$r(A)$, the spectral radius of $A$.

**Proof.** By theorem 1.3.4 and lemma 1.3.4 we have that
$$[r(A)]^n = r(A^n) \le \|A_n\|_{\#} \Rightarrow r(A) \le \|A_n\|_{\#}^{1/n}, n \in \mathbb{N} \Rightarrow r(A) \le \#\text{-lim } \inf_{n \to \infty} \|A_n\|_{\#}^{1/n}.$$

For the other direction, examine again the function from lemma 1.3.5, but this time
restricted to $\Omega = \{z \in \mathbb{C}_c^\# | z > r(A)\}$. We know that $fAy$ is analytic in $\Omega \subset \varphi(A)$. So
it has laurent expansion $\text{Ext}-\sum_{n \in \mathbb{N}} a_n z^n$ and also that $\#\text{-lim}_{z \to \infty} f_{A_\gamma}(z) = 0$. So in
fact, we have laurent expansion $\text{Ext}-\sum_{n \in \mathbb{N}} \frac{a_n}{z^n}$. To determine the coefficients we

know that for $z \in \Omega$, $\left\| \frac{A}{z} \right\|_{\#} < 1$ and hence, by lemma 1.3.1 one obtains
$$ \frac{1}{z-A} = \frac{1}{z(1-z^{-1}A)} = \frac{1}{z} \left( \text{Ext} - \sum_{n=0}^{\infty} \frac{A^n}{z^n} \right) = \text{Ext} - \sum_{n=0}^{\infty} \frac{A^n}{z^{n+1}} = \text{Ext} - \sum_{n=1}^{\infty} \frac{A^{n-1}}{z^n} $$

Hence, $f_{A_\gamma}(z) = \gamma(\frac{1}{z-A}) = \gamma(\text{Ext}-\sum_{n=1}^{\infty} \frac{A^{n-1}}{z^n}) = \text{Ext}-\sum_{n=1}^{\infty} \frac{\gamma(A^{n-1})}{z^n}$.

So we have $\lim_{n \to \infty} \frac{\gamma(A^{n-1})}{z^n} = 0$, for all functionals $\gamma \in H^{\#*}$. It follows that

$\#\text{-lim}_{n \to \infty} \frac{\|A^{n-1}\|_{\#}}{|z|^n} = 0$ and so $\forall z \in \Omega |z| > \#\text{-lim sup}_{n \to \infty} \|A_n\|_{\#}^{1/n}$

then, $\forall z \in \#\bar{\Omega} |z| \ge \#\text{-lim sup}_{n \to \infty} \|A_n\|_{\#}^{1/n}$. In particular then,
$r(A) \ge \#\text{-lim sup}_{n \to \infty} \|A_n\|_{\#}^{1/n}$ and so we are done.

**Remark 1.3.1.** If $A$ is self adjoint, $\|A^2\|_{\#} = \|A\|_{\#}^2$ so by induction $\|A^{2n}\|_{\#} = \|A\|_{\#}^{2n}$ and therefore $r(A) = \#\text{-lim}_{n \to \infty} \|A^{2n}\|_{\#}^{1/2n} = \|A\|$.

## §1.4. $l_2^\#(G)$ and $B(l_2^\#(G))$.

**Definition 1.4.1.** Let $G$ be a discrete, $*$-countable group. Then define

$$l_2^\#(G) = \{f:G \to \mathbb{C}_c^\# \mid \text{Ext}-\sum_{g \in G} \langle f(g) \rangle^2 < \infty\} \quad (1.4.1)$$

This is a non-Archimedean Hilbert space with respect to the inner product

$$\langle f | h \rangle_{\#} = \text{Ext}-\sum_{g \in G} f(g) \bar{h}(g). \quad (1.4.2)$$

**Lemma 1.4.1.** Let $g \in G, f \in l_2^\#(G)$ then define $g * f \in l_2^\#(G)$ by $g * f(h) = f(g^{-1}h)$. This defines a group action on $l_2^\#(G)$.

**Proof.** Fix $f \in l_2^\#(G)$. We verify directly, $\forall h, g_1, g_2 \in G$ :
$$(g_1 \cdot g_2) * f(h) = f((g_1 \cdot g_2)^{-1}h) = f(g_2^{-1}g_1^{-1}h) = g_2 * f(g_1^{-1}h) = g_1 * (g_2 * f(h)).$$

**Definition 1.4.1.** Let $g \in G$, we define $T_g \in B(l_2^\#(G))$ as $T_g:l_2^\#(G) \to l_2^\#(G)$, $f \mapsto g * f$.
Where $g * f$ is the group action as in lemma 1.4.1.

**Lemma 1.4.2.** $T_g$ has the following properties:

(i) $T_{g_1} \cdot T_{g_2} = T_{g_1 \cdot g_2}$ (ii) $T_g^* = T_{g^{-1}}$.

**Proof.** (i) This follows clearly from lemma 1.4.1.

(ii) Let $f, h \in l_2^\#(G)$. Then,
$$ \begin{aligned} \langle T_g f | h \rangle_{\#} &= \text{Ext}-\sum_{a \in G} T_g f(a) \overline{h(a)} = \text{Ext}-\sum_{a \in G} g * f(a) \overline{h(a)} = \\ &= \left[ \text{Ext}-\sum_{a \in G} f(g^{-1}(ga)) \overline{h(a)} \right] \left[ \text{Ext}-\sum_{a \in G} f(g^{-1}a) \overline{h((ga))} \right] = \\ &= \text{Ext}-\sum_{a \in G} f(a) \overline{g^{-1} * h(a)} = \text{Ext}-\sum_{a \in G} f(a) \overline{T_{g^{-1}}h(a)} = \langle f | T_{g^{-1}}h \rangle_{\#}. \end{aligned} $$

## §1.5. Topologies on $B(H^\#)$.

In order to study a non-Archimedean von Neumann algebras, one needs to look into
useful topologies on $B(H^{\#})$. Since all operators are bounded in $\tilde{\mathbb{R}}_c^{\#}$ we have the operator #-norm and therefore the induced topology.

**Definition 1.5.1.** #-Norm Topology.

Using this norm, we can define a metric topology, using the induced $\tilde{\mathbb{R}}_c^{\#}$-valued metric
$d:B(H^{\#}) \times B(H^{\#}) \to R, d(T_1, T_2) = \|T_1 - T_2\|_{\#}.$

This topology is useful for many reasons, but for the purposes of looking at non-
Archimedean von Neumann Algebras is somehow too “fine”. We need coarser
topologies to enable us to have nice examples.

**Definition 1.5.2.** Strong Operator Topology (s.o.t.)

We define the strong operator topology as the coarsest topology such that $\forall v \in H^\#$
the map $\psi_v:B(H^\#) \to \tilde{\mathbb{R}}_c^\#$, $T \mapsto \|Tv\|_\#$ is #-continuous.

**Example 1.5.1.** For $H^\# = l_2(N)$, let $T_n:H^\# \to H^\#$, $v \mapsto (v, e_n)e_n$. We have that $T_n \to_\# 0$ in the s.o.t. but not in the #-norm topology.

$T_n \to 0$ in #-norm sense, since $\| T_n \|_{\#} \ge \| T_n(e_n) \|_{\#} = \| e_n \| = 1 \forall n$.

However in the strong operator topology, we have that

$$
T_n \to_\# 0 \Leftrightarrow \psi_v(T_n) \to_\# 0 \quad \forall v \in H^\#
$$

In this case, $v \in '2$, so in particular the entries of $v$ always tend to zero, i.e.

$$
\psi_v(T_n) \to_\# 0 \quad \forall v \in H^\#.
$$

This distinguishes the strong and the #-norm topologies. Making use of the adjoint, we
define a finer topology

**Definition 1.5.3.** Strong-\*Operator Topology (s\*.o.t.)

We define the strong-\* operator topology as the coarsest topology such that $\forall v \in H^\#$
the two maps

$$
\psi_v:B(H^\#) \to \tilde{\mathbb{R}}_c^\#, T \mapsto \|Tv\|_\# \quad (1.5.1)
$$

and

$$
\psi_v^*:B(H^\#) \to \tilde{\mathbb{R}}_c^\#, T \mapsto \|T^*v\|_\# \quad (1.5.2)
$$

are both #-continuous.

And finally, making use of the inner product we define the weak operator topology:

**Definition 1.5.4.** Weak Operator Topology (w.o.t.)

We define the weak operator topology (w.o.t.) as the coarsest topology such that
$\forall v, w \in H^\#$ the map $\psi_{vw}:B(H^\#) \to \tilde{\mathbb{R}}_c^\#$, $T \mapsto |\langle v, Tw \rangle_\#|$ is #-continuous.

**Lemma 1.5.1.** For the topologies as in Definitions 1.5.1, 1.5.2, 1.5.3 and 1.5.4 we have that w.o.t. < s.o.t. < s-\*.o.t < #-norm topology.

