\otimes\theta)\circ\Delta\circ\nabla$
$\displaystyle=(\pi\otimes\theta)\circ(\nabla\otimes\nabla)\circ(\iota\otimes\tau\otimes\iota)\circ(\Delta\otimes\Delta)$
(3.24)
$\displaystyle=(\nabla\otimes\nabla)\circ(\pi\otimes\pi\otimes\theta\otimes\theta)\circ(\iota\otimes\tau\otimes\iota)\circ(\Delta\otimes\Delta)$
$\displaystyle=(\nabla\otimes\nabla)\circ(\pi\otimes\theta\otimes\pi\otimes\theta)\circ(\Delta\otimes\Delta)$
$\displaystyle=(\nabla\otimes\nabla)\circ(((\pi\otimes\theta)\circ\Delta)\otimes((\pi\otimes\theta)\circ\Delta))$

The third property is verified by:

$\displaystyle(\pi\otimes\theta)\circ\Delta\circ{\star}$
$\displaystyle=(\pi\otimes\theta)\circ({\star}\otimes{\star})\circ\Delta$
(3.25)
$\displaystyle=((\pi\circ{\star})\otimes(\theta\circ{\star}))\circ\Delta$
$\displaystyle=(({\star}\circ\pi)\otimes({\star}\circ\theta))\circ\Delta$
$\displaystyle=({\star}\otimes{\star})\circ(\pi\otimes\theta)\circ\Delta$

The commutation relations between the $\ast$-algebra and $\ast$-coalgebra thus
ensure that $(\pi\otimes\theta)\circ\Delta$ is a representation.

Consider the copositive states $\mathsf{y,z}:\mathbb{C}\leftarrow\mathsf{V}$.
By the Gelfand-Naimark-Segal construction, these states can be expressed as:

$\displaystyle\mathinner{\llangle{\mathsf{y}|\mathsf{a}}\rrangle}$
$\displaystyle=\mathinner{({v|\theta\circ\mathsf{a}|v})}$ (3.26)
$\displaystyle\mathinner{\llangle{\mathsf{z}|\mathsf{a}}\rrangle}$
$\displaystyle=\mathinner{({u|\pi\circ\mathsf{a}|u})}$

for the observable $\mathsf{a}\in\mathsf{V}$, where
$\pi:\mathcal{O}[\mathsf{H}]\leftarrow\mathsf{V}$ and
$\theta:\mathcal{O}[\mathsf{K}]\leftarrow\mathsf{V}$ are representations of
the observables as operators on the Hilbert spaces $\mathsf{H}$ and
$\mathsf{K}$, and $u\in\mathsf{H}$ and $v\in\mathsf{K}$ are vectors that
define the copositive states as pure states. The product state
$\mathsf{yz}:\mathbb{C}\leftarrow\mathsf{V}$ is defined by:

$\mathsf{yz}=(\mathsf{y}\otimes\mathsf{z})\circ\Delta$ (3.27)

The product state is expressed in terms of the representations as:

$\mathinner{\llangle{\mathsf{yz}|\mathsf{a}}\rrangle}=\mathinner{({v\otimes
u|(\theta\otimes\pi)\circ\Delta\circ\mathsf{a}|v\otimes u})}$ (3.28)

The result on the product of representations implies that the product state is
also copositive. This result underpins the construction of the conditional
valuation.

####  3.2 Positive functions and copositive measures

In this section, observables are modelled as functions and states are modelled
as measures on a measurable group. The integral is not defined for all
combinations of measures and functions; it is, however, defined for all
combinations of measures and test functions. The Gelfand-Naimark-Segal
construction then relates the copositive measure with a projection-valued
measure on the group.

######  Definition 10 (Copositive measure)

The measure $\mathsf{z\in U}[\mathcal{A}]$ is copositive if it is coreal and
the valuation map
$\mathinner{\llangle{\mathsf{z}}|}:\mathbb{C}\leftarrow\mathsf{V}_{0}[\mathcal{A}]$
on the $\ast$-algebra of test functions is copositive:

$\mathinner{\llangle{\mathsf{z}|\mathsf{b}^{*}\mathsf{b}}\rrangle}\geq 0$
(3.29)

for all test functions $\mathsf{b\in V}_{0}[\mathcal{A}]$. The function
$\mathsf{a\in V}[\mathcal{A}]$ is positive if it is real and satisfies:

$\mathinner{\llangle{\mathsf{z}|\mathsf{a}}\rrangle}\geq 0$ (3.30)

for all copositive measures $\mathsf{z\in U}_{1}[\mathcal{A}|\mathsf{a}]$.

The Gelfand-Naimark-Segal construction identifies the copositive measure with
a pure state derived from a representation of the test functions, and this
allows a simple characterisation of the positive function. The copositive
measure $\mathsf{z\in U}[\mathcal{A}]$ satisfies:

$\mathinner{\llangle{\mathsf{z}|\mathsf{a}}\rrangle}=\mathinner{({u|\pi\circ\mathsf{a}|u})}$
(3.31)

for the test function $\mathsf{a\in V}_{0}[\mathcal{A}]$. In this expression,
$u\in\mathsf{H}$ is a vector in a Hilbert space $\mathsf{H}$ and
$\pi:\mathcal{O}[\mathsf{H}]\leftarrow\mathsf{V}_{0}[\mathcal{A}]$ is a
representation of the $\ast$-algebra of test functions as (possibly unbounded)
operators on the Hilbert space.

The measure is explicitly identified by applying the representation to the
test function $\mathsf{d}[X]\mathsf{\in V}_{0}[\mathcal{A}]$:

$\mathsf{z}[X]=\mathinner{({u|\pi[X]|u})}$ (3.32)

for the subset $X\subset\mathcal{A}$, where:

$\pi[X]=\pi\circ\mathsf{d}[X]$ (3.33)

The properties required of the $\ast$-algebra representation translate to:

$\displaystyle\pi[\mathcal{A}]$ $\displaystyle=1$ (3.34)
$\displaystyle\pi[X\cap Y]$ $\displaystyle=\pi[X]\pi[Y]$
$\displaystyle\pi[X]^{\ast}$ $\displaystyle=\pi[X]$

for the subsets $X,Y\subset\mathcal{A}$. The representation thus defines a
projection-valued measure on the group $\mathcal{A}$, and the positive
function $\mathsf{a\in V}[\mathcal{A}]$ is characterised by the property that:

$\pi\circ\mathsf{a}=\int_{x\in\mathcal{A}}\pi[dx]\,\mathsf{a}[x]$ (3.35)

is a positive operator for any projection-valued measure $\pi$ on
$\mathcal{A}$.

#####  3.2.1 Numeraire change

The numeraire change of a measure by a function is the dual of the positivity-
preserving multiplier action on functions. The function $\mathsf{a}$ acts on
the function $\mathsf{b}$ via the two-sided product
$\mathsf{a}^{\ast}\mathsf{ba}$, so defined as to preserve positivity. The
coaction of the function $\mathsf{a}$ on the measure $\mathsf{z}$ is then
defined as the dual of this action, whose integral with the function
$\mathsf{b}$ is given by
$\mathinner{\llangle{\mathsf{z}|\mathsf{a}^{*}\mathsf{b}\mathsf{a}}\rrangle}$.

######  Theorem 11 (Numeraire change)

The function $\mathsf{a\in V}[\mathcal{A}]$ generates the numeraire change
action on functions and coaction on measures:

$\displaystyle\mathinner{\langle{\mathsf{a}}]}$
$\displaystyle=\nabla\circ(\mathinner{\llangle{\mathsf{a}^{*}}|}\otimes\iota\otimes\mathinner{\llangle{\mathsf{a}}|}):\mathsf{V}[\mathcal{A}]\leftarrow\mathsf{V}[\mathcal{A}]$
(3.36) $\displaystyle\mathinner{[{\mathsf{a}}\rangle}$
$\displaystyle=\Delta\circ(\mathinner{|{\mathsf{a}^{*}}\rrangle}\otimes\iota\otimes\mathinner{|{\mathsf{a}}\rrangle}):\mathsf{U}_{1}[\mathcal{A}|\mathsf{a}^{\ast}\mathsf{a}]\rightarrow\mathsf{U}[\mathcal{A}]$

Numeraire change preserves positivity and copositivity.

  * •

If the function $\mathsf{b}\in\mathsf{V}[\mathcal{A}]$ is positive, then the
function $\mathinner{\langle{\mathsf{a}}]}\circ\mathsf{b\in V}[\mathcal{A}]$
generated from the action of $\mathsf{a}$ is positive.

  * •

If the measure
$\mathsf{z}\in\mathsf{U}_{1}[\mathcal{A}|\mathsf{a}^{\ast}\mathsf{a}]$ is
copositive, then the measure
$\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle}\in\mathsf{U}[\mathcal{A}]$
generated from the coaction of $\mathsf{a}$ is copositive.

Proof. To verify the second result, note that for the function $\mathsf{b}$:

$\mathinner{\llangle{\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle}|\mathsf{b}^{*}\mathsf{b}}\rrangle}=\mathinner{\llangle{\mathsf{z}|(\mathsf{b}\mathsf{a})^{*}(\mathsf{b}\mathsf{a})}\rrangle}$
(3.37)

so that copositivity of $\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle}$
follows from copositivity of $\mathsf{z}$. To verify the first result, note
that for the function $\mathsf{b}$:

$\mathinner{\llangle{\mathsf{z}|\mathinner{\langle{\mathsf{a}}]}\circ\mathsf{b}}\rrangle}=\mathinner{\llangle{\mathsf{z}\circ\Delta|\mathsf{a}^{*}\otimes\mathsf{b}\otimes\mathsf{a}}\rrangle}=\mathinner{\llangle{\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle}|\mathsf{b}}\rrangle}$
(3.38)

so that positivity of $\mathinner{\langle{\mathsf{a}}]}\circ\mathsf{b}$
follows from positivity of $\mathsf{b}$ and copositivity of
$\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle}$ when $\mathsf{z}$ is
copositive.

This abstract characterisation of numeraire change is made explicit by
considering the evaluation at elements and subsets of the measurable group.
Numeraire change by the function $\mathsf{a}\in\mathsf{V}[\mathcal{A}]$
corresponds to the following operations on the function
$\mathsf{b}\in\mathsf{V}[\mathcal{A}]$ and the measure
$\mathsf{z}\in\mathsf{U}_{1}[\mathcal{A}|\mathsf{a}^{\ast}\mathsf{a}]$:

$\displaystyle(\mathinner{\langle{\mathsf{a}}]}\circ\mathsf{b})[x]$
$\displaystyle=|\mathsf{a}[x]|^{2}\mathsf{b}[x]$ (3.39)
$\displaystyle(\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle})[X]$
$\displaystyle=\int_{x\in X}\mathsf{z}[dx]\,|\mathsf{a}[x]|^{2}$

for the element $x\in\mathcal{A}$ and the subset $X\subset\mathcal{A}$.
Numeraire change of the function $\mathsf{b}$ amounts to scaling the function
by the positive function $|\mathsf{a}|^{2}$. Numeraire change of the measure
$\mathsf{z}$ applies the positive function $|\mathsf{a}|^{2}$ as an integral
kernel in the measure.

When the measure $\mathsf{z}$ is copositive, the numeraire change measure
$\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle}$ may be related to the same
projection-valued measure on the group as the original function:

$\displaystyle\mathsf{z}[X]$ $\displaystyle=\mathinner{({u|\pi[X]|u})}$ (3.40)
$\displaystyle(\mathsf{z}\circ\mathinner{[{\mathsf{a}}\rangle})[X]$
$\displaystyle=\mathinner{({v|\pi[X]|v})}$

where:

$v=(\pi\circ\mathsf{a})u$ (3.41)

In this case, numeraire change is implemented by modifying the vector of the
pure state.

#####  3.2.2 Conditional valuation

The conditional valuation of a function by a measure is the dual of the right
multiplier action on measures. The measure $\mathsf{z}$ acts on the measure
$\mathsf{y}$ via the product $\mathsf{yz}$. The coaction of the measure
$\mathsf{z}$ on the function $\mathsf{a}$ is then defined as the dual of this
action, whose integral with the measure $\mathsf{y}$ is given by
$\mathinner{\llangle{\mathsf{y}\mathsf{z}|\mathsf{a}}\rrangle}$.

######  Theorem 12 (Conditional valuation)

The measure $\mathsf{z\in U}[\mathcal{A}]$ generates the conditional valuation
action on measures and coaction on functions:

$\displaystyle\ket{\mathsf{z}}$
$\displaystyle=(\iota\otimes\mathinner{|{\mathsf{z}}\rrangle})\circ\nabla:\mathsf{U}[\mathcal{A}]\rightarrow\mathsf{U}[\mathcal{A}]$
(3.42) $\displaystyle\bra{\mathsf{z}}$
$\displaystyle=(\iota\otimes\mathinner{\llangle{\mathsf{z}}|})\circ\Delta:\mathsf{V}[\mathcal{A}]\leftarrow\mathsf{V}_{1}[\mathsf{z}|\mathcal{A}]$

Conditional valuation preserves copositivity and positivity when the measure
is copositive.

  * •

For the copositive measure $\mathsf{z\in U}[\mathcal{A}]$, if the measure
$\mathsf{y}\in\mathsf{U}[\mathcal{A}]$ is copositive, then the measure
$\mathsf{y}\circ\ket{\mathsf{z}}\in\mathsf{U}[\mathcal{A}]$ generated from the
action of $\mathsf{z}$ is copositive.

  * •

For the copositive measure $\mathsf{z\in U}[\mathcal{A}]$, if the function
$\mathsf{a}\in\mathsf{V}_{1}[\mathsf{z}|\mathcal{A}]$ is positive, then the
function $\bra{\mathsf{z}}\circ\mathsf{a\in V}[\mathcal{A}]$ generated from
the coaction of $\mathsf{z}$ is positive.

Proof. The first result is verified by appeal to the Gelfand-Naimark-Segal
construction. Represent the copositive measures $\mathsf{y}$ and $\mathsf{z}$
by:

$\displaystyle\mathsf{y}[X]$
$\displaystyle=\mathinner{({v|\theta\circ\mathsf{d}[X]|v})}$ (3.43)
$\displaystyle\mathsf{z}[X]$
$\displaystyle=\mathinner{({u|\pi\circ\mathsf{d}[X]|u})}$

for the subset $X\subset\mathcal{A}$. The map $(\theta\otimes\pi)\circ\Delta$,
constructed as the convolution of the representations $\theta$ and $\pi$ of
the $\ast$-algebra of test functions, is also a representation by virtue of
the properties of the $\ast$-bialgebra. Furthermore:

$(\mathsf{y}\circ\ket{\mathsf{z}})[X]=\mathinner{({v\otimes
u|(\theta\otimes\pi)\circ\Delta\circ\mathsf{d}[X]|v\otimes u})}$ (3.44)

which demonstrates copositivity for the measure
$\mathsf{y}\circ\ket{\mathsf{z}}$. To verify the second result, note that for
the function $\mathsf{a}$:

$\mathinner{\llangle{\mathsf{y}|\bra{\mathsf{z}}\circ\mathsf{a}}\rrangle}=\mathinner{\llangle{\mathsf{y}\otimes\mathsf{z}|\Delta\circ\mathsf{a}}\rrangle}=\mathinner{\llangle{\mathsf{y}\circ\ket{\mathsf{z}}|\mathsf{a}}\rrangle}$
(3.45)

so that positivity of $\bra{\mathsf{z}}\circ\mathsf{a}$ follows from
positivity of $\mathsf{a}$ and copositivity of
$\mathsf{y}\circ\ket{\mathsf{z}}$ when $\mathsf{z}$ is copositive.

This abstract characterisation of conditional valuation is made explicit by
considering the evaluation at subsets and elements of the measurable group.
Conditional valuation by the measure $\mathsf{z}\in\mathsf{U}[\mathcal{A}]$
corresponds to the following operations on the measure
$\mathsf{y}\in\mathsf{U}[\mathcal{A}]$ and the function
$\mathsf{a}\in\mathsf{V}_{1}[\mathsf{z}|\mathcal{A}]$:

$\displaystyle(\mathsf{y}\circ\ket{\mathsf{z}})[X]$ $\displaystyle=\int_{xy\in
X}\mathsf{y}[dx]\mathsf{z}[dy]$ (3.46)
$\displaystyle(\bra{\mathsf{z}}\circ\mathsf{a})[x]$
$\displaystyle=\int_{y\in\mathcal{A}}\mathsf{z}[dy]\,\mathsf{a}[xy]$

for the subset $X\subset\mathcal{A}$ and the element $x\in\mathcal{A}$.
Conditional valuation of the measure $\mathsf{y}$ amounts to the right
convolution with the measure $\mathsf{z}$. Conditional valuation of the
function $\mathsf{a}$ is the right translation of the function integrated with
measure $\mathsf{z}$ over the translation.

####  3.3 Positive measures and copositive functions

In this section, observables are modelled as measures and states are modelled
as functions on a measurable group. The integral is not defined for all
combinations of functions and measures; it is, however, defined for all
combinations of functions and test measures. The Gelfand-Naimark-Segal
construction then relates the copositive function with a unitary
representation of the group.

######  Definition 13 (Copositive function)

The function $\mathsf{a\in V}[\mathcal{A}]$ is copositive if it is coreal and
the valuation map
$\mathinner{|{\mathsf{a}}\rrangle}:\mathsf{U}_{0}[\mathcal{A}]\rightarrow\mathbb{C}$
on the $\ast$-algebra of test measures is copositive:

$\mathinner{\llangle{\mathsf{y}^{*}\mathsf{y}|\mathsf{a}}\rrangle}\geq 0$
(3.47)

for all test measures $\mathsf{y\in U}_{0}[\mathcal{A}]$. The measure
$\mathsf{z\in U}[\mathcal{A}]$ is positive if it is real and satisfies:

$\mathinner{\llangle{\mathsf{z}|\mathsf{a}}\rrangle}\geq 0$ (3.48)

for all copositive functions $\mathsf{a\in V}_{1}[\mathsf{z}|\mathcal{A}]$.

The Gelfand-Naimark-Segal construction identifies the copositive function with
a pure state derived from a representation of the test measures, and this
allows a simple characterisation of the positive measure. The copositive
function $\mathsf{a\in V}[\mathcal{A}]$ satisfies:

$\mathinner{\llangle{\mathsf{z}|\mathsf{a}}\rrangle}=\mathinner{({u|\mathsf{z}\circ\pi|u})}$
(3.49)

for the test measure $\mathsf{z\in U}_{0}[\mathcal{A}]$. In this expression,
$u\in\mathsf{H}$ is a vector in a Hilbert space $\mathsf{H}$ and
$\pi:\mathsf{U}_{0}[\mathcal{A}]\rightarrow\mathcal{O}[\mathsf{H}]$ is a
representation of the $\ast$-algebra of test measures as (possibly unbounded)
operators on the Hilbert space.

The function is explicitly identified by applying the representation to the
test measure $\mathsf{d}[x]\mathsf{\in U}_{0}[\mathcal{A}]$:

$\mathsf{a}[x]=\mathinner{({u|\pi[x]|u})}$ (3.50)

for the element $x\in\mathcal{A}$, where:

$\pi[x]=\mathsf{d}[x]\circ\pi$ (3.51)

The properties required of the $\ast$-algebra representation translate to:

$\displaystyle\pi[1]$ $\displaystyle=1$ (3.52) $\displaystyle\pi[xy]$
$\displaystyle=\pi[x]\pi[y]$ $\displaystyle\pi[x]^{\ast}$
$\displaystyle=\pi[x]^{-1}$

for the elements $x,y\in\mathcal{A}$. The representation thus defines a
unitary representation of the group $\mathcal{A}$, and the positive measure
$\mathsf{z\in U}[\mathcal{A}]$ is characterised by the property that:

$\mathsf{z}\circ\pi=\int_{x\in\mathcal{A}}\mathsf{z}[dx]\,\pi[x]$ (3.53)

is a positive operator for any unitary representation $\pi$ of $\mathcal{A}$.

#####  3.3.1 Numeraire change

The numeraire change of a function by a measure is the dual of the positivity-
preserving multiplier action on measures. The measure $\mathsf{z}$ acts on the
measure $\mathsf{y}$ via the two-sided product $\mathsf{z}^{\ast}\mathsf{yz}$,
so defined as to preserve positivity. The coaction of the measure $\mathsf{z}$
on the function $\mathsf{a}$ is then defined as the dual of this action, whose
integral with the measure $\mathsf{y}$ is given by
$\mathinner{\llangle{\mathsf{z}^{*}\mathsf{y}\mathsf{z}|\mathsf{a}}\rrangle}$.

######  Theorem 14 (Numeraire change)

The measure $\mathsf{z\in U}[\mathcal{A}]$ generates the numeraire change
action on measures and coaction on functions:

$\displaystyle\mathinner{[{\mathsf{z}}\rangle}$
$\displaystyle=(\mathinner{|{\mathsf{z}^{*}}\rrangle}\otimes\iota\otimes\mathinner{|{\mathsf{z}}\rrangle})\circ\nabla:\mathsf{U}[\mathcal{A}]\rightarrow\mathsf{U}[\mathcal{A}]$
(3.54) $\displaystyle\mathinner{\langle{\mathsf{z}}]}$
$\displaystyle=(\mathinner{\llangle{\mathsf{z}^{*}}|}\otimes\iota\otimes\mathinner{\llangle{\mathsf{z}}|})\circ\Delta:\mathsf{V}[\mathcal{A}]\leftarrow\mathsf{V}_{1}[\mathsf{z}^{\ast}\mathsf{z}|\mathcal{A}]$

Numeraire change preserves positivity and copositivity.

  * •

If the measure $\mathsf{y}\in\mathsf{U}[\mathcal{A}]$ is positive, then the
measure $\mathsf{y}\circ\mathinner{[{\mathsf{z}}\rangle}\mathsf{\in
U}[\mathcal{A}]$ generated from the action of $\mathsf{z}$ is positive.