**Lemma 1.5.2.** A basis for the strong operator topology is given by

$$
\beta = \{N(A, \{v_i\}_{i=1}^N), A \in B(H^\#), v_i \in H^\# > 0\}
$$

$$
\text{where } N(A, \{v_i\}_{i=1}^N) := \{B \in B(H^\#) \mid \|(A-B)v_i\|_\# < i = 1, 2, \dots, N\}
$$

**Proof.** First we need to check it is a basis for a topology.

(i) It covers $B(H^\#)$ since for example

$$
N(A, \{0\}, 1) = \{B \in B(H^{\#}) \mid \|(A - B)0\|_{\#} < 1\} = \{B \in B(H^{\#}) \mid \|0\|_{\#} < 1\} = B(H^{\#}).
$$

(ii) It is closed under intersection since for $C \in N(A, \{v_i\}_{i=1}^N, \varepsilon) \cap N(B, \{\hat{v}_i\}_{i=1}^N, \hat{\varepsilon})$.

We have that $C \in N(C, \{v_i\}_{i=1}^N \cup \{\hat{v}_i\}_{i=1}^N, \min\{\varepsilon - \|(C-A)v_i\|_\#, \hat{\varepsilon}\} - \|(C-B)\hat{v}_i\|_\#)$).

The only thing we need to verify for this is that (w.l.o.g.)

$$
\forall D \in N(C, \{v_i\}_{i=1}^N \cup \{\hat{v}_i\}_{i=1}^N, \min\{\varepsilon - \|(C-A)v_i\|_\#, \hat{\varepsilon}\} - \|(C-B)\hat{v}_i\|_\#),
$$

$$
D \in N(A, \{v_i\}_{i=1}^N, \varepsilon). \text{ This is clear since for all } i
$$
$$ \|(D-A)v_i\|_{\#} \le \|(D-C)v_i\|_{\#} + \|(C-A)v_i\|_{\#} \le \varepsilon - \|(C-A)v_i\|_{\#} + \|(C-A)v_i\|_{\#} = \varepsilon. $$

Now we need to show that for all topologies such that $\forall v \in H^\# \text{ the map } \psi_v:B(H^\#) \to *\tilde{\mathbb{R}}_c^\#, T \mapsto \|Tv\|_{\#} \text{ is #-continuous, subsets of this form are #-open.}$

Noting that $N(A, \{v_i\}_{i=1}^N, \lambda) = \bigcap_{i=1}^N \psi_{v_i}^{-1}(\|Av_i\|_{\#} - \varepsilon, \|Av_i\|_{\#} + \varepsilon)$. This is clear.

**Lemma 1.5.3.** A basis for the weak operator topology is given by:

$$ \beta = \{N(A, \{v_n\}_{n \le N}, \{w_n\}_{n \le N}, \varepsilon) \mid A \in B(H^\#), \{v_n\}_{n \le N}, \{w_n\}_{n \le N} \subset H^\#, \varepsilon > 0\}. $$

$$ \text{Where } N(A, \{v_n\}_{n \le N}, \{w_n\}_{n \le N}, \varepsilon) = \{B \in B(H^\#) \mid \|(B-A)v_n, w_n\|_{\#} < \varepsilon \text{ for } n \le N\}. $$

We omit the proof of this result. It is similar to the proof of the basis of the SOT.

**Lemma 1.5.4.** Let $f:B(H^\#) \to *\tilde{\mathbb{C}}_c^\#$ be a linear functional. The following are equivalent:

(i) $\exists \{v_n\}_{n \le N}, \{w_n\}_{n \le N} \subset H^\#$, such that $f(A) = \text{Ext-}\sum_{n \le N} \langle Av_n, w_n \rangle_{\#} \quad \forall A \in B(H^\#)$.

(ii) $f$ is #-continious in the weak sense

(iii) $f$ is #-continious in the strong sense

**Proof.** It is clear that the first implies the second, and by lemma 1.5.1 that the second implies the third. Hence all we must show is that for all $f$ is #-continious in the strong sense then we can find $\{v_n\}_{n \le N}, \{w_n\}_{n \le N} \subset H^\#$, such that $\forall A \in B(H^\#)$:

$$ \begin{gather*} f(A) = \text{Ext-}\sum_{n \le N} \langle Av_n, w_n \rangle_{\#} \quad \forall A \in B(H^\#). \text{ Suppose } f \text{ is #-continious in the strong sense,} \\ \text{then the inverse image of the #-open ball in } *\tilde{\mathbb{C}}_c^\# \text{ is #-open in the strong operator} \\ \text{topology. Considering our basis elements, then there is some constant } \kappa > 0 \text{ and} \\ \{v_n\}_{n \le N} \text{ such that } |f(A)|^2 \le \kappa (\text{Ext-}\sum_n \|Tv_n\|_{\#}^2). \text{ Now consider the subspace of} \\ \text{Ext-}H^\# \oplus H^\# \dots \oplus H^\# \text{ given by } \bigoplus_{n \le N} Av_n | A \in B(H^\#) \text{ we can define a linear functional} \\ \text{on this set by } \bigoplus_{n \le N} Av_n \mapsto f(A). \text{ Then by the generalized Riesz representation theorem,} \\ \exists \{w_n\}_{n \le N} \text{ such that } f(A) = \text{Ext-}\sum_{n \le N} (Av_n, w_n) \quad \forall A \in B(H^\#) \text{ as required.} \end{gather*} $$

#2. Non-Archemedean Banach algebras and $C_{\#}^*$-Algebras.

## §2.1. Initial Definitions and #-Continious Functional Calculus.

von Neumann Algebras are a specific type of $C_{\#}^*$ algebra, and so it is important to understand well the theory of $C_{\#}^*$ algebras before non-Archemedean von Neumann Algebras.

**Definition 2.1.1.** A non-Archemedean Banach algebra $A_{\#}$ is a complex algebra over field

$*\tilde{\mathbb{C}}_c^*$ which is a non-Archemedean Banach space under a $*\tilde{\mathbb{R}}_c^*$-valued #-norm which is submultiplicative:

$$ \|xy\|_{\#} \le \|x\|_{\#} \|y\|_{\#} \qquad (2.1.1) $$

for all $x, y \in A_{\#}$.

**Definition 2.1.2.** An involution on a non-Archemedean Banach algebra $A_{\#}$ is a conjugate-linear #-isometric antiautomorphism of order two, usually denoted $x \mapsto x^*$. In other words,

$$ 1.(x^*)^* = x, \|x^*\| = \|x\| $$

$$ 2.(x+y)^* = x^* + y^*, $$

$$ 3.(xy)^* = y^*x^*, $$

$$ 4.(\lambda x)^* = \bar{\lambda}x^*, \text{ for all } x, y \in A, \lambda \in *\tilde{\mathbb{C}}_c^*. $$
**Definition 2.1.3.** Spectrum (of an element of some a non-Archemedean algebra)
Let $A_\#$ be some a non-Archemedean algebra and $a \in A_\#$ we define
$$ \sigma(a) = \{\lambda \in \tilde{\mathbb{C}}_c^\# | a - \lambda 1 \text{ is not invertible} \}. $$

**Definition 2.1.4.** A Banach #-algebra is a non-Archemedean Banach algebra $A_\#$ with an involution. An $C_\#$-algebra is a Banach #-algebra $A_\#$ satisfying the $C_\#$-axiom: for all $x \in A_\#$

$$ \|x^*x\|_{\#} = \|x\|_{\#}^2. \qquad (2.1.2) $$

**Example 2.1.1.** $B(H^\#)$ is a $C_\#^*$ Algebra

We see this is an immediate consequence of lemma 1.4

**Lemma 2.1.1.** $K \subset B(H^\#)$ is a $C_\#^*$ Algebra iff

(i) K is an algebra over $\tilde{\mathbb{C}}_c^\#$

(ii) $K = K^*$

(iii) K is #-closed with respect to the #-norm topology.

**Proof.** It is clear that if $K$ is a $C_\#^*$ algebra it must be closed with respect to the #-norm topology and an algebra. To see the other direction, we note that the only conditions we must check are conditions of #-closure by lemma 1.4 all of the operations work algebraically as they should. We have

1. K is #-closed under taking sums, scalar multiples and products as it is an algebra.

2. K is #-closed under taking adjoints by the second bullet point

3. K is #-closed with respect to the #-norm topology by the third bullet point.

Therefore, K is a $C_\#^*$ algebra.

**Example 2.2.** $K(B(H^\#))$ is a $C_\#^*$ algebra. This follows clearly from lemma 2.1.1 and theorem

1.22, as $K(B(H^\#)) = \{A \in B(H^\#) | \Re(A) < *\infty\} = \mathcal{F}\Re(B(H^\#))$ and $\mathcal{F}\Re(B(H^\#)) = \{A \in B(H^\#) | \Re(A) < *\infty\}$ is a *-algebra.

**Example 2.1.3.** The set $\mathcal{F}\Re(B(H^\#))$ is in general a *-subalgebra of $B(H^\#)$ but is not a $C_\#^*$ algebra if $H^\#$ is hyper infinite. This can be seen by considering an orthonormal basis $\{e_i\}_{i \in X}$ and

considering $p_i$ to be the orthonormal projection into the line spanned by $e_i(p_i(e_j) = \delta_{ij})$ then the hyper infinite sequence $(q_N)_{N \in *\mathbb{N}}$ where $q_N = \text{Ext}-\sum_{i=1}^N p_i$ #-converges in #-norm to the identity, which would not be hyperfinite rank.