  * •

If the function
$\mathsf{a}\in\mathsf{V}_{1}[\mathsf{z}^{\ast}\mathsf{z}|\mathcal{A}]$ is
copositive, then the function
$\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a}\in\mathsf{V}[\mathcal{A}]$
generated from the coaction of $\mathsf{z}$ is copositive.

Proof. To verify the second result, note that for the measure $\mathsf{y}$:

$\mathinner{\llangle{\mathsf{y}^{*}\mathsf{y}|\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a}}\rrangle}=\mathinner{\llangle{(\mathsf{y}\mathsf{z})^{*}(\mathsf{y}\mathsf{z})|\mathsf{a}}\rrangle}$
(3.55)

so that copositivity of $\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a}$
follows from copositivity of $\mathsf{a}$. To verify the first result, note
that for the measure $\mathsf{y}$:

$\mathinner{\llangle{\mathsf{y}\circ\mathinner{[{\mathsf{z}}\rangle}|\mathsf{a}}\rrangle}=\mathinner{\llangle{\mathsf{z}^{*}\otimes\mathsf{y}\otimes\mathsf{z}|\Delta\circ\mathsf{a}}\rrangle}=\mathinner{\llangle{\mathsf{y}|\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a}}\rrangle}$
(3.56)

so that positivity of $\mathsf{y}\circ\mathinner{[{\mathsf{z}}\rangle}$
follows from positivity of $\mathsf{y}$ and copositivity of
$\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a}$ when $\mathsf{a}$ is
copositive.

This abstract characterisation of numeraire change is made explicit by
considering the evaluation at subsets and elements of the measurable group.
Numeraire change by the measure $\mathsf{z}\in\mathsf{U}[\mathcal{A}]$
corresponds to the following operations on the measure
$\mathsf{y}\in\mathsf{U}[\mathcal{A}]$ and the function
$\mathsf{a}\in\mathsf{V}_{1}[\mathsf{z}^{\ast}\mathsf{z}|\mathcal{A}]$:

$\displaystyle(\mathsf{y}\circ\mathinner{[{\mathsf{z}}\rangle})[X]$
$\displaystyle=\int_{y^{-1}xz\in
X}\mathsf{z}[dy]^{\ast}\mathsf{y}[dx]\mathsf{z}[dz]$ (3.57)
$\displaystyle(\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a})[x]$
$\displaystyle=\int_{y,z\in\mathcal{A}}\mathsf{z}[dy]^{\ast}\mathsf{z}[dz]\,\mathsf{a}[y^{-1}xz]$

for the subset $X\subset\mathcal{A}$ and the element $x\in\mathcal{A}$.
Numeraire change of the measure $\mathsf{y}$ amounts to the positivity-
preserving convolution of the measure with the measure $\mathsf{z}$. Numeraire
change of the function $\mathsf{a}$ is the two-sided translation of the
function integrated with measure $\mathsf{z}$ over the translations.

When the function $\mathsf{a}$ is copositive, the numeraire change function
$\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a}$ may be related to the same
unitary representation of the group as the original function:

$\displaystyle\mathsf{a}[x]$ $\displaystyle=\mathinner{({u|\pi[x]|u})}$ (3.58)
$\displaystyle(\mathinner{\langle{\mathsf{z}}]}\circ\mathsf{a})[x]$
$\displaystyle=\mathinner{({v|\pi[x]|v})}$

where:

$v=(\mathsf{z}\circ\pi)u$ (3.59)

In this case, numeraire change is implemented by modifying the vector of the
pure state.

#####  3.3.2 Conditional valuation

The conditional valuation of a measure by a function is the dual of the left
multiplier action on functions. The function $\mathsf{a}$ acts on the function
$\mathsf{b}$ via the product $\mathsf{ab}$. The coaction of the function
$\mathsf{a}$ on the measure $\mathsf{z}$ is then defined as the dual of this
action, whose integral with the function $\mathsf{b}$ is given by
$\mathinner{\llangle{\mathsf{z}|\mathsf{a}\mathsf{b}}\rrangle}$.

######  Theorem 15 (Conditional valuation)

The function $\mathsf{a\in V}[\mathcal{A}]$ generates the conditional
valuation action on functions and coaction on measures:

$\displaystyle\bra{\mathsf{a}}$
$\displaystyle=\nabla\circ(\mathinner{\llangle{\mathsf{a}}|}\otimes\iota):\mathsf{V}[\mathcal{A}]\leftarrow\mathsf{V}[\mathcal{A}]$
(3.60) $\displaystyle\ket{\mathsf{a}}$
$\displaystyle=\Delta\circ(\mathinner{|{\mathsf{a}}\rrangle}\otimes\iota):\mathsf{U}_{1}[\mathcal{A}|\mathsf{a}]\rightarrow\mathsf{U}[\mathcal{A}]$

Conditional valuation preserves copositivity and positivity when the function
is copositive.

  * •

For the copositive function $\mathsf{a\in V}[\mathcal{A}]$, if the function
$\mathsf{b}\in\mathsf{V}[\mathcal{A}]$ is copositive, then the function
$\bra{\mathsf{a}}\circ\mathsf{b}\in\mathsf{V}[\mathcal{A}]$ generated from the
action of $\mathsf{a}$ is copositive.

  * •

For the copositive function $\mathsf{a\in V}[\mathcal{A}]$, if the measure
$\mathsf{z}\in\mathsf{U}_{1}[\mathcal{A}|\mathsf{a}]$ is positive, then the
measure $\mathsf{z}\circ\ket{\mathsf{a}}\mathsf{\in U}[\mathcal{A}]$ generated
from the coaction of $\mathsf{a}$ is positive.

Proof. The first result is verified by appeal to the Gelfand-Naimark-Segal
construction. Represent the copositive functions $\mathsf{b}$ and $\mathsf{a}$
by:

$\displaystyle\mathsf{b}[x]$
$\displaystyle=\mathinner{({v|\mathsf{d}[x]\circ\theta|v})}$ (3.61)
$\displaystyle\mathsf{a}[x]$
$\displaystyle=\mathinner{({u|\mathsf{d}[x]\circ\pi|u})}$

for the element $x\subset\mathcal{A}$. The map
$\Delta\circ(\pi\otimes\theta)$, constructed as the convolution of the
representations $\theta$ and $\pi$ of the $\ast$-algebra of test measures, is
also a representation by virtue of the properties of the $\ast$-bialgebra.
Furthermore:

$(\bra{\mathsf{a}}\circ\mathsf{b})[x]=\mathinner{({u\otimes
v|\mathsf{d}[x]\circ\Delta\circ(\pi\otimes\theta)|u\otimes v})}$ (3.62)

which demonstrates copositivity for the function
$\bra{\mathsf{a}}\circ\mathsf{b}$. To verify the second result, note that for
the measure $\mathsf{z}$:

$\mathinner{\llangle{\mathsf{z}\circ\ket{\mathsf{a}}|\mathsf{b}}\rrangle}=\mathinner{\llangle{\mathsf{z}\circ\Delta|\mathsf{a}\otimes\mathsf{b}}\rrangle}=\mathinner{\llangle{\mathsf{z}|\bra{\mathsf{a}}\circ\mathsf{b}}\rrangle}$
(3.63)

so that positivity of $\mathsf{z}\circ\ket{\mathsf{a}}$ follows from
positivity of $\mathsf{z}$ and copositivity of
$\bra{\mathsf{a}}\circ\mathsf{b}$ when $\mathsf{a}$ is copositive.

This abstract characterisation of conditional valuation is made explicit by
considering the evaluation at elements and subsets of the measurable group.
Conditional valuation by the function $\mathsf{a}\in\mathsf{V}[\mathcal{A}]$
corresponds to the following operations on the function
$\mathsf{b}\in\mathsf{V}[\mathcal{A}]$ and the measure
$\mathsf{z}\in\mathsf{U}_{1}[\mathcal{A}|\mathsf{a}]$:

$\displaystyle(\bra{\mathsf{a}}\circ\mathsf{b})[x]$
$\displaystyle=\mathsf{a}[x]\mathsf{b}[x]$ (3.64)
$\displaystyle(\mathsf{z}\circ\ket{\mathsf{a}})[X]$ $\displaystyle=\int_{x\in
X}\mathsf{z}[dx]\,\mathsf{a}[x]$

for the element $x\in\mathcal{A}$ and the subset $X\subset\mathcal{A}$.
Conditional valuation of the function $\mathsf{b}$ amounts to scaling by the
function $\mathsf{a}$. Conditional valuation of the measure $\mathsf{z}$
applies the function $\mathsf{a}$ as an integral kernel in the measure.

###  Chapter 4 Economics

The principles of replicability and the absence of arbitrage require that
price is a linear and positive map on the securities. Implicit in this is the
requirement that the securities form a $\ast$-algebra, allowing derivative
securities to be constructed as convex combinations of underlying securities
with a well defined sector of positive securities. Linearity and positivity is
assured if the security $\mathsf{a}$ is mapped to the price $\lambda$ in the
form:

$\lambda=\mathinner{\llangle{\mathsf{w}|\mathsf{q}^{*}\mathsf{a}\mathsf{q}}\rrangle}$
(4.1)

where the deflator $\mathsf{q}$ is an invertible observable that discounts the
security and the driver $\mathsf{w}$ is a copositive state that maps an
observable to its expected value. Adapting the model to the discrete time
context, the deflator and driver are extended to processes that reference a
partially ordered set, provisioning the economy with a concept of time. The
principles impose the requirement that a deflated price process is a
martingale, which is used to identify the prices of derivative securities.

The dual $\ast$-Hopf algebras associated with a measurable group provide two
possible $\ast$-algebras that could be utilised to represent economic
observables. When the observables are taken to be the functions on the
measurable group, the resulting economy has commuting observables but the
states will be noncommuting if the group is noncommutative. When the
observables are taken to be the measures on the measurable group, the
resulting economy has commuting states but the observables will be
noncommuting if the group is noncommutative. The guiding economic principles
are applicable in either case, generating pricing models with distinct
phenomenology.

####  4.1 Commutative economics

In the commutative approach to economics, the observables are taken to be the
functions and the states are taken to be the measures on a measurable group.
The elements of the group effectively form a basis for the states, inheriting
the (possibly noncommutative) product from the group, with the measures
generalising formal linear combinations of these basis states. The functions
then become observables by virtue of the pairing provided by the integral. The
noncommutativity of states implies that the order in which the states are
applied is important. This feature can be used to model dynamics where
accumulated change is contingent on the order in which information arrives.

Consider the model constructed from a measurable group $\mathcal{A}$. The
driving process provides a set of states
$\mathsf{w}_{ij}\in\mathsf{U}[\mathcal{A}]$ defined for each ordered pair of
times $i\leq j$ and satisfying the semigroup property:

$\displaystyle\mathsf{w}_{ik}$ $\displaystyle=\mathsf{w}_{ij}\mathsf{w}_{jk}$
(4.2) $\displaystyle\mathsf{w}_{ii}$ $\displaystyle=1$

for the ordered sequence of times $i\leq j\leq k$. The deflator process for
currency $a$ provides a set of invertible observables
$\mathsf{q}_{i}^{a}\in\mathsf{V}[\mathcal{A}]$ defined for each time $i$. The
process $\mathsf{a}_{i}^{a}\in\mathsf{V}[\mathcal{A}]$ is then a price process
in currency $a$ when it satisfies the martingale condition:

$\mathsf{q}_{i}^{a\ast}\mathsf{a}_{i}^{a}\mathsf{q}_{i}^{a}=\bra{\mathsf{w}_{ij}}\circ\mathsf{q}_{j}^{a\ast}\mathsf{a}_{j}^{a}\mathsf{q}_{j}^{a}$
(4.3)

for each ordered pair of times $i\leq j$.

#####  4.1.1 Discount factors and foreign exchange rates

The core financial processes are the discount factor, modelled in currency $a$
at set time $i$ and maturity time $j$ by the observable:

$\mathsf{p}_{ij}^{a}\in\mathsf{V}[\mathcal{A}]$ (4.4)

and the foreign exchange rate, modelled in foreign currency $a$ and domestic
currency $b$ at time $i$ by the observable:

$\mathsf{s}_{i}^{ab}\in\mathsf{V}[\mathcal{A}]$ (4.5)

Expressions for these processes are derived from the martingale condition:

$\displaystyle\mathsf{q}_{i}^{a\ast}\mathsf{p}_{ij}^{a}\mathsf{q}_{i}^{a}$
$\displaystyle=\bra{\mathsf{w}_{ij}}\circ\mathsf{q}_{j}^{a\ast}\mathsf{q}_{j}^{a}$
(4.6)
$\displaystyle\mathsf{q}_{i}^{b\ast}\mathsf{s}_{i}^{ab}\mathsf{q}_{i}^{b}$
$\displaystyle=\mathsf{q}_{i}^{a\ast}\mathsf{q}_{i}^{a}$

These expressions for the discount factor and foreign exchange rate imply the
normalisation for the deflator:

$\mathinner{\llangle{\mathsf{w}_{i}|\mathsf{q}^{a*}_{i}\mathsf{q}^{a}_{i}}\rrangle}=s^{a}p_{i}^{a}$
(4.7)

where $p_{i}^{a}\in\mathbb{R}^{+}$ is the initial discount factor in currency
$a$ at maturity time $i$ and $s^{a}\in\mathbb{R}^{+}$ is the initial foreign
exchange rate in foreign currency $a$ and fixed reference domestic currency.

#####  4.1.2 Foreign exchange options

The spatial dimension of the economy is currency, and the foreign exchange
option is the option to exchange different currencies at the same time.
Consider the option to receive one unit of foreign currency $a$ at time $i$ in
exchange for $\kappa$ units of domestic currency $b$ at time $i$, with a
single opportunity to exercise the option at time $i$. Exercise of the option
at expiry is indicated by a projection observable
$\mathsf{e}_{i}\in\mathsf{U}[\mathcal{A}]$ with digital spectrum
$\mathop{\mathrm{s}pec}[\mathsf{e}_{i}]=\\{0,1\\}$. The price in the fixed
reference domestic currency at initial time is then:

$\mathinner{\llangle{\mathsf{w}_{i}|\mathsf{q}^{a*}_{i}\mathsf{e}_{i}\mathsf{q}^{a}_{i}}\rrangle}-\kappa\mathinner{\llangle{\mathsf{w}_{i}|\mathsf{q}^{b*}_{i}\mathsf{e}_{i}\mathsf{q}^{b}_{i}}\rrangle}$
(4.8)

The optimal exercise for the option corresponds to the exercise projection
that maximises this price.

#####  4.1.3 Interest rate options

The temporal dimension of the economy is time, and the interest rate option is
the option to exchange the same currency at different times. Consider the
option to receive one unit of currency $a$ at set time $i$ in exchange for
$(1+\kappa\delta_{ij})$ units of currency $a$ at maturity time $j$, where
$\delta_{ij}$ is the accrual fraction for the interval $i\leq j$, with a
single opportunity to exercise the option at time $i$. Exercise of the option
at expiry is indicated by a projection observable
$\mathsf{e}_{i}\in\mathsf{U}[\mathcal{A}]$ with digital spectrum
$\mathop{\mathrm{s}pec}[\mathsf{e}_{i}]=\\{0,1\\}$. The price in the fixed
reference domestic currency at initial time is then:

$\mathinner{\llangle{\mathsf{w}_{i}|\mathsf{q}^{a*}_{i}\mathsf{e}_{i}\mathsf{q}^{a}_{i}}\rrangle}-(1+\kappa\delta_{ij})\mathinner{\llangle{\mathsf{w}_{i}\otimes\mathsf{w}_{ij}|(\Delta\circ\mathsf{q}^{a*}_{j})(\mathsf{e}_{i}\otimes
1)(\Delta\circ\mathsf{q}^{a}_{j})}\rrangle}$ (4.9)

The optimal exercise for the option corresponds to the exercise projection
that maximises this price.

####  4.2 Cocommutative economics

In the cocommutative approach to economics, the observables are taken to be
the measures and the states are taken to be the functions on a measurable
group. The elements of the group effectively form a basis for the observables,
inheriting the (possibly noncommutative) product from the group, with the
measures generalising formal linear combinations of these basis observables.
The functions then become states by virtue of the pairing provided by the
integral. Noncommuting observables cannot be simultaneously measured with
accuracy, as to do so would imply the price map is a group homomorphism. This
is a nontrivial source of volatility even in the simplest pricing models.

Consider the model constructed from a measurable group $\mathcal{A}$. The
driving process provides a set of states
$\mathsf{w}_{ji}\in\mathsf{V}[\mathcal{A}]$ defined for each ordered pair of
times $j\geq i$ and satisfying the semigroup property:

$\displaystyle\mathsf{w}_{ki}$ $\displaystyle=\mathsf{w}_{kj}\mathsf{w}_{ji}$
(4.10) $\displaystyle\mathsf{w}_{ii}$ $\displaystyle=1$

for the ordered sequence of times $k\geq j\geq i$. The deflator process for
currency $a$ provides a set of invertible observables
$\mathsf{q}_{i}^{a}\in\mathsf{U}[\mathcal{A}]$ defined for each time $i$. The
process $\mathsf{a}_{i}^{a}\in\mathsf{U}[\mathcal{A}]$ is then a price process
in currency $a$ when it satisfies the martingale condition:

$\mathsf{q}_{i}^{a\ast}\mathsf{a}_{i}^{a}\mathsf{q}_{i}^{a}=\mathsf{q}_{j}^{a\ast}\mathsf{a}_{j}^{a}\mathsf{q}_{j}^{a}\circ\ket{\mathsf{w}_{ji}}$
(4.11)

for each ordered pair of times $j\geq i$.

#####  4.2.1 Discount factors and foreign exchange rates

The core financial processes are the discount factor, modelled in currency $a$
at set time $i$ and maturity time $j$ by the observable:

$\mathsf{p}_{ji}^{a}\in\mathsf{U}[\mathcal{A}]$ (4.12)

and the foreign exchange rate, modelled in foreign currency $a$ and domestic
currency $b$ at time $i$ by the observable:

$\mathsf{s}_{i}^{ab}\in\mathsf{U}[\mathcal{A}]$ (4.13)

Expressions for these processes are derived from the martingale condition:

$\displaystyle\mathsf{q}_{i}^{a\ast}\mathsf{p}_{ji}^{a}\mathsf{q}_{i}^{a}$
$\displaystyle=\mathsf{q}_{j}^{a\ast}\mathsf{q}_{j}^{a}\circ\ket{\mathsf{w}_{ji}}$
(4.14)
$\displaystyle\mathsf{q}_{i}^{b\ast}\mathsf{s}_{i}^{ab}\mathsf{q}_{i}^{b}$
$\displaystyle=\mathsf{q}_{i}^{a\ast}\mathsf{q}_{i}^{a}$

These expressions for the discount factor and foreign exchange rate imply the
normalisation for the deflator:

$\mathinner{\llangle{\mathsf{q}^{a*}_{i}\mathsf{q}^{a}_{i}|\mathsf{w}_{i}}\rrangle}=s^{a}p_{i}^{a}$
(4.15)

where $p_{i}^{a}\in\mathbb{R}^{+}$ is the initial discount factor in currency
$a$ at maturity time $i$ and $s^{a}\in\mathbb{R}^{+}$ is the initial foreign
exchange rate in foreign currency $a$ and fixed reference domestic currency.

#####  4.2.2 Foreign exchange options

The spatial dimension of the economy is currency, and the foreign exchange
option is the option to exchange different currencies at the same time.
Consider the option to receive one unit of foreign currency $a$ at time $i$ in
exchange for $\kappa$ units of domestic currency $b$ at time $i$, with a
single opportunity to exercise the option at time $i$. Exercise of the option
at expiry is indicated by a projection observable
$\mathsf{e}_{i}\in\mathsf{U}[\mathcal{A}]$ with digital spectrum
$\mathop{\mathrm{s}pec}[\mathsf{e}_{i}]=\\{0,1\\}$. The price in the fixed
reference domestic currency at initial time is then:

$\mathinner{\llangle{\mathsf{q}^{a*}_{i}\mathsf{e}_{i}\mathsf{q}^{a}_{i}|\mathsf{w}_{i}}\rrangle}-\kappa\mathinner{\llangle{\mathsf{q}^{b*}_{i}\mathsf{e}_{i}\mathsf{q}^{b}_{i}|\mathsf{w}_{i}}\rrangle}$
(4.16)

The optimal exercise for the option corresponds to the exercise projection
that maximises this price.

#####  4.2.3 Interest rate options

The temporal dimension of the economy is time, and the interest rate option is
the option to exchange the same currency at different times. Consider the
option to receive one unit of currency $a$ at set time $i$ in exchange for
$(1+\kappa\delta_{ji})$ units of currency $a$ at maturity time $j$, where
$\delta_{ji}$ is the accrual fraction for the interval $j\geq i$, with a
single opportunity to exercise the option at time $i$. Exercise of the option
at expiry is indicated by a projection observable
$\mathsf{e}_{i}\in\mathsf{U}[\mathcal{A}]$ with digital spectrum
$\mathop{\mathrm{s}pec}[\mathsf{e}_{i}]=\\{0,1\\}$. The price in the fixed
reference domestic currency at initial time is then:

$\mathinner{\llangle{\mathsf{q}^{a*}_{i}\mathsf{e}_{i}\mathsf{q}^{a}_{i}|\mathsf{w}_{i}}\rrangle}-(1+\kappa\delta_{ji})\mathinner{\llangle{(\Delta\circ\mathsf{q}^{a*}_{j})(1\otimes\mathsf{e}_{i})(\Delta\circ\mathsf{q}^{a}_{j})|\mathsf{w}_{ji}\otimes\mathsf{w}_{i}}\rrangle}$
(4.17)

The optimal exercise for the option corresponds to the exercise projection
that maximises this price.