As promised, we return to spectral theory, with a more general version of theorem

1.34.

**Theorem 2.1.1.** #-Continious functional calculus

Let $K_1, K_2$ be $C_\#^*$ algebras and $A \in K_1$ normal, then we have:

(i) The map $\psi: C^\#(\sigma(A)) \to K_1f \mapsto f(A)$ is a homomorphism.

(ii) For all $f \in C^\#(\sigma(A))$ we have $\sigma(f(A)) = f(\sigma(A))$

If $\Psi: K_1 \to K_2$ is a $C_\#^*$-homomorphism, then $\Psi(f(A)) = f(\Psi(A))$

This of course raises a few questions, how for example, would one take the square root of

an operator? For the purposes of these notes we don't look too deeply into this, but one

way to define this we can take any sequence $f_n \in *\tilde{\mathbb{C}}_c^\#[z]$ which approximates $f$ locally uniformly well, and take $f(A) := \#\text{-lim}_{n \to *\infty} f_n(A)$.
Most of these definitions we get are intuitive, for example for $f(z) = |z|^2$, we take
$f(A) = A^*A$

§ 2.2 *$\tilde{C}_c^\#$-valued States

**Definition 2.2.1.** If $K$ is a $*$ algebra, a state is a linear $^{*}\tilde{C}_c^\#$-valued functional that is positive and normalised. That is: $\omega:K \to ^{*}\tilde{C}_c^\#$ such that:

(i) $\omega(A^*A) \geq 0 \forall A \in K$

(ii) $\omega(1) = 1$.

**Notation 2.2.1.** We denote the space of all states on A by S#(A).

Throughout the rest of this subsection, $K$ will refer to a $C^\#$ algebra and we will consider states on $K$.

**Example 2.2.2.** Let $K = M_n(*\tilde{C}_c^\#)$, $n \in *\mathbb{N}$ the $n \times n$ matrices with complex coefficients.

Then for all $A$ positive, $\omega(A):K \to *\tilde{C}_c^\#$, $B \mapsto \frac{\mathrm{Ext}\text{-}\mathrm{Tr}(AB)}{\mathrm{Ext}\text{-}\mathrm{Tr}(A)}$.

Where $\mathrm{Ext-Tr}(C)$ is the external sum of the diagonal entries of $C$ (or equivalently the external sum of the eigenvalues of $C$). Indeed, since $\mathrm{Ext-Tr}(AB) = \mathrm{Ext-Tr}(BA)$ and $\mathrm{Ext-Tr}(A) \ge 0$ if $A$ is positive, letting $A = C^*C$ we see also

$$
\mathrm{Ext-Tr}(AB^*B) = \mathrm{Ext-Tr}(BAB^*) = \mathrm{Ext-Tr}(BC^*CB^*) = \mathrm{Ext-Tr}((CB^*)^*(CB^*)) \ge 0.
$$

So $\omega(A)$ is positive, it is also normalised clearly and therefore a state.

**Definition 2.2.2.** We say that a linear $^{*}\tilde{C}_c^\#$-valued functional $\psi$ is hermitian if

$$
\forall A \in K[\psi(A^*) = \overline{\psi(A)}].
$$

We for some state $\omega$ are interested in the bilinear form $f_\omega:K \times K \to *\tilde{C}_c^\#$,

$(A, B) \mapsto \omega(B^*A).$

This is because it has many properties similar to an inner product. The first we show
is that states are hermitian, which implies something similar to conjugate symmetry
for $f_\omega$.

**Lemma 2.2.1.** Let $\omega \in S^\#(A)$ then $\omega$ is hermitian.

**Proof.** First suppose $A = A^*$ i.e. $A$ is self \#-adjoint. Then let

$$
A_+ = \mathrm{Ext} - \sum_{\lambda \in \sigma(A), \lambda > 0} \lambda p_\lambda, A_- = \mathrm{Ext} - \sum_{\lambda \in \sigma(A), \lambda < 0} (-\lambda) p_\lambda
$$

Noting that both of these are positive, we have that

$$
\omega(A) = \omega(A_+ - A_-) \in *\mathbb{R}_c^\# \Rightarrow \omega(A^*) = \omega(A) = \overline{\omega(A)}.
$$

Then for any $A \in K$ we can write $A = A_1 + iA_2$ where $A_1, A_2$ are both self \#-adjoint.

Then we have

$$
\omega(A^*) = \omega(A_1) - i\omega(A_2) = \overline{\omega(A_1) + i\omega(A_2)} = \overline{\omega(A_1) + i\omega(A_2)} = \overline{\omega(A)}.
$$

**Corollary 2.2.1.** Let $f_\omega$ be the bilinear form as defined before. Then it is conjugate symmetric i.e. $f_\omega(A, B) = f_\omega(B, A)$.

**Proof.** Using that states are hermitian we see clearly

$$
f_{\omega}(BA) = \omega(A^*B) = \omega((A^*B)^*) = \omega(B^*A) = f_{\omega}(A, B).
$$

Next, we show the cauchy schwarz for states.

**Lemma 2.2.2.** (Cauchy Schwarz)

Let $\omega \in S^\#(K)$, then we have, $|\omega(AB^*)|^2 \le \omega(A^*A)\omega(B^*B)$.

**Proof.** If $B=0$ this is clear. Otherwise, by positivity we have for

$$
C = \omega(BB^*)A - \omega(AB^*)B :
$$

$$
\begin{align*}
0 \le \omega(CC^*) &= \omega((\omega(BB^*)A - \omega(AB^*)B)(\omega(BB^*)A^* - \omega(AB^*)B^*)) = \dots = \\
&= \omega(BB^*)(\omega(BB^*)\omega(AA^*) - \omega(AB^*)\omega(BA^*) - \omega(AB^*)\omega(AB^*) + \omega(AB^*)\omega(AB^*)).
\end{align*}
$$
Then using that states are hermitian (lemma 2.13) we can simplify
$$ \dots = \omega(BB^*)(\omega(BB^*)\omega(AA^*) - |\omega(AB^*)|^2). $$

Then by positivity, $\omega(BB^*) \ge 0$ and so $\omega(BB^*)\omega(AA^*) - |\omega(AB^*)|^2 \ge 0$ as required.

**Corollary 2.2.2.** $|f_{\omega}(A, B)| \le \sqrt{f_{\omega}(A, A)f_{\omega}(B, B)}$.

We see now that $f_{\omega}$ is very similar to an inner product, but fails on positive definiteness, as seen in the following example.

**Example 2.2.3.** In $M_2(*\tilde{\mathbb{C}}_c^#)$, we can set $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$. Then as $A$ is positive we can define the state $\omega_A$ as before: $\omega_A:M_2(*\tilde{\mathbb{C}}_c^#) \to *\tilde{\mathbb{C}}_c^#$, $B \mapsto \text{Ext-Tr}(AB)$.

Then for $B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \neq 0$ we have $\omega_A(B^*B) = \omega_A(B) = \text{Ext-Tr}(AB) = \text{Ext-Tr}(0) = 0$.

This motivates the next definition.

**Definition 2.2.3.** We define for each $\omega \in S^#(K)$

$$ J_{\omega} = \{A \in K \mid \omega(A^*A) = f_{\omega}(A, A) = 0\}. $$

The fact that our candidate is not positive definite is not an issue, so long as we can use some devices from abstract algebra (namely quotient objects) to “forget” about the problem areas. For this, we need to find an appropriate ideal of $K$.

**Lemma 2.2.3.** Let $J_{\omega}$ be as before. Then $J_{\omega}$ is a left ideal.

**Proof.** Say $A, B \in J_{\omega}$. Then

$$ (A+B)^*(A+B) \le (A+B)^*(A+B) + (A-B)^*(A-B) = 2A^*A + 2B^*B $$

so $0 \le f_{\omega}(A+B, A+B) \le 2f_{\omega}(A) + 2f_{\omega}(B) = 0$. So $J_{\omega}$ is a #-closed linear subspace.

We also see that for all $A \in J_{\omega}$ and $B \in K(BA)^*(BA) \le \|B\|_#^2 A^*A$, and so $BA \in J_{\omega}$

**Lemma 2.2.4.** If $\omega$ is a positive linear functional on $K$, then the operator #-norm of $\omega$,

$$ \|\omega\|_{\#} = \sup_{A \in K \setminus \{0\}} \frac{\omega(A)}{\|\omega\|_{\#}} \text{ satisfies } \|\omega\|_{\#} = \omega(1). $$

**Proof.** We know that $\|\omega\|_{\#} \ge \omega(1)$ since $\|\mathbf{1}\|_{\#} = 1$. Now let $A \in K \setminus \{0\}$. We have that $\|A+A^*\|_{\#} \mathbf{1} - (A+A^*) \ge 0$ and so $\omega(A+A^*) \le \|A+A^*\|_{\#}\omega(1)$. But also, we have

$$ \left| \omega\left(\frac{A+A^*}{2}\right) \right| \le \frac{|\omega(A)| + |\omega(A^*)|}{2} = |\omega(A)| = \frac{\omega(A+A^*)}{2} + \frac{\omega(A-A^*)}{2} \le \frac{\omega(A+A^*)}{2} = \\ = \left| \frac{\omega(A+A^*)}{2} \right|. \text{ And so we have equality everywhere and that } |\omega(A)| = \left| \frac{\omega(A+A^*)}{2} \right|. $$

Putting this together we have

$$ |\omega(A)| = \left| \frac{\omega(A+A^*)}{2} \right| \le \frac{\|A+A^*\|_{\#} \omega(\mathbf{1})}{2} \le \frac{\|A\|_{\#} + \|A^*\|_{\#} \omega(\mathbf{1})}{2} = \|A\|_{\#} \omega(\mathbf{1}). $$

And so $\|\omega\|_{\#} \le \omega(1)$. In fact this relationship is equivalent.

**Lemma 2.2.5.** Let $\omega$ be a linear functional on $K$. The following statements are equivalent

1. $\omega$ is positive

2. $\|\omega\|_{\#} = \omega(1)$.

**Proof.** 1. ⇒ 2. is lemma 2.18• 2. ⇒ 1.

Let $A$ be positive, and say $\omega(A) = a + ib, a, b \in *\tilde{\mathbb{R}}_c^#$. Then for all $t \in *\tilde{\mathbb{R}}_c^#$ we have: $a^2 + (b+t\|\omega\|_{\#})^2 = \|A+it\|_{\#}^2 \le \|A+it\|_{\#}^2 \|\omega\|_{\#}^2 \le (\|A\|_{\#}^2 + t^2) \|\omega\|_{\#}^2$.

Subtracting $2t \|\omega\|_{\#}$ from both sides we have $2bt \le \|A\|_{\#}^2$ and hence $b = 0$. Then, $\|A\|_{\#} \|\omega\|_{\#} - a = \omega(\|A\|_{\#} - A) = \|\omega\|_{\#} \|\|A\|_{\#} - A\|_{\#} \le \|\omega\|_{\#} \|A\|_{\#}$. So $a \ge 0$.
From the theory so far, we can relate the spectrum of some element $A \in K$ to some
states on $K$.

**Lemma 2.2.6.** Let $A \in K$ then for each $\lambda \in \sigma(A)$ there exists a state $\omega_{A\lambda}: K \to *\tilde{\mathbb{C}}_c^\#$
such that $\omega_{A\lambda}(A) = \lambda$.

**Proof.** We define the linear functional on the subspace $*\tilde{\mathbb{C}}_c^\# \cdot A + *\tilde{\mathbb{C}}_c^\# \cdot 1$ by
$\omega_0(aA + b_1) = a\lambda + b$. It is clear then that $\omega_0(aA + b_1) = a\lambda + b \in \sigma(aA + b1)$ and
hence by lemma **1.29** $1 = \omega_0(1) \le \|\omega_0\|_\# = \sup_{a,b \in *\mathbb{C}_c^\#} \left[ \frac{|a\lambda + b|}{\|aA + b1\|_\#} \le 1 \right]$.

Then by generalized Hahn-Banach theorem, there exists an extension of $\omega_0$ to $K$, $\omega_{A\lambda}$
with $\|\omega_{A\lambda}\|_\# = 1 = \omega_{A\lambda}(1)$ by **lemma 2.19**, $\omega_{A\lambda}$ is a state.

The next lemma shows us how even though we don't have positive definiteness, we
can conclude an equivalence between $A = 0$ and $\omega(A) = 0 \forall \omega$.

**Lemma 2.21.** Let $K$ be a $\mathbb{C}_\#^*$ algebra, and $A \in K$. Then we have
$A = 0 \Leftrightarrow \omega(A) = 0 \forall \omega \in S^\#(K)$.

**Proof.** $\Rightarrow$ is clear by linearity of $\omega$.
$\Leftarrow$ can be seen by the string of implications
$\omega(A) = 0 \forall \omega \in S^\#(K) \Rightarrow \sigma(A) = \{0\} \Rightarrow A = 0$.

We see in fact there are a huge number of results in analogy to those discussed in
subsection **2.1** using that $\sigma(A) \subset \{\omega(A) \mid \omega \in S^\#(K)\}$. For example

**Lemma 2.22.** Let $K$ be a $\mathbb{C}_\#^*$ algebra and let $A \in K$.
(i) $A = A^* \Leftrightarrow \omega(A) \in *\mathbb{R}_c^\# \forall \omega \in S^\#(K)$

(ii) $A \ge 0 \Leftrightarrow \omega(A) \ge 0 \forall \omega \in S^\#(K)$.

**Proof.** (i)-(ii) $\Leftarrow$ follows since $\sigma(A) \subset \{\omega(A) \mid \omega \in S^\#(K)\}$ and $\Rightarrow$ since $\omega$ is hermitian.
(i)-(ii) $\Leftarrow$ follows since $\sigma(A) \subset \{\omega(A) \mid \omega \in S^\#(K)\}$ and $\Rightarrow$ since $\omega$ is positive.

## §2.3. Representations and the generalized Gelfand-Naimark-Segal Construction.

**Definition 2.3.1.** Let $K$ be a $\mathbb{C}_\#^*$ algebra. A representation is a $*$-homomorphism
$\pi:K \to B(H_1^\#)A \mapsto \pi[A]$

**Definition 2.3.2.** Let $K$ be a $\mathbb{C}_\#^*$ algebra, represented in $B(H^\#)$ by $\pi$. Suppose further
that $H_0^\# \subset H^\#$ is a subspace such that $\{\pi[A]H_0^\#\}A \in K \subset H_0^\#$ (i.e. $\pi$ is stable in $H_0^\#$).
Then the restriction of $\pi$ to this subspace, $\pi_0:K \to B(H_0^\#)A \mapsto \pi[A]$ is called a
subrepresentation.

**Example 2.3.1.** For all representations $\pi$ we always have the trivial subrepresentations
where we restrict the domain of $\pi[A]$ to $\{0\}$ or $H^\#$

**Definition 2.3.3.** A representation $\pi:K \to B(H_1^\#)A \mapsto \pi[A]$ is called irreducible if the only
subrepresentations are the restrictions to $\{0\}$ or $H^\#$ there are no nontrivial
subrepresentations.

**Definition 2.3.4.** (Equivalent Representations)
Say $\pi_1:K \to B(H_1^\#)A \mapsto \pi_1[A]\pi_2:K \to B(H_2^\#)A \mapsto \pi_2[A]$

Are two representations of the same $\mathbb{C}_\#^*$ algebra such that there exists a unitary linear
map $U:H_1^\# \to H_2^\#$ such that $\forall A \in K: U\pi_1(A) = \pi_2 U(A)$
Then they are called equivalent.

**Example 2.3.2.** (Direct Sum over representations)
Say $\pi_i:K \to B(H_i^\#)$, $i \le N \in *\mathbb{N}$ are a finite or hyperfinite family of representations.
Then we can define a representation $\pi:K \to B(Ext-\oplus_i H_i^\#) A \mapsto \pi[A]$ where if $v$ is
uniquely decomposed into $Ext-\sum_i v_i$ (where each $v_i \in H_i^\#$) we have
$\pi[A](v) = \sum_i \pi_i[A](v_i)$. Then for each $i$ we have a subrepresentation equivalent to the
representation in $H_i^\#$, given by the restriction of $\pi$ to the subspace
$0 \oplus \dots \oplus 0 \oplus H_i^\# \oplus 0 \oplus \dots \oplus 0$. We can imagine representations like this in terms of
hyperfinite matrices:

$$
\pi[A](v) =
\begin{bmatrix}
\pi_1[A] & 0 & \dots & 0 \\
0 & \pi_2[A] & \dots & 0 \\
0 & 0 & \dots & 0 \\
0 & 0 & \dots & \pi_N[A]
\end{bmatrix}
\begin{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_N
\end{bmatrix}
$$

We explore this concept later in greater detail.

**Lemma 2.3.1.** Let $\pi:K \to B(H^\#)$ be a representation and say $v \in H^\#$ has #-norm 1.
Then the map $\omega_v:K \to *C_c^\#A \mapsto (\pi[A]v,v)H^\#$ defines a state on K

**Proof.** It is clear that $\omega_v$ is linear and by cauchy schwarz we have

$$
|\omega_v(A)| \le \sqrt{\|\pi[A]v\|_{\#} \|v\|_{\#}} \le \sqrt{\|\pi[A]v\|_{\#} \|v\|_{\#}^2} = \sqrt{\|\pi[A]v\|_{\#}} \|v\|_{\#}.
$$

So $\omega_v$ is bounded in $*\mathbb{R}_c^\#$. It is positive since $\langle\pi[A^*A]v, v\rangle = \langle\pi[A]v, \pi[A]v\rangle = \|\pi[A]v\|_\#^2$.
And so by **Lemma 2.19** we have

$$
\|\omega_v\|_{\#} = \langle \pi[1]v, v \rangle_{\#} = \langle v, v \rangle_{\#} = \|v\|_{\#}^2 = 1 \text{ as required.}
$$

In fact, every state on K arises in this fashion, as shown in the GNS construction.
We break down the proof of the GNS construction into a few lemmas.