###  Chapter 5 The Quadratic Gauss model

The Quadratic Gauss economy takes the observables to be functions and the
states to be measures on a finite-dimensional vector space, and so provides an
example of a commutative economy. The defining feature of the Quadratic Gauss
model is the form of the observables, assumed to be exponential-quadratic
functions, and the states, assumed to be Gauss measures.

Whether describing the function that represents the observable or the measure
that represents the state, the parametrisation combines a vector with a
symmetric matrix. The finite-dimensional vector space $\mathcal{A}$ has dual
finite-dimensional vector space $\mathcal{A}^{\ast}$, where the pairing is
here denoted by the dot product. The symmetric matrix is described by a linear
map $\zeta:\mathcal{A}\rightarrow\mathcal{A}^{\ast}$, which is symmetric if it
satisfies:

$x\cdot\zeta y=y\cdot\zeta x$ (5.1)

for all vectors $x,y\in\mathcal{A}$. The symmetric matrix is nonnegative if
$x\cdot\zeta x\geq 0$ for all vectors $x\in\mathcal{A}$, and positive if
$x\cdot\zeta x>0$ for all nonzero vectors $x\in\mathcal{A}$.

####  5.1 Quadratic Gauss observables and states

The exponential-quadratic functions on the finite-dimensional real vector
space $\mathcal{A}$ form a basis for the observables in the Quadratic Gauss
model. This family of observables is closed under the operations of the
$\ast$-Hopf algebra of functions, as can be verified by direct application.

######  Theorem 16 (Quadratic Gauss observables)

The observables of the Quadratic Gauss model are generated from the
exponential-quadratic functions
$\mathsf{e}[\phi|\zeta]\in\mathsf{V}[\mathcal{A}]$ on a finite-dimensional
real vector space $\mathcal{A}$, parametrised by a vector
$\phi\in\mathcal{A}^{\ast}$, the linear coefficients, and a symmetric matrix
$\zeta:\mathcal{A}\rightarrow\mathcal{A}^{\ast}$, the quadratic coefficients,
where:

$\mathsf{e}[\phi|\zeta][x]=\exp[x\cdot\phi+\frac{1}{2}x\cdot\zeta x]$ (5.2)

for the vector $x\in\mathcal{A}$. The Quadratic Gauss observables generate a
sub $\ast$-Hopf algebra of functions on $\mathcal{A}$.

  * •

The operations of the $\ast$-algebra are:

$\displaystyle\eta\circ 1$ $\displaystyle=\mathsf{e}[0|0]$ (5.3)
$\displaystyle\nabla\circ\mathsf{e}\\!\\!\left[\\!\\!\begin{array}[]{c|cc}\phi_{1}&\zeta_{11}&\zeta_{12}\\\
\phi_{2}&\zeta_{21}&\zeta_{22}\end{array}\\!\\!\right]$
$\displaystyle=\mathsf{e}[\phi_{1}+\phi_{2}|\zeta_{11}+\zeta_{12}+\zeta_{21}+\zeta_{22}]$
(5.6) $\displaystyle{\star}\circ\mathsf{e}[\phi|\zeta]$
$\displaystyle=\mathsf{e}[\phi|\zeta]$

  * •

The operations of the $\ast$-coalgebra are:

$\displaystyle\epsilon\circ\mathsf{e}[\phi|\zeta]$ $\displaystyle=1$ (5.7)
$\displaystyle\Delta\circ\mathsf{e}[\phi|\zeta]$
$\displaystyle=\mathsf{e}\\!\\!\left[\\!\\!\begin{array}[]{c|cc}\phi&\zeta&\zeta\\\
\phi&\zeta&\zeta\end{array}\\!\\!\right]$ (5.10)
$\displaystyle{\rotatebox[origin={c}]{180.0}{$\star$}}\circ\mathsf{e}[\phi|\zeta]$
$\displaystyle=\mathsf{e}[-\phi|\zeta]$

  * •

The antipode of the $\ast$-Hopf algebra is:

$s\circ\mathsf{e}[\phi|\zeta]=\mathsf{e}[-\phi|\zeta]$ (5.11)

The Gauss measures on the finite-dimensional real vector space $\mathcal{A}$
form a basis for the states in the Quadratic Gauss model. This family of
states is closed under the operations of the $\ast$-Hopf algebra of measures,
as can be verified by direct application.

######  Theorem 17 (Quadratic Gauss states)

The states of the QuadraticGauss model are generated from the Gauss measures
$\mathsf{n}[\mu|\nu]\in\mathsf{U}[\mathcal{A}]$ on a finite-dimensional real
vector space $\mathcal{A}$, parametrised by a vector $\mu\in\mathcal{A}$, the
mean, and a nonnegative symmetric matrix
$\nu:\mathcal{A}^{\ast}\rightarrow\mathcal{A}$, the covariance, where:

$\mathsf{n}[\mu|\nu][X]=\frac{\mathop{\displaystyle\int}_{\mu+\nu\phi\in
X}\mathsf{h}[\nu
d\phi]\,\exp[-\dfrac{1}{2}\nu\phi\cdot\phi]}{\mathop{\displaystyle\int}_{\mu+\nu\phi\in\mathcal{A}}\mathsf{h}[\nu
d\phi]\,\exp[-\dfrac{1}{2}\nu\phi\cdot\phi]}$ (5.12)

for the subset $X\subset\mathcal{A}$. Here, $\mathsf{h}$ is a Haar measure on
$\mathcal{A}$, unique up to a scale factor that cancels in this definition.
The Quadratic Gauss states generate a sub $\ast$-Hopf algebra of measures on
$\mathcal{A}$.

  * •

The operations of the $\ast$-algebra are:

$\displaystyle 1\circ\eta$ $\displaystyle=\mathsf{n}[0|0]$ (5.13)
$\displaystyle\mathsf{n}\\!\\!\left[\\!\\!\begin{array}[]{c|cc}\mu_{1}&\nu_{11}&\nu_{12}\\\
\mu_{2}&\nu_{21}&\nu_{22}\end{array}\\!\\!\right]\circ\nabla$
$\displaystyle=\mathsf{n}[\mu_{1}+\mu_{2}|\nu_{11}+\nu_{12}+\nu_{21}+\nu_{22}]$
(5.16) $\displaystyle\mathsf{n}[\mu|\nu]\circ{\star}$
$\displaystyle=\mathsf{n}[-\mu|\nu]$

  * •

The operations of the $\ast$-coalgebra are:

$\displaystyle\mathsf{n}[\mu|\nu]\circ\epsilon$ $\displaystyle=1$ (5.17)
$\displaystyle\mathsf{n}[\mu|\nu]\circ\Delta$
$\displaystyle=\mathsf{n}\\!\\!\left[\\!\\!\begin{array}[]{c|cc}\mu&\nu&\nu\\\
\mu&\nu&\nu\end{array}\\!\\!\right]$ (5.20)
$\displaystyle\mathsf{n}[\mu|\nu]\circ{\rotatebox[origin={c}]{180.0}{$\star$}}$
$\displaystyle=\mathsf{n}[\mu|\nu]$

  * •

The antipode of the $\ast$-Hopf algebra is:

$\mathsf{n}[\mu|\nu]\circ s=\mathsf{n}[-\mu|\nu]$ (5.21)

The pairing of the Quadratic Gauss state
$\mathsf{n}[\mu|\nu]\in\mathsf{U}[\mathcal{A}]$ with the positive observable
$\mathsf{a}^{\ast}\mathsf{a}\in\mathsf{V}[\mathcal{A}]$ is given by:

$\mathinner{\llangle{\mathsf{n}[\mu|\nu]|\mathsf{a}^{*}\mathsf{a}}\rrangle}=\frac{\mathop{\displaystyle\int}_{\mu+\nu\phi\in\mathcal{A}}\mathsf{h}[\nu
d\phi]\,\exp[-\dfrac{1}{2}\nu\phi\cdot\phi]\left|\mathsf{a}[\mu+\nu\phi]\right|^{2}}{\mathop{\displaystyle\int}_{\mu+\nu\phi\in\mathcal{A}}\mathsf{h}[\nu
d\phi]\,\exp[-\dfrac{1}{2}\nu\phi\cdot\phi]}\geq 0$ (5.22)

This demonstrates that the Quadratic Gauss state is copositive.

The closure properties of the observables and states allow many of the key
operations of the Quadratic Gauss model to be expressed algebraically.
Furthermore, the obvious similarities between the operations of the respective
$\ast$-Hopf algebras is an early indication of the duality manifested in the
model. The pairing in the Quadratic Gauss model, given by the integral of the
exponential-quadratic function in the Gauss measure, has simple expression.

######  Theorem 18 (Quadratic Gauss pairing)

The pairing of the QuadraticGauss state $\mathsf{n}[\mu|\nu]$ with the
Quadratic Gauss observable $\mathsf{e}[\phi|\zeta]$ is defined when the
eigenvalues of the matrix $1-\nu\zeta:\mathcal{A}\rightarrow\mathcal{A}$ are
positive, and is given by:

$\displaystyle\mathinner{\llangle{\mathsf{n}[\mu|\nu]|\mathsf{e}[\phi|\zeta]}\rrangle}=$
(5.23)
$\displaystyle\dfrac{1}{\sqrt{\det[1-\nu\zeta]}}\exp[\dfrac{1}{2}(1-\nu\zeta)^{-1}(\mu+\nu\phi)\cdot\phi+\dfrac{1}{2}\mu\cdot(1-\zeta\nu)^{-1}(\phi+\zeta\mu)]$

Dual expressions for numeraire change and conditional valuation in the
Quadratic Gauss model are similarly derived.

######  Theorem 19 (Numeraire change)

For the observable $\mathsf{e}[\phi|\zeta]$ and the state
$\mathsf{n}[\mu|\nu]$, the numeraire change state is given by:

$\displaystyle\mathsf{n}[\mu|\nu]\circ\ket{\mathsf{e}[\phi|\zeta]}$
$\displaystyle=$ (5.24)
$\displaystyle\mathinner{\llangle{\mathsf{n}[\mu|\nu]|\mathsf{e}[\phi|\zeta]}\rrangle}\mathsf{n}[(1-\nu\zeta)^{-1}(\mu+\nu\phi)|(1-\nu\zeta)^{-1}\nu]$

######  Theorem 20 (Conditional valuation)

For the observable $\mathsf{e}[\phi|\zeta]$ and the state
$\mathsf{n}[\mu|\nu]$, the conditional valuation observable is given by:

$\displaystyle\bra{\mathsf{n}[\mu|\nu]}\circ\mathsf{e}[\phi|\zeta]$
$\displaystyle=$ (5.25)
$\displaystyle\mathinner{\llangle{\mathsf{n}[\mu|\nu]|\mathsf{e}[\phi|\zeta]}\rrangle}\mathsf{e}[(1-\zeta\nu)^{-1}(\phi+\zeta\mu)|(1-\zeta\nu)^{-1}\zeta]$

Armed with algebraic representations for the pairing and the dual operations
of numeraire change and conditional valuation, it is straightforward to
construct models for the valuation of derivative securities with simple
expressions for the core financial processes.