**Lemma 2.3.2.** The non-Archimedean Hilbert space completion of the space $K/J_\omega$
with respect to the $^*C_c^\#$-valued inner product

$$
\langle \cdot | \cdot \rangle_{\#} : K/J_{\omega} \times K/J_{\omega} \to {}^*C_{c}^{\#}([A], [B]) \mapsto \omega(B^{*}A) = f_{\omega}(A, B)
$$

is a non-Archimedean Hilbert space.

**Proof.** We have seen in lemma 2.17 that $J_\omega$ is a left ideal. Therefore this quotient
object makes sense, and furthermore the inner product is well defined. It is clearly
linear in the first argument, as well as positive definite by lemma 2.14 and conjugate
symmetric by corollary 2.13.1. Therefore it is an inner product on the quotient
space, and the Hilbert space completion defines a Hilbert space clearly.

**Definition 2.3.5.** Given a $C_{\#}^*$ algebra $K$ and a state $\omega$, we define the Hilbert space completion of $K/J_{\omega}$ with respect to the inner product to be $L_2^{\#}(K, \omega)$.

**Lemma 2.3.3.** Given $K, \omega$ as before, we can define a representation $\pi:K \to B(L_2^\#(K, \omega))$
Such that $\omega(A) = (\pi[A]1_\omega \mid 1_\omega)$. Where $1_\omega \in L_2^\#(K, \omega)$ is the unit cyclic vector

**Proof.** For $A \in K$ we consider the map $\pi_0(A):K/J_\omega \to K/J_\omega[B] \mapsto [AB]$.

It is clear to see since $J_\omega$ is a left ideal that this is well defined and since
$\|\pi_0[A](B)\|_\#^2 = \omega((AB)^*(AB)) \le \|A\|_\#^2\omega(B^*B)$ this extends to a bounded in $^*\mathbb{R}_c^-$ operator
$\pi(A) \in B(L_2^\#(K, \omega))$. Then we have that the map $\pi:K \to B(L_2^\#(K, \omega)), A \mapsto \pi(A)$.

Is a homomorphism clearly but moreover for all $C,D \in K, ([C]\omega|\pi(A^*)(D]\omega) =$
$= \omega(D^*A^*C) = (\pi(A)[C]\omega|[D]\omega)$. And so $\pi$ is a *-homomorphism. Also, since
$1_\omega = [1]_\omega \in K/J_\omega$, we have $(\pi(A)1_\omega \mid 1_\omega) = \omega(1_\omega^*A1_\omega) = \omega(A)$.

**Theorem 2.33.** (The non-Archimedean GNS construction)

Let $K$ be a $C_{\#}^*$ algebra. For every state $\omega \in S^{\#}(K)$ then there is a
non-Archimedean Hilbert space $L_2^{\#}(K, \omega)$ and a unique (up to equivalence)
representation $\pi:K \to B(L_2^{\#}(K, \omega))$ such that $\omega(A) = (\pi[A]1_{\omega} \mid 1_{\omega}), A \in K$.
Where $1_{\omega} \in L_2^{\#}(K, \omega)$ is the unit cyclic vector.

**Proof.** By lemma 2.32 it remains to show uniqueness.

Say $\rho:A \rightarrow B(H^{\#})$ is representation with $\iota \in H^{\#}$ cyclic and $\omega(A) = \langle\rho(A)\iota, \iota\rangle_{\#}$ then we can consider the map $U_0:K/J_{\omega} \rightarrow H^{\#}, [A] \mapsto \rho(A)\iota$. We then would have $(U_0(A), U_0(B)) = (\rho(A)\iota, \rho(B)\iota) = (\rho(B^*A)\iota, \iota) = \omega(B^*A) = ([A][B])$.

So $U_0$ is well defined, and an isometry. Furthermore for any $A, B \in K$ we have $U_0(\pi(A)[B]\omega) = U_0([AB]\omega) = \rho(AB)\iota = \rho(A)U_0([B])$.

So $U_0$ extends to an isometry

$U:L_2^{\#}(K, \omega) \rightarrow H^{\#}$ suth that $\forall A \in K: U\pi(A) = \rho(A)U$

Since $\iota$ is cyclic, and we have $\rho(K)\iota \subset U(L_2^{\#}(K, \omega))$ it follows that $U$ must be unitary. The following corollary tells us that we can think of any $C_{\#}^{*}$ algebra as a subset of $B(H^{\#})$ for some $H^{\#}$.

**Corollary 2.33.1.** Let $K$ be a $C_{\#}^{*}$ algebra. Then there exists a faithful representation of $K$.

**Proof.** Let $\pi$ be the direct sum over all GNS representations corresponding to states. Then by **lemma 2.21** this representation is faithful.

This result is very deep, and shows that there is a one to one correspondence.

## §2. Non-Archemedean Banach algebras and $C_{\#}^{*}$-Algebras.

# Chapter XII. Operators on a non-Archimedean Hilbert space over non-Archimedean field* $\tilde{\mathbb{R}}_{C^{+}}^{\#}$.

## §1. Introduction

We study the sum $A + B$ of two \#-selfadjoint operators on a non-Archimedean Banach spaces, and we find sufficient conditions for $C = A + B$ to be \#-selfadjoint.

Our technique is to approximate $B$ by a hyperinfinite sequence of bounded \#-selfadjoint operators $B_n, n \in \mathbb{N}$ and so to approximate $C$ by \#-selfadjoint operators $C_n = A + B_n$. We answer three questions separately:

1. When do the operators $C_n$ have a \#-lim $C$? 2. When is $C$ a \#-selfadjoint operator?

3. When is $C = A + B$?

In Theorem 8 we give a set of estimates on the relative size of $A$ and $B$ which ensure a positive answer to all three questions. Hence these estimates show that $A + B = C$ is \#-selfadjoint. In another paper [5], we use Theorem 2.8 to prove the existence of a self-interacting, causal quantum field in 4-dimensional space-time. Formally this field theory is Lorentz covariant and has non-trivial scattering; this application was the motivation for the present work.

In order to investigate the meaning of \#-lim$_{n \to \infty}$ $C_n$, we give a new definition for the strong \#-convergence of a hyperinfinite sequence of operators. Consequences of this definition are worked out in Section 2. In Section 3 we give estimates on operators $C_n$ which are sufficient to ensure that the \#-lim$_{n \to \infty}$ $C_n = C$ exists and that $C$ is maximal symmetric or \#-selfadjoint. This result is given in Theorem 5 and Corollary 6.

In Section 4 we investigate whether \#-lim$_{n \to \infty}$ $C_n = C$ is equal to $A + B$.

We combine this work in Theorem 8, our second main theorem, where $B$ is
a singular, but nearly positive #-selfadjoint perturbation of a positive #-selfadjoint operator A. To illustrate this theorem, let $A \ge I$ and let B be essentially #-selfadjoint on

$$D^{\#} = \bigcap_{n \in \mathbb{N}} D(A^n). \qquad (1.0)$$

Assume now that, for some $\beta > 0$ and some $\alpha$,

$$A^{-(1-\beta)}BA^{-(1-\beta)} \text{ and } A^{\beta}BA^{\alpha} \qquad (1.1)$$

are #-densely defined, bounded operators. Also, for some positive $a, \varepsilon \in *\tilde{\mathbb{R}}_{c+}^{\#}$ satisfying $2a + \varepsilon < 1$, suppose that there is a constant $b \in *\tilde{\mathbb{R}}_{c}^{\#}$ such that, as bilinear forms on $D \times D$,

$$0 \le aA + B + b \qquad (1.2)$$

and

$$0 \le \varepsilon A^2 + [A^{1/2}, [A^{1/2}, B]] + b. \qquad (1.3)$$

Then $A + B$ is #-selfadjoint.

We see from this example that neither the operator B nor the bilinear form B need be bounded relative to A.

While it may not appear evident, the conditions (1.1)-(1.3) are closely related to a more easily understandable estimate on $D^{\#} \times D^{\#}$,

$$A^2 + B^2 c(A+B)^2 + c. \qquad (1.4)$$

In fact, estimates (1.1)-(1.3) are chosen because they allow us not only to prove (1.4), but also the similar inequality where B is replaced by $B_n$.

Let us now see that if $A + B$ is #-selfadjoint, then (1.4) must hold for every vector in $D(A + B) = D(A) \cap D(B)$.

**Proposition 1.1.** Let A and B be #-closed operators. Then A + B is #-closed if and only if there is a constant $c \in *\tilde{\mathbb{R}}_{c}^{\#}$ such that for all $\psi \in D(A + B)$

$$\|A\psi\|_{\#} + \|B\psi\|_{\#} \le \|(A+B)\psi\|_{\#} + c\|\psi\|_{\#} \qquad (1.5)$$

and (1.5) is equivalent to (1.4) on $D(A + B) \times D(A + B)$.