####  5.2 The Quadratic Gauss economy

In the Quadratic Gauss economy, the driving process is modelled using Gauss
measures and the deflator process is modelled using exponential-quadratic
functions on a finite-dimensional real vector space $\mathcal{A}$.

  * •

For the ordered pair of times $i\leq j$, the driving state
$\mathsf{w}_{ij}\in\mathsf{U}[\mathcal{A}]$ is modelled by the Gauss measure:

$\mathsf{w}_{ij}=\mathsf{n}[\mu_{ij}|\nu_{ij}]$ (5.26)

for the mean vector $\mu_{ij}\in\mathcal{A}$ and the covariance nonnegative
symmetric matrix $\nu_{ij}:\mathcal{A}^{\ast}\rightarrow\mathcal{A}$. The
semigroup condition is satisfied when the mean and covariance satisfy:

$\displaystyle\mu_{ik}$ $\displaystyle=\mu_{ij}+\mu_{jk}$ (5.27)
$\displaystyle\nu_{ik}$ $\displaystyle=\nu_{ij}+\nu_{jk}$

  * •

For the currency $a$ and the time $i$, the normed square of the deflator
observable
$\mathsf{q}_{i}^{a\ast}\mathsf{q}_{i}^{a}\in\mathsf{V}[\mathcal{A}]$ is
modelled by the exponential-quadratic function:

$\mathsf{q}_{i}^{a\ast}\mathsf{q}_{i}^{a}=s^{a}p_{i}^{a}\frac{1}{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}\rrangle}}\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]$
(5.28)

for the linear coefficients vector $\phi_{i}^{a}\in\mathcal{A}^{\ast}$ and the
quadratic coefficients symmetric matrix
$\zeta_{i}^{a}:\mathcal{A}\rightarrow\mathcal{A}^{\ast}$.

Utilising the general results on Quadratic Gauss observables and states, the
martingale condition for a Quadratic Gauss price process can be expressed as
algebraic relationships among the parameters of the process.

######  Theorem 21 (Quadratic Gauss model)

The Quadratic Gauss observable:

$\mathsf{a}_{i}^{a}=\alpha_{i}^{a}\mathsf{e}[\psi^{a}_{i}|\xi^{a}_{i}]$ (5.29)

satisfies the martingale condition for a price in currency $a$ when the
normalisation $\alpha_{i}^{a}\in\mathbb{R}$ satisfies:

$\alpha_{i}^{a}=\alpha_{j}^{a}\frac{p_{j}^{a}}{p_{i}^{a}}\frac{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}\rrangle}\mathinner{\llangle{\mathsf{n}[\mu_{ij}|\nu_{ij}]|\mathsf{e}[\phi^{a}_{j}+\psi^{a}_{j}|\zeta^{a}_{j}+\xi^{a}_{j}]}\rrangle}}{\mathinner{\llangle{\mathsf{n}[\mu_{j}|\nu_{j}]|\mathsf{e}[\phi^{a}_{j}|\zeta^{a}_{j}]}\rrangle}}$
(5.30)

and the linear coefficients $\psi_{i}^{a}\in\mathcal{A}^{\ast}$ and quadratic
coefficients $\xi_{i}^{a}:\mathcal{A}\rightarrow\mathcal{A}^{\ast}$ satisfy:

$\displaystyle\psi_{i}^{a}$
$\displaystyle=(1-(\zeta_{j}^{a}+\xi_{j}^{a})\nu_{ij})^{-1}((\phi_{j}^{a}+\psi_{j}^{a})+(\zeta_{j}^{a}+\xi_{j}^{a})\mu_{ij})-\phi_{i}^{a}$
(5.31) $\displaystyle\xi_{i}^{a}$
$\displaystyle=(1-(\zeta_{j}^{a}+\xi_{j}^{a})\nu_{ij})^{-1}(\zeta_{j}^{a}+\xi_{j}^{a})-\zeta_{i}^{a}$

for each ordered pair of times $i\leq j$.

Proof. The result follows from the relations:

$\displaystyle\mathsf{q}_{i}^{a\ast}\mathsf{a}_{i}^{a}\mathsf{q}_{i}^{a}$
$\displaystyle=\alpha_{i}^{a}s^{a}p_{i}^{a}\frac{1}{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}\rrangle}}\mathsf{e}[\hat{\psi}^{a}_{ii}|\hat{\xi}^{a}_{ii}]$
(5.32)
$\displaystyle\bra{\mathsf{w}_{ij}}\circ\mathsf{q}_{j}^{a\ast}\mathsf{a}_{j}^{a}\mathsf{q}_{j}^{a}$
$\displaystyle=\alpha_{j}^{a}s^{a}p_{j}^{a}\frac{\mathinner{\llangle{\mathsf{n}[\mu_{ij}|\nu_{ij}]|\mathsf{e}[\phi^{a}_{j}+\psi^{a}_{j}|\zeta^{a}_{j}+\xi^{a}_{j}]}\rrangle}}{\mathinner{\llangle{\mathsf{n}[\mu_{j}|\nu_{j}]|\mathsf{e}[\phi^{a}_{j}|\zeta^{a}_{j}]}\rrangle}}\mathsf{e}[\hat{\psi}^{a}_{ij}|\hat{\xi}^{a}_{ij}]$

where:

$\displaystyle\hat{\psi}_{ij}^{a}$
$\displaystyle=(1-(\zeta_{j}^{a}+\xi_{j}^{a})\nu_{ij})^{-1}((\phi_{j}^{a}+\psi_{j}^{a})+(\zeta_{j}^{a}+\xi_{j}^{a})\mu_{ij})$
(5.33) $\displaystyle\hat{\xi}_{ij}^{a}$
$\displaystyle=(1-(\zeta_{j}^{a}+\xi_{j}^{a})\nu_{ij})^{-1}(\zeta_{j}^{a}+\xi_{j}^{a})$

The martingale condition requires that these two terms are equal.

####  5.3 Foreign exchange options

The martingale condition implies that the foreign exchange rate
$\mathsf{s}_{i}^{ab}\in\mathsf{V}[\mathcal{A}]$ in foreign currency $a$ and
domestic currency $b$ at time $i$ is an exponential-quadratic observable:

$\mathsf{s}_{i}^{ab}=\frac{s^{a}p_{i}^{a}}{s^{b}p_{i}^{b}}\frac{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{b}_{i}|\zeta^{b}_{i}]}\rrangle}}{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}\rrangle}}\mathsf{e}[\phi^{ab}_{i}|\zeta^{ab}_{i}]$
(5.34)

where the linear and quadratic coefficients are given by:

$\displaystyle\phi_{i}^{ab}$ $\displaystyle=\phi_{i}^{a}-\phi_{i}^{b}$ (5.35)
$\displaystyle\zeta_{i}^{ab}$ $\displaystyle=\zeta_{i}^{a}-\zeta_{i}^{b}$

For the option to receive one unit of foreign currency $a$ at time $i$ in
exchange for $\kappa$ units of domestic currency $b$ at time $i$, expiring at
time $i$, the price is:

$o_{i}^{ab}[\kappa]=s^{b}p_{i}^{b}\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|(F\mathsf{A}-K\mathsf{B})^{+}}\rrangle}$
(5.36)

parametrised by the forward $F$, strike $K$ and normalised exponential-
quadratic observables $\mathsf{A}$ and $\mathsf{B}$:

$\displaystyle F$ $\displaystyle=\frac{s^{a}p_{i}^{a}}{s^{b}p_{i}^{b}}$ (5.37)
$\displaystyle K$ $\displaystyle=\kappa$ $\displaystyle\mathsf{A}$
$\displaystyle=\frac{\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}\rrangle}}$
$\displaystyle\mathsf{B}$
$\displaystyle=\frac{\mathsf{e}[\phi^{b}_{i}|\zeta^{b}_{i}]}{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{b}_{i}|\zeta^{b}_{i}]}\rrangle}}$

Optimal exercise is achieved by exercising when the observable
$F\mathsf{A}-K\mathsf{B}$, a spread between two exponential-quadratic
observables, is positive.

####  5.4 Interest rate options

The martingale condition implies that the discount factor
$\mathsf{p}_{ij}^{a}\in\mathsf{V}[\mathcal{A}]$ in currency $a$ at set time
$i$ and maturity time $j$ is an exponential-quadratic observable:

$\mathsf{p}_{ij}^{a}=\frac{p_{j}^{a}}{p_{i}^{a}}\frac{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}\rrangle}\mathinner{\llangle{\mathsf{n}[\mu_{ij}|\nu_{ij}]|\mathsf{e}[\phi^{a}_{j}|\zeta^{a}_{j}]}\rrangle}}{\mathinner{\llangle{\mathsf{n}[\mu_{j}|\nu_{j}]|\mathsf{e}[\phi^{a}_{j}|\zeta^{a}_{j}]}\rrangle}}\mathsf{e}[\phi^{a}_{ij}|\zeta^{a}_{ij}]$
(5.38)

where the linear and quadratic coefficients are given by:

$\displaystyle\phi_{ij}^{a}$
$\displaystyle=(1-\zeta_{j}^{a}\nu_{ij})^{-1}(\phi_{j}^{a}+\zeta_{j}^{a}\mu_{ij})-\phi_{i}^{a}$
(5.39) $\displaystyle\zeta_{ij}^{a}$
$\displaystyle=(1-\zeta_{j}^{a}\nu_{ij})^{-1}\zeta_{j}^{a}-\zeta_{i}^{a}$

For the option to receive one unit of currency $a$ at set time $i$ in exchange
for $(1+\kappa\delta_{ij})$ units of currency $a$ at maturity time $j$,
expiring at time $i$, the price is:

$o_{ij}^{a}[\kappa]=s^{a}p_{j}^{a}\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|(F\mathsf{A}-K\mathsf{B})^{+}}\rrangle}$
(5.40)

parametrised by the forward $F$, strike $K$ and normalised exponential-
quadratic observables $\mathsf{A}$ and $\mathsf{B}$:

$\displaystyle F$ $\displaystyle=\frac{p_{i}^{a}}{p_{j}^{a}}$ (5.41)
$\displaystyle K$ $\displaystyle=1+\kappa\delta_{ij}$
$\displaystyle\mathsf{A}$
$\displaystyle=\frac{\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{i}|\zeta^{a}_{i}]}\rrangle}}$
$\displaystyle\mathsf{B}$
$\displaystyle=\frac{\mathsf{e}[\phi^{a}_{ij}|\zeta^{a}_{ij}]}{\mathinner{\llangle{\mathsf{n}[\mu_{i}|\nu_{i}]|\mathsf{e}[\phi^{a}_{ij}|\zeta^{a}_{ij}]}\rrangle}}$

Optimal exercise is achieved by exercising when the observable
$F\mathsf{A}-K\mathsf{B}$, a spread between two exponential-quadratic
observables, is positive.

###  Chapter 6 The Linear Dirac model

The Linear Dirac economy takes the observables to be measures and the states
to be functions on a compact group, and so provides an example of a
cocommutative economy. The defining feature of the Linear Dirac model is the
form of the observables, assumed to be Dirac measures, and the states, assumed
to be pure states derived from a finite-dimensional unitary representation of
the group.