**Proof:** Certainly (1.5) implies that $A + B$ is #-closed. Conversely, assume that $A + B$ is #-closed and introduce the #-norms on $D(A + B) = D(A) \cap D(B)$,

$$\|\psi\|_{\#1} \triangleq \|\psi\|_{\#} + \|A\psi\|_{\#} + \|B\psi\|_{\#} \qquad (1.6)$$

and

$$\|\psi\|_{\#2} \triangleq \|\psi\|_{\#} + \|(A+B)\psi\|_{\#} \qquad (1.7)$$

Then $D(A + B)$, $\|\cdot\|_{\#2}$ is a non-Archimedean Banach space because $A + B$ is #-closed. The identity map from $D(A + B)$, $\|\cdot\|_{\#2}$ to $D(A + B)$, $\|\cdot\|_{\#1}$ has a #-closed graph because $A, B$, and $A + B$ are #-losed. By the #-closed graph theorem, the identity map is #-continuous; hence

$$\|\psi\|_{\#1} \le c\|\psi\|_{\#2}. \qquad (1.7')$$

**Proposition 1.2.** Let $A \ge I, B$ be #-selfadjoint operators with $D^{\#} \subset D(B)$ and suppose (1.2) and (1.3) hold. Then (1.4) is valid on $D^{\#} \times D^{\#}$.

**Proof** The operators $A^2, B^2, AB, BA$, and $A^{1/2}BA^{1/2}$ define bilinear forms on $D^{\#} \times D^{\#}$. Using (1.2) and (1.3), we have the inequality:

$$A^2 + B^2 = (A + B)^2 - 2A^{1/2}BA^{1/2} - [A^{1/2}, [A^{1/2}, B]] \le (A + B)^2 + (2a + \varepsilon)A^2 + 2Ab + b \quad \text{which establishes (1.4).}$$
## §2. Strong #-Convergence of Operators

Let $\mathcal{L}(C)$ be the graph of the operator $C$. For any hyperinfinite sequence $\{C_n\}$, $n \in *\mathbb{N}$ of #-densely defined operators we define

$$ \mathcal{L}^{*\infty}(C) = \{\phi, \chi | \phi = \#\lim_{n \to *\infty} \phi_n, \phi_n \in D(C_n), \chi = \#\lim_{n \to *\infty} C_n \phi_n\}. \quad (8) $$

In general, $\mathcal{L}^{*\infty}$ will not be the graph of an operator. If the hyperinfinite sequence $\{C_n^*\}$, $n \in *\mathbb{N}$ #-converges strongly on a #-dense domain $D$ to an operator $C^*$, namely,

$$ C^*\psi = \#\lim_{n \to *\infty} C_n^*\psi, \psi \in D, $$

then $\mathcal{L}^{*\infty}$ is the graph of some operator $C^*$. In particular, if each $C_n$ is self #-adjoint, and if the $C_n$ #-converge on a #-dense set $D$ to an operator $C$ defined on $D$, then $\mathcal{L}^{*\infty} = \mathcal{L}^{*\infty}(C^{*\infty})$ and $C^{*\infty}$ is a symmetric extension of $C$.

**Definition 2.1.** G #-CONVERGENCE. The hyperinfinite sequence of operators $C_n, n \in *\mathbb{N}$ #-converge strongly to $C^{*\infty}$ in the sense of graphs, written

$$ C_n \to_{\#G} C^{*\infty} \quad (8') $$

if $\mathcal{L}^{*\infty}$ is the graph of a #-densely defined operator $C^{*\infty}$.

**Remark 2.1.** Note that for a hyperinfinite sequence of uniformly bounded operators $\{C_n^*\}_{n \in *\mathbb{N}}$ such that $C_n \to_{\#G} C^{*\infty}$, $C^{*\infty}$ is the usual strong #-limit of the operators $C_n, n \in *\mathbb{N}$ and is everywhere defined.

**Definition 2.2.** R #-CONVERGENCE. Let the resolvents $R_n(z) = (C_n - z)^{-1}, n \in *\mathbb{N}$ exist for some $z \in *\tilde{C}_c^\#$, and be uniformly bounded in $n$. The operators $C_n$ #-converge strongly to $C^{*\infty}$ in the sense of resolvents, written

$$ C_n \to_{\#R} C^{*\infty} \quad (8'') $$

if the resolvents $R_n(z)$ #-converge strongly to an operator $R(z)$, which has a #-densely defined inverse.

**Remark 2.2.** Note that in that case, the operator $C^{*\infty} = R^{-1}(z) + z$ exists for all $z \in *\tilde{C}_c^\#$ for which the strong #-limit of the $R_n(z)$ exists, and $R^{-1}(z) + z$ is independent of $z$.

**Remark 2.3.** Note that G #-convergence is weaker than R #-convergence, in the case $C_n = C_n^*$ at least, because, as we shall show, in this case $C_n \to_{\#R} C^{*\infty}$ implies $C_n \to_{\#G} C^{*\infty}$. It seems likely that G #-convergence is strictly weaker than R #-convergence; this could be established by giving an example for which $C_n^* = C_n \to_{\#G} C^{*\infty}$ with $C^{*\infty}$ not maximal symmetric. The importance of G #-convergence is that it is technically easier to verify-and gives less information about the #-limit-than R #-convergence, while automatically selecting the correct domain in the case that R #-convergence also holds. The most familiar examples of G #-convergence occur where there is $C_n$ strong #-convergence on a #-dense domain. A less trivial example occurs where there is $D(C_n)$ is independent of $n$, but apparently

$$ D(C) \cap D(C_n) = \{0\}. $$

We have the following connection between G and R #-convergence for a hyperinfinite sequence of #-selfadjoint operators.

**Proposition 3.** Let $C_n, n \in *\mathbb{N}$ be #-selfadjoint.

(a) The domain $D^{*\infty} = \{\phi | \{\phi, \chi\} \in \mathcal{L}^{*\infty} \text{ for some } \chi\}$ is #-dense in H and only if $C_n \to_{\#G} C^{*\infty}$, and in this case $C^{*\infty}$ is necessarily symmetric.
(b) If $R_n(z) = (C_n - z)^{-1}$, $n \in *\mathbb{N}$ #-converges to a bounded operator $R(z)$ for an unbounded set of $z$'s with $\|zR(z)\|_{\#}$ bounded uniformly in $z \in *\mathbb{C}^{\#}_c$ and $n \in *\mathbb{N}$ and if $C_n \to_{\#G} C^{*}_{\infty}$, then each $R(z)$ is invertible.

(c) If $R_n(z)$ #-converges to an invertible $R(z)$, then $C_n \to_{\#R} C$.

(d) If $C_n \to_{\#R} C$, then $C_n \to_{\#G} C^{*}_{\infty}$, $\mathcal{L}^{*}_{\infty} = \mathcal{L}(C)$, and $C$ is maximal symmetric.

(e) Conversely, if $C_n \to_{\#G} C$, where $C$ is maximal symmetric, then $C_n \to_{\#R} C$.

**Proof.** (a) Suppose that $D^{*}_{\infty}$ is #-dense and let $\{0, \psi\} \in \mathcal{L}^{*}_{\infty}$. Then, for some $\theta_n$ in each $D(C_n)$, $\theta_n \to_{\#} 0$, $C_n\theta_n \to_{\#} \psi$. Let $\phi \in \mathcal{L}^{*}_{\infty}$, $\phi = \#\lim_{n \to *\infty} \phi_n$, $\chi = \#\lim_{n \to *\infty} C_n\phi_n$. Then, $0 = \langle 0, \psi \rangle_{\#} = \#\lim_{n \to *\infty} \langle \theta_n, C_n\phi_n \rangle_{\#} = \#\lim_{n \to *\infty} \langle C_n\theta_n, \phi_n \rangle_{\#} = \langle \psi, \phi \rangle_{\#}$, so $\psi \in D^{\perp}_{\infty}$ and $\psi = 0$. Thus $\mathcal{L}^{*}_{\infty}$ is the graph of an operator $C$ with domain $D^{*}_{\infty}$, and since $D^{*}_{\infty}$ #-dense by hypothesis, $C_n \to_{\#G} C^{*}_{\infty}$. Conversely, if $C_n \to_{\#G} C^{*}_{\infty}$, we have $D^{*}_{\infty} = D(C^{*}_{\infty})$ and, since $C$, is assumed to be #-densely defined, $D^{*}_{\infty}$ is dense. The expectation values

$$ \langle \phi, C^{*}_{\infty} \phi \rangle_{\#} = \#\lim_{n \to *\infty} \langle \phi_n, C_n \phi_n \rangle_{\#} $$

are real and so $C^{*}_{\infty}$ is symmetric.

(b) Let $z_1$ and $z_2$ be any two $z$'s for which $\#\lim_{n \to *\infty} R_n(z)$ exists. Then $R(z)$ satisfies the resolvent equation

$$ R(z_1) - R(z_2) = (z_1 - z_2)R(z_1)R(z_2) $$

from which it follows that $R(z)$ has a range and a null space which are independent of $z$. If the null space is zero, $R(z)$ is invertible, and it is sufficient to show that, for large $|z|,

$$ \|zR(z)\psi + \psi\|_{\#} < \|\psi\|_{\#} $$

Since $D^{*}_{\infty}$ is #-dense, we choose $\phi \in D^{*}_{\infty}$ so that $\|\psi - \phi\|_{\#} < \varepsilon \approx 0$. Then,

$$ \|zR(z)\psi + \psi\|_{\#} \le \|zR(z)(\psi - \phi)\|_{\#} + \|zR(z)\phi + \phi\|_{\#} + \|\phi - \psi\|_{\#} $$

For the $z$ under consideration, $\|zR_n(z)\|_{\#}$ and $\|zR(z)\|_{\#}$ are uniformly bounded in $\tilde{\mathbb{R}}^{\#}_c$ by a constant $M \in *\tilde{\mathbb{R}}^{\#}_{c^+}$. Thus,

$$ \|zR(z)\psi + \psi\|_{\#} \le (M+1)\varepsilon + \|zR(z)\phi + \phi\|_{\#} $$

Since $\{\phi, \chi\} \in \mathcal{L}^{*}_{\infty}$, there exists $\phi_n \in D(C_n)$ such that $\|\phi_n - \phi\|_{\#} \to_{\#} 0$ and $\|C_n\phi_n - \chi\|_{\#} \to_{\#} 0$. Thus $\|C_n\phi_n\|_{\#}$ is uniformly bounded in $n \in *\mathbb{N}$. Thus,

$$ \|zR(z)\phi - \phi\|_{\#} \le \|z(R(z) - R_n(z))\phi\|_{\#} + \|zR_n(z)(\phi_n - \phi)\|_{\#} + \|zR_n(z)\phi_n + \phi_n\|_{\#} + \\ + \|\phi_n - \phi\|_{\#} \le |z|\|(R(z) - R_n(z))\phi\|_{\#} + (M+1)\|\phi - \phi_n\|_{\#} + \|(C_n - z)^{-1}C_n\phi_n\|_{\#} $$

We can choose $|z|$ sufficiently large so that $\|(C_n - z)^{-1}\|_{\#} \le M|z|$ is small, then the last term above is small, uniformly in $n \in *\mathbb{N}$. With this fixed value of $z$, we choose $n$ large enough so that the first two terms are small, and we conclude that $\|zR(z)\phi + \phi\|_{\#}$ is arbitrarily small for large $|z|$ and that the null space of $R(z)$ is zero.

(c) In order to show that $C_n \to_{\#R} C$, we need only show that $C = R(z)^{-1} + z$ is #-densely defined. We show that

$$ \{\mathrm{range}R(z)\}^{\perp} \subset \mathrm{null}R(z) $$

which implies (c). To prove the inclusion we may suppose that $y = \mathrm{Im}(z) \ne 0$, because $R_n(z) + R(z)$ for $z$ in an #-open subset of the complex plane $*\tilde{\mathbb{C}}^{\#}_{c^+}$.
If $\psi \perp \{\text{range}R(z)\}$ and if $\psi \notin \text{null}R(z)$, then

$\|\psi\|_{\#} < \|iyR(z)\psi + \psi\|_{\#} = \#\text{lim}_{n \to *\infty} \|iyR_n(z)\psi + \psi\|_{\#} \le \|\psi\|_{\#}$, which is a contradiction.

(d) Let $C_n \to_{\#R} C$, and let $\{\phi, \theta\} \in \mathcal{L}(C)$. Then, for some $\chi \in H$ and some $z$,

$$ \phi = R(z)\chi = \#\text{lim}_{n \to *\infty} \phi_n = \#\text{lim}_{n \to *\infty} R_n(z)\chi, $$

and

$$ C_n\phi_n = (C_n - z)\phi_n + z\phi_n = \chi + z\phi_n \underset{n \to *\infty}{\to} \chi + z\phi = C\phi. $$

Thus, $\mathcal{L}(C) \subset \mathcal{L}_{*\infty}$ so $D(C) \subset D_{*\infty}$, and since $D(C)$ is \#-dense by assumption, $D_{*\infty}$ is \#-dense also. By

(a), $C_n \to_{\#G} C_{*\infty}$ and $C_{*\infty}$ is a symmetric extension of $C$. However, \#- is closed and it has a resolvent $R(z)$; therefore, $C$ has defect indices $(0, n)$ and is maximal symmetric. Thus, $C_{*\infty} = C$.

(e) Suppose that $\mathcal{L}_{*\infty} = \mathcal{L}(C_{*\infty})$ is the graph of a maximal symmetric operator. For non-real $z$ we have $\|R_n(z)\|_{\#}$ bounded uniformly in $n \in *\mathbb{N}$. Let $\chi \in H^{\#}$, and write $\chi$ in the form $\chi = (C_{*\infty} \pm i)\psi$ for some $\psi \in D(C_{*\infty})$. Then,

$$ \psi = \#\text{lim}_{n \to *\infty} \psi_n, C_{*\infty}\psi = \#\text{lim}_{n \to *\infty} C_n\psi_n, $$

and

$$ \chi = \#\text{lim}_{n \to *\infty} (C_{*\infty} \pm i)\psi_n = \#\text{lim}_{n \to *\infty} \psi_n. $$

Therefore

$$ \#\text{lim}_{n \to *\infty} R_n(\mp i)\chi = \#\text{lim}_{n \to *\infty} R_n(\mp i)\chi_n = \psi = (C_{*\infty} \pm i)^{-1}\chi $$

and so $C_n \to_{\#R} C_{*\infty}$

**Remark 2.4.** In case the \#-limit of the $C_n, n \in *\mathbb{N}$ is actually \#-selfadjoint, there are further connections between $G$ and $R$ \#-convergence.

**Theorem 4.**

(a) $C_n \to_{\#G} C$, and $C = C^*$.

(b) $C_n \to_{\#R} C$, and $C = C^*$.

(c) The hyper infinite sequences $\{R_n(z)\}$ and $\{[R_n(z)]^*\}, n \in *\mathbb{N}$ \#-converge strongly and \#-lim$_{n \to *\infty} R_n(z)$ is invertible for some $z$.

(d) Statement (c) holds for all non-real $z \in *\tilde{C}_c^\#$

### §3. Estimates on a $G$ \#-convergent hyper infinite sequence

In this section we give estimates which are sufficient to assure that it $G$ \#-convergent sequence of operators is $R$ \#-convergent, and that the limit is maximal symmetric or selfadjoint. In order to measure the rate of \#-convergence, we introduce a selfadjoint operator $N \ge I$ and the associated non-Archimedean Hilbert spaces $H_\lambda$ with the scalar product

$$ \langle \psi, \psi \rangle_{\#\lambda} = \langle N^{\lambda/2}\psi, N^{\lambda/2}\psi \rangle_{\#}. \qquad (3.1) $$

By standard identifications we have for $\lambda \ge 0: H_\lambda \subset H_0 \subset H_{-1}$ and $H_0 = H$. If $D: H_\alpha \to H_\beta$ is a \#-densely defined, bounded operator from $H_\alpha$ to $H_\beta$, we let $\|D\|_{\#\alpha,\beta}$ denote its \#-norm. Setting $\|D\|_{\#} = \|D\|_{\#0,0}$ we obtain
$$ \|D\|_{\#\alpha,\beta} = \|N^{\beta/2}DN^{-\alpha/2}\|. \qquad (3.2) $$

Let $C_n, n \in *\mathbb{N}$ be a hyper infinite sequence of #-selfadjoint operators, and consider the following three conditions.

(i) Suppose that $C_n - C_m$ is a #-densely defined, bounded operator from $H_\lambda$ to $H_{-\lambda}$, for some $\lambda$, and that as $n, m \to *\infty$

$$ \|C_n - C_m\|_{\#\lambda, -\lambda} \to 0. \qquad (3.3) $$

(ii) Suppose that, for some $p$ and for an unbounded set of $z = x + iy \in *\mathbb{C}^c$ in the sector $|x| \le \text{const} \times |y|$,

$$ \|R_n(z)\|_{\#\mu,\lambda} \le M(z), \qquad (3.4) $$

where the bound $M(z)$ is uniform in $n \in *\mathbb{N}$.

(iii) Suppose that, for the above $z$'s,

$$ \|R_n(\bar{z})\|_{\#\mu,\lambda} \le M(z). \qquad (3.5) $$

**Theorem 5.** Let $C_n, n \in *\mathbb{N}$ be a hyper infinite sequence of #-selfadjoint operators with a common domain, such that

$$ C_n \to_{\#G} C. $$

If conditions (i) and (ii) hold, then

$$ C_n \to_{\#R} C $$

and $C$ is maximal symmetric.

**Corollary 6.** If in addition to the hypothesis of Theorem 5, condition (iii) also holds, then $C$ is #-selfajoint.

**Remark 3.1.** (1) If $\mu = 0$ in (ii), then the resolvents #-converge uniformly.

(2) If the $C_n$ are uniformly semibounded from below, then we may choose the $z$ in condition (ii) to be infinite large negative numbers. In that case the conclusion of Theorem 5 is that $C_n \to_{\#R} C = C^*$.

# § 4. Estimates for singular perturbations

In this section we consider a singular perturbation $B$ of a #-selfadjoint operator $A$. We give estimates on $B$ which ensure that the sum $A + B$ is #-selfadjoint.

**Abbreviation 4.1.** We abbreviate $A^{\#}$ instead $-\bar{A}$.

**Definition 4.1.** A #-core of an operator $C$ is a domain $D$ contained in $D(C)$ such that $C = (C \upharpoonright D)^{\#}$.

**Lemma 7.** Let $A, A_n, n \in *\mathbb{N}$, $B, B_n, n \in *\mathbb{N}$ and $C_n = A + B_n, n \in *\mathbb{N}$ be #-selfadjoint operators with a common #-core $D$. Assume the hypotheses of Theorem 5 and Corollary 6 for $C_n, n \in *\mathbb{N}$ and suppose also that, for $\theta \in D$,

$$ \|(A - A_n)\theta\|_{\#} + \|(B - B_n)\theta\|_{\#} \to 0 \text{ as } n \to *\infty \qquad (4.9) $$

and

$$ \|A_n\theta\|_{\#}^2 + \|B_n\theta\|_{\#}^2 \le \text{const.} \times \|\theta\|_{\#}^2 + \text{const.} \times \|C_n\theta\|_{\#}^2, \qquad (4.10) $$

with constants independent of $n$. Then $A + B$ is #-selfadjoint and $C_n \to_{\#R} A + B$.

**Remark 4.1.** As hypothesis for our next theorem, our second main result, we assume that $N \le A$ and that $N$ and $A$ commute. Let

$$ D^{*\infty}(A) = \bigcap_{n \in *\mathbb{N}} D(A^n) \qquad (4.11) $$

the elements of $D^{*\infty}(A)$ are called $C^{*\infty}$ vectors for $A$. Assume that $D^{*\infty}(A)$ is a #-core
for the #-selfadjoint operator $B$. Also assume that, as bilinear forms on $D^{*\infty} \times D^{*\infty}$, and for some $\alpha$ and $\varepsilon$ in the indicated ranges,

$$0 \le \alpha N + B + \text{const.}, 0 \le \alpha < 1/2 \qquad (4.12)$$

and

$$0 \le \varepsilon A^2 + \text{const} \times B + [A^{1/2}, [A^{1/2}, B]] + \text{const.}, 2\alpha + \varepsilon < 1. \qquad (4.13)$$

Let $B$ be a bounded operator from $H_v$ to $H_{-v}$ and from $H_\alpha$ to $H_\beta$ for some $\alpha, \beta$ and $v, \beta > 0$ ($H_\alpha$ is defined following Theorem 4.) If $v \ge 2$, assume that for all $\varepsilon > 0$

$$0 \le \varepsilon N^{\mu+2} + [N^{(\mu+1)/2}, [N^{(\mu+1)/2}, B]] + \text{const.} \qquad (4.14)$$

as bilinear forms on $D^{*\infty} \times D^{*\infty}$, for some $\mu > v - 2$.

**Theorem 8.** Under the above hypothesis, $A + B$ is #-selfadjoint.

# Chapter IX.

## §1. Free scalar field

Let $\mathbf{H}^{\#}$ be a #-complex Hilbert space over field $\mathbb{C}^{\#}$ and let $\mathcal{F}(\mathbf{H}^{\#}) = \bigoplus_{n=0}^{\infty} \mathbf{H}_{\#}^{(n)}$

(where $\mathbf{H}_{\#}^{(n)} = \bigoplus_{k=1}^{n} \mathbf{H}^{\#}$) be the Fock space over $\mathbf{H}^{\#}$. Our goal is to

define the abstract free field on $\mathcal{F}_s(\mathbf{H}^{\#})$, the Boson subspace of $\mathcal{F}(\mathbf{H}^{\#})$; to do this we need to introduce several other families of operators and some terminology. Let $f \in \mathbf{H}^{\#}$ be

fixed. For vectors in $\mathbf{H}_{\#}^{(n)}$ of the form $\eta = \psi_1 \otimes \psi_2 \otimes \cdots \otimes \psi_n$ we define a map $b^{-}(f) : \mathbf{H}_{\#}^{(n)} \to \mathbf{H}_{\#}^{(n-1)}$ by

$$b^{-}(f)\eta = (f, \psi_{1})(\psi_{2} \otimes \cdots \otimes \psi_{n}) \qquad (1)$$

$b^{-}(f)$ extends by linearity to finite linear combinations of such $\eta$, the extension is well defined, and $\|b^{-}(f)\eta\| \le \|f\| \times \|\eta\|$. Thus $b^{-}(f)$ extends to a bounded map (of norm $\|f\|$) of $\mathbf{H}_{\#}^{(n)}$ into $\mathbf{H}_{\#}^{(n-1)}$. Since this is true for each $n$ (except for $n = 0$ in which case we define $b^{-}(f) : \mathbf{H}_{\#}^{(0)} \to 0$), $b^{-}(f)$ is in a natural way a bounded operator of norm $\|f\|$ from $\mathcal{F}(\mathbf{H}^{\#})$

to $\mathcal{F}(\mathbf{H}^{\#})$. It is easy to check that $b^{+}(f) = (b^{-}(f))^{*}$ takes each $\mathbf{H}_{\#}^{(n)}$ into $\mathbf{H}_{\#}^{(n+1)}$ with the action

$$b^{+}(f)\eta = f \otimes \psi_{1} \otimes \psi_{2} \otimes \cdots \otimes \psi_{n} \qquad (2)$$

on product vectors. Notice that the map $f \mapsto b^{+}(f)$ is linear, but $f \mapsto b^{-}(f)$ is antilinear.

Let $S_n$ be the symmetrization operators introduced in Section II.4. Then $S = \bigoplus_{n=0}^{\infty} S_n$ is the projection onto the symmetric Fock space $\mathcal{F}_s(\mathbf{H}^{\#}) = \bigoplus_{n=0}^{\infty} S_n \mathbf{H}^{\#(n)}$. We will write $S_n \mathbf{H}^{\#(n)} = \mathbf{H}_s^{\#(n)}$ and call $\mathbf{H}_s^{\#(n)}$ the $n$-particle subspace of $\mathcal{F}_s(\mathbf{H}^{\#})$. Notice that $b^{-}(f)$ takes $\mathcal{F}_s(\mathbf{H}^{\#})$ into itself, but that $b^{+}(f)$ does not. A vector $\Psi = \{\psi^{(n)}\}_{n=1}^{\infty}$ for which $\psi^{(n)} = 0$ for all except finitely many $n$ is called a finite particle vector. We will denote the set of finite particle vectors by $F_0$. The vector $\Omega_0 = \langle 1, 0, 0, \dots \rangle$ plays a special role; it is called the vacuum.
Let A be any self-adjoint operator on $\mathbf{H}^\#$ with domain of essential selfadjointness $D$.

Let $D_A = \{\Psi \in F_0 | \Psi^{(n)} \otimes_{k=1}^n D \text{ for each } n \in \mathbb{N}^\#\}$ and define $d\Gamma^\#(A)$ on $D_A \cap \mathbf{H}_s^{\#(n)}$ as

$$d\Gamma^\#(A) = A \otimes I \cdots \otimes I + I \otimes A \otimes \cdots \otimes I + \cdots + \otimes I \cdots \otimes I \otimes A. \quad (3)$$

Note that $d\Gamma^\#(A)$ is essentially self-adjoint on $D_A$; $d\Gamma^\#(A)$ is called the second quantization of $A$. For example, let $A = I$. Then its second quantization $N = d\Gamma^\#(I)$ is essentially self-adjoint on $F_0$ and for $\psi \in \mathbf{H}_s^{\#(n)}$, $N\psi = n\psi$. $N$ is called the number operator. If $U$ is a unitary operator on $\mathbf{H}^\#$, we define $d\Gamma^\#(U)$ to be the unitary operator on $\mathcal{F}_s(\mathbf{H}^\#)$ which equals $\text{Ext}-\otimes_{k=1}^n U$ when restricted to $\mathbf{H}_s^{\#(n)}$ for $n > 0$, and which equals

the identity on $\mathbf{H}_s^{\#(0)}$. If $\text{Ext}-\exp(itA)$ is a $\#$-continuous unitary group on $\mathbf{H}^\#$, then $\Gamma^\#(\text{Ext}-\exp(itA))$ is the group generated by $d\Gamma^\#(A)$, i.e., $\Gamma^\#(\text{Ext}-\exp(itA)) = \text{Ext}-\exp[itd\Gamma^\#(A)]$.

**Definition 1.1.** We define the annihilation operator $a^-(f)$ on $\mathcal{F}_s(\mathbf{H}^\#)$ with domain $F_0$ by

$$a^-(f) = \sqrt{N+1} b^-(f) \quad (4)$$

$a^-(f)$ is called an annihilation operator because it takes each $(n+1)$-particle subspace into the $n$-particle subspace. For each $\psi$ and $\eta$ in $F_0$,

$$(\sqrt{N+1} b^-(f)\psi, \eta) = (\psi, Sb^+(f)\sqrt{N+1}). \quad (5)$$

Then Eq.(5) implies that

$$(a^-(f))^* \upharpoonright F_0 = Sb^+(f) \sqrt{N+1} \quad (6)$$

The operator $(a^-(f))^*$ is called a creation operator. Both $a^-(f)$ and $(a^-(f))^*$ $\upharpoonright F_0$ are $\#$-closable; we denote their $\#$-closures by $a^-(f)$ and