The construction exploits the fact that a representation of the compact group
$\mathcal{A}$ as unitary operators on a Hilbert space can be extended to a
representation of the group $\ast$-algebra of $\mathcal{A}$, generated by the
Dirac measures.

####  6.1 Linear Dirac observables and states

The Dirac measures on the compact group $\mathcal{A}$ form a basis for the
observables in the Linear Dirac model. This family of observables is closed
under the operations of the $\ast$-Hopf algebra of measures, as can be
verified by direct application.

######  Theorem 22 (Linear Dirac observables)

The observables of the LinearDirac model are generated from the Dirac measures
$\mathsf{d}[x]\in\mathsf{U}[\mathcal{A}]$ on a compact group $\mathcal{A}$,
parametrised by an element $x\in\mathcal{A}$, where:

$\mathsf{d}[x][X]=(x\in X)$ (6.1)

for the subset $X\subset\mathcal{A}$. The Linear Dirac observables generate a
sub $\ast$-Hopf algebra of measures on $\mathcal{A}$.

  * •

The operations of the $\ast$-algebra are:

$\displaystyle 1\circ\eta$ $\displaystyle=\mathsf{d}[1]$ (6.2)
$\displaystyle\mathsf{d}[x,y]\circ\nabla$ $\displaystyle=\mathsf{d}[xy]$
$\displaystyle\mathsf{d}[x]\circ{\star}$ $\displaystyle=\mathsf{d}[x^{-1}]$

  * •

The operations of the $\ast$-coalgebra are:

$\displaystyle\mathsf{d}[x]\circ\epsilon$ $\displaystyle=1$ (6.3)
$\displaystyle\mathsf{d}[x]\circ\Delta$ $\displaystyle=\mathsf{d}[x,x]$
$\displaystyle\mathsf{d}[x]\circ{\rotatebox[origin={c}]{180.0}{$\star$}}$
$\displaystyle=\mathsf{d}[x]$

  * •

The antipode of the $\ast$-Hopf algebra is:

$\mathsf{d}[x]\circ s=\mathsf{d}[x^{-1}]$ (6.4)

The representative functions on the compact group $\mathcal{A}$ form a basis
for the states in the Linear Dirac model. This family of states is closed
under the operations of the $\ast$-Hopf algebra of functions, as can be
verified by direct application.

######  Theorem 23 (Linear Dirac states)

The states of the Linear Dirac model are generated from the representative
functions $\mathinner{({u|\pi|v})}\in\mathsf{V}[\mathcal{A}]$ on a compact
group $\mathcal{A}$, parametrised by a representation
$\pi:\mathcal{A}\rightarrow\mathcal{O}[\mathsf{H}]$ of the group as unitary
operators on a finite-dimensional Hilbert space $\mathsf{H}$ and a pair of
vectors $u,v\in\mathsf{H}$, where:

$\mathinner{({u|\pi|v})}[x]=\mathinner{({u|\pi[x]|v})}$ (6.5)

for the element $x\in\mathcal{A}$. The Linear Dirac states generate a sub
$\ast$-Hopf algebra of functions on $\mathcal{A}$.

  * •

The operations of the $\ast$-algebra are:

$\displaystyle\eta\circ 1$ $\displaystyle=\mathinner{({1|1|1})}$ (6.6)
$\displaystyle\nabla\circ\mathinner{({u|\pi|v})}$
$\displaystyle=\mathinner{({u|\Delta\circ\pi|v})}$
$\displaystyle{\star}\circ\mathinner{({u|\pi|v})}$
$\displaystyle=\mathinner{({u|\pi|v})}^{\ast}$

  * •

The operations of the $\ast$-coalgebra are:

$\displaystyle\epsilon\circ\mathinner{({u|\pi|v})}$
$\displaystyle=\mathinner{({u|v})}$ (6.7)
$\displaystyle\Delta\circ\mathinner{({u|\pi|v})}$
$\displaystyle=\sum\nolimits_{i}\mathinner{({u|\pi|w_{i}})}\otimes\mathinner{({w_{i}|\pi|v})}$
$\displaystyle{\rotatebox[origin={c}]{180.0}{$\star$}}\circ\mathinner{({u|\pi|v})}$
$\displaystyle=\mathinner{({v|\pi|u})}$

where the vectors $w_{i}\in\mathsf{H}$ form an orthonormal basis of
$\mathsf{H}$.

  * •

The antipode of the $\ast$-Hopf algebra is:

$s\circ\mathinner{({u|\pi|v})}=\mathinner{({v|\pi|u})}^{\ast}$ (6.8)

The pairing of the Linear Dirac state
$\mathinner{({u|\pi|v})}\in\mathsf{V}[\mathcal{A}]$ with the positive
observable $\mathsf{z}^{\ast}\mathsf{z}\in\mathsf{U}[\mathcal{A}]$ is given
by:

$\mathinner{\llangle{\mathsf{z}^{*}\mathsf{z}|\mathinner{({u|\pi|v})}}\rrangle}=\mathinner{({(\mathsf{z}\circ\pi)u|(\mathsf{z}\circ\pi)v})}$
(6.9)

where $\mathsf{z}\circ\pi\in\mathcal{O}[\mathsf{H}]$ is the operator:

$\mathsf{z}\circ\pi=\int_{x\in\mathcal{A}}\mathsf{z}[dx]\,\pi[x]$ (6.10)

This demonstrates that the Linear Dirac state is copositive when $u=v$, as in
this case:

$\mathinner{\llangle{\mathsf{z}^{*}\mathsf{z}|\mathinner{({u|\pi|u})}}\rrangle}=\left\|(\mathsf{z}\circ\pi)u\right\|^{2}\geq
0$ (6.11)

The closure properties of the observables and states allow many of the key
operations of the Linear Dirac model to be expressed algebraically. The
pairing in the Linear Dirac model, given by the valuation of the
representative function at the group element, has simple expression.

######  Theorem 24 (Linear Dirac pairing)

The pairing of the Linear Dirac observable
$\mathsf{d}[x]\in\mathsf{U}[\mathcal{A}]$ with the Linear Dirac state
$\mathinner{({u|\pi|v})}\in\mathsf{V}[\mathcal{A}]$ is given by:

$\mathinner{\llangle{\mathsf{d}[x]|\mathinner{({u|\pi|v})}}\rrangle}=\mathinner{({u|\pi[x]|v})}$
(6.12)

Dual expressions for numeraire change and conditional valuation in the Linear
Dirac model are similarly derived.

######  Theorem 25 (Numeraire change)

For the observable $\mathsf{d}[x]$ and the state$\mathinner{({u|\pi|v})}$, the
numeraire change state is given by:

$\mathinner{\langle{\mathsf{d}[x]}]}\circ\mathinner{({u|\pi|v})}=\mathinner{({\pi[x]u|\pi|\pi[x]v})}$
(6.13)

Proof. The valuation of the numeraire change state
$\mathinner{\langle{\mathsf{d}[x]}]}\circ\mathinner{({u|\pi|v})}$ at the
element $y$ is:

$\displaystyle(\mathinner{\langle{\mathsf{d}[x]}]}\circ\mathinner{({u|\pi|v})})[y]$
$\displaystyle=\mathinner{\llangle{\mathsf{d}[y]|\mathinner{\langle{\mathsf{d}[x]}]}\circ\mathinner{({u|\pi|v})}}\rrangle}$
(6.14)
$\displaystyle=\mathinner{\llangle{\mathsf{d}[x]^{*}\mathsf{d}[y]\mathsf{d}[x]|\mathinner{({u|\pi|v})}}\rrangle}$
$\displaystyle=\mathinner{\llangle{\mathsf{d}[x^{-1}yx]|\mathinner{({u|\pi|v})}}\rrangle}$
$\displaystyle=\mathinner{({u|\pi[x^{-1}yx]|v})}$
$\displaystyle=\mathinner{({\pi[x]u|\pi[y]|\pi[x]v})}$
$\displaystyle=\mathinner{({\pi[x]u|\pi|\pi[x]v})}[y]$

This demonstrates the statement for numeraire change.

######  Theorem 26 (Conditional valuation)

For the observable $\mathsf{d}[x]$ and the state $\mathinner{({u|\pi|v})}$,
the conditional valuation observable is given by:

$\mathsf{d}[x]\circ\ket{\mathinner{({u|\pi|v})}}=\mathinner{({u|\pi[x]|v})}\mathsf{d}[x]$
(6.15)

Proof. The valuation of the conditional valuation observable
$\mathsf{d}[x]\circ\ket{\mathinner{({u|\pi|v})}}$ at the subset $X$ is:

$\displaystyle(\mathsf{d}[x]\circ\ket{\mathinner{({u|\pi|v})}})[X]$
$\displaystyle=\mathinner{\llangle{\mathsf{d}[x]\circ\ket{\mathinner{({u|\pi|v})}}|\mathsf{d}[X]}\rrangle}$
(6.16)
$\displaystyle=\mathinner{\llangle{\mathsf{d}[x]|\mathinner{({u|\pi|v})}\mathsf{d}[X]}\rrangle}$
$\displaystyle=\mathinner{({u|\pi[x]|v})}(x\in X)$
$\displaystyle=\mathinner{({u|\pi[x]|v})}\mathsf{d}[x][X]$

This demonstrates the statement for conditional valuation.

####  6.2 The Linear Dirac economy

In the Linear Dirac economy, the driving process is modelled using
representative functions and the deflator process is modelled using Dirac
measures on a compact group $\mathcal{A}$.

  * •

For the ordered pair of times $j\geq i$, the driving state
$\mathsf{w}_{ji}\in\mathsf{V}[\mathcal{A}]$ is modelled by the representative
function:

$\mathsf{w}_{ji}=\mathinner{({u_{ji}|\pi_{ji}|u_{ji}})}$ (6.17)

for the unitary representation
$\pi_{ji}:\mathcal{A}\rightarrow\mathcal{O}[\mathsf{H}_{ji}]$ on the finite-
dimensional Hilbert space $\mathsf{H}_{ji}$ and the unit vector
$u_{ji}\in\mathsf{H}_{ji}$ with $\left\|u_{ji}\right\|=1$. The semigroup
condition is satisfied when the representations and vectors satisfy:

$\displaystyle\mathsf{H}_{ki}$
$\displaystyle=\mathsf{H}_{kj}\otimes\mathsf{H}_{ji}$ (6.18)
$\displaystyle\pi_{ki}$ $\displaystyle=\Delta\circ(\pi_{kj}\otimes\pi_{ji})$
$\displaystyle u_{ki}$ $\displaystyle=u_{kj}\otimes u_{ji}$

  * •

For the currency $a$ and the time $i$, the deflator observable
$\mathsf{q}_{i}^{a}\in\mathsf{U}[\mathcal{A}]$ is modelled by the Dirac
measure:

$\mathsf{q}_{i}^{a}=\sqrt{s^{a}p_{i}^{a}}\,\mathsf{d}[x_{i}^{a}]$ (6.19)

for the element $x_{i}^{a}\in\mathcal{A}$.

Utilising the general results on Linear Dirac observables and states, the
martingale condition for a Linear Dirac price process can be expressed as
algebraic relationships among the parameters of the process.

######  Theorem 27 (Linear Dirac model)

The Linear Dirac observable:

$\mathsf{a}_{i}^{a}=\alpha_{i}^{a}\mathsf{d}[y_{i}^{a}]$ (6.20)

satisfies the martingale condition for a price in currency $a$ when the
normalisation $\alpha_{i}^{a}\in\mathbb{R}$ satisfies:

$\alpha_{i}^{a}=\alpha_{j}^{a}\frac{p_{j}^{a}}{p_{i}^{a}}\mathinner{({\pi_{ji}[x^{a}_{j}]u_{ji}|\pi_{ji}[y^{a}_{j}]|\pi_{ji}[x^{a}_{j}]u_{ji}})}$
(6.21)

and the element $y_{i}^{a}\in\mathcal{A}$ satisfies:

$y_{i}^{a}=(x_{j}^{a}x_{i}^{a}{}^{-1})^{-1}y_{j}^{a}(x_{j}^{a}x_{i}^{a}{}^{-1})$
(6.22)

for each ordered pair of times $j\geq i$.

Proof. The result follows from the relations:

$\displaystyle\mathsf{q}_{i}^{a\ast}\mathsf{a}_{i}^{a}\mathsf{q}_{i}^{a}$
$\displaystyle=\alpha_{i}^{a}s^{a}p_{i}^{a}\mathsf{d}[x_{i}^{a}{}^{-1}y_{i}^{a}x_{i}^{a}]$
(6.23)
$\displaystyle\mathsf{q}_{j}^{a\ast}\mathsf{a}_{j}^{a}\mathsf{q}_{j}^{a}\circ\ket{\mathsf{w}_{ji}}$
$\displaystyle=\alpha_{j}^{a}s^{a}p_{j}^{a}\mathinner{({\pi_{ji}[x^{a}_{j}]u_{ji}|\pi_{ji}[y^{a}_{j}]|\pi_{ji}[x^{a}_{j}]u_{ji}})}\mathsf{d}[x_{j}^{a}{}^{-1}y_{j}^{a}x_{j}^{a}]$

The martingale condition requires that these two terms are equal.

####  6.3 Foreign exchange options

The martingale condition implies that the foreign exchange rate
$\mathsf{s}_{i}^{ab}\in\mathsf{U}[\mathcal{A}]$ in foreign currency $a$ and
domestic currency $b$ at time $i$ is a scalar:

$\mathsf{s}_{i}^{ab}=\frac{s^{a}p_{i}^{a}}{s^{b}p_{i}^{b}}$ (6.24)

The apparent absence of volatility for the foreign exchange rate is
misleading, as foreign exchange options acquire value through the
noncommutativity of the deflators associated with the foreign and domestic
currencies.

For the option to receive one unit of foreign currency $a$ at time $i$ in
exchange for $\kappa$ units of domestic currency $b$ at time $i$, expiring at
time $i$, the price is:

$s^{b}p_{i}^{b}(f\mathinner{({u^{a}_{i}|E|u^{a}_{i}})}-\kappa\mathinner{({u^{b}_{i}|E|u^{b}_{i}})})$
(6.25)

with:

$f=\frac{s^{a}p_{i}^{a}}{s^{b}p_{i}^{b}}$ (6.26)

where the unit vectors $u_{i}^{a},u_{i}^{b}\in\mathsf{H}_{i}$ are given by:

$\displaystyle u_{i}^{a}$ $\displaystyle=\mathsf{\pi}_{i}[x_{i}^{a}]u_{i}$
(6.27) $\displaystyle u_{i}^{b}$
$\displaystyle=\mathsf{\pi}_{i}[x_{i}^{b}]u_{i}$

and the projection $E\in\mathcal{O}[\mathsf{H}_{i}]$ indicates exercise.
Optimal exercise maximises the option price:

$o_{i}^{ab}[k]=s^{b}p_{i}^{b}\sup\\{f\mathinner{({u^{a}_{i}|E|u^{a}_{i}})}-\kappa\mathinner{({u^{b}_{i}|E|u^{b}_{i}})}\\}$
(6.28)

where the supremum ranges over all the projections in the weak closure of the
image of the representation
$\pi_{i}[\mathcal{A}]^{\prime\prime}\subset\mathcal{O}[\mathsf{H}_{i}]$.

The precise formula for the option price depends on the range of projections
available in the von Neumann algebra $\pi_{i}[\mathcal{A}]^{\prime\prime}$.
The supremum over all the projections in $\mathcal{O}[\mathsf{H}_{i}]$ can be
calculated explicitly, however, as it depends only on the two dimensional
subspace spanned by the unit vectors $u_{i}^{a}$ and $u_{i}^{b}$. This gives
an upper bound for the option price:

$\displaystyle\frac{o_{i}^{ab}[k]}{s^{b}p_{i}^{b}}$
$\displaystyle\leq\mathop{\mathrm{t}r}[(F\mathinner{|{u^{a}_{i}})}\mathinner{({u^{a}_{i}}|}-K\mathinner{|{u^{b}_{i}})}\mathinner{({u^{b}_{i}}|})^{+}]$
(6.29)
$\displaystyle=\frac{1}{2}(F-K)+\frac{1}{2}\sqrt{(F-K)^{2}+4FK\sigma^{2}}$

parametrised by the forward $F$, strike $K$ and volatility $\sigma$:

$\displaystyle F$ $\displaystyle=f$ (6.30) $\displaystyle K$
$\displaystyle=\kappa$ $\displaystyle\sigma^{2}$
$\displaystyle=1-\left|\mathinner{({u^{a}_{i}|u^{b}_{i}})}\right|^{2}$

The option has zero volatility when the vectors are parallel and maximum
volatility when the vectors are perpendicular, and in general the volatility
is the absolute sine of the angle between the vectors.

####  6.4 Interest rate options

The martingale condition implies that the discount factor
$\mathsf{p}_{ji}^{a}\in\mathsf{U}[\mathcal{A}]$ in currency $a$ at set time
$i$ and maturity time $j$ is a scalar:

$\mathsf{p}_{ji}^{a}=\frac{p_{j}^{a}}{p_{i}^{a}}$ (6.31)

The apparent absence of volatility for the discount factor is misleading, as
interest rate options acquire value through the noncommutativity of the
deflators associated with the set and maturity times.

For the option to receive one unit of currency $a$ at set time $i$ in exchange
for $(1+\kappa\delta_{ji})$ units of currency $a$ at maturity time $j$,
expiring at time $i$, the price is:

$s^{a}p_{j}^{a}((1+f\delta_{ji})\mathinner{({u^{a}_{i}|E|u^{a}_{i}})}-(1+\kappa\delta_{ji})\mathinner{({u^{a}_{ji}|E|u^{a}_{ji}})})$
(6.32)

with:

$f=\frac{1}{\delta_{ji}}(\frac{p_{i}^{a}}{p_{j}^{a}}-1)$ (6.33)

where the unit vectors $u_{i}^{a},u_{ji}^{a}\in\mathsf{H}_{i}$ are given by:

$\displaystyle u_{i}^{a}$ $\displaystyle=\mathsf{\pi}_{i}[x_{i}^{a}]u_{i}$
(6.34) $\displaystyle u_{ji}^{a}$
$\displaystyle=\mathsf{\pi}_{i}[x_{j}^{a}]u_{i}$

and the projection $E\in\mathcal{O}[\mathsf{H}_{i}]$ indicates exercise.
Optimal exercise maximises the option price:

$o_{ji}^{a}[k]=s^{a}p_{j}^{a}\sup\\{(1+f\delta_{ji})\mathinner{({u^{a}_{i}|E|u^{a}_{i}})}-(1+\kappa\delta_{ji})\mathinner{({u^{a}_{ji}|E|u^{a}_{ji}})}\\}$
(6.35)

where the supremum ranges over all the projections in the weak closure of the
image of the representation
$\pi_{i}[\mathcal{A}]^{\prime\prime}\subset\mathcal{O}[\mathsf{H}_{i}]$.

The precise formula for the option price depends on the range of projections
available in the von Neumann algebra $\pi_{i}[\mathcal{A}]^{\prime\prime}$.
The supremum over all the projections in $\mathcal{O}[\mathsf{H}_{i}]$ can be
calculated explicitly, however, as it depends only on the two dimensional
subspace spanned by the unit vectors $u_{i}^{a}$ and $u_{ji}^{a}$. This gives
an upper bound for the option price:

$\displaystyle\frac{o_{ji}^{a}[k]}{s^{a}p_{j}^{a}}$
$\displaystyle\leq\mathop{\mathrm{t}r}[(F\mathinner{|{u^{a}_{i}})}\mathinner{({u^{a}_{i}}|}-K\mathinner{|{u^{a}_{ji}})}\mathinner{({u^{a}_{ji}}|})^{+}]$
(6.36)
$\displaystyle=\frac{1}{2}(F-K)+\frac{1}{2}\sqrt{(F-K)^{2}+4FK\sigma^{2}}$

parametrised by the forward $F$, strike $K$ and volatility $\sigma$:

$\displaystyle F$ $\displaystyle=1+f\delta_{ji}$ (6.37) $\displaystyle K$
$\displaystyle=1+\kappa\delta_{ji}$ $\displaystyle\sigma^{2}$
$\displaystyle=1-\left|\mathinner{({u^{a}_{i}|u^{a}_{ji}})}\right|^{2}$

The option has zero volatility when the vectors are parallel and maximum
volatility when the vectors are perpendicular, and in general the volatility
is the absolute sine of the angle between the vectors.

##  Part II Quantum duality

###  Chapter 7 Quantum groups

The elements of the model of information as presented in the first part of
this thesis comprise a pair of dual $\ast$-Hopf algebras, the measures and
functions on a measurable group, brought together by a partially defined
pairing, the integral of the function in the measure. These $\ast$-Hopf
algebras do not exist in isolation, but instead form the objects of symmetric
monoidal categories generated as the images of functors on the cartesian
monoidal category of measurable groups. The coherence of the categorification
ensures that the static model can be expanded to the discrete time setting
and, via appropriately constructed limits, the continuous time setting.

Abstracting this categorification, the theory is generalised to arbitrary
symmetric monoidal categories. More precisely, the financial calculus is
constructed from a pair of dual functors from a cartesian monoidal category,
the _factor_ category, to a symmetric monoidal category, the _observable_
category. The factor category is further assumed to include natural
transformations that represent the $\ast$-Hopf algebra, and the component
arrows of these natural transformations are carried to the observable category
via the functors. The pairing is then a natural transformation from the
monoidal product of the dual functors to the trivial functor. The financial
calculus is thus identified with the category of quantum groups.

In the finance application, one functor is utilised to generate the
observables and one functor is utilised to generate the states. The model for
pricing derivative securities is constructed from a martingale condition that
depends on a driving process drawn from the states and a deflator process
drawn from the observables. The principles of replicability and absence of
arbitrage are respected, and the further properties of the pricing model
derive from, and in turn illuminate, the properties of the corresponding
quantum group.

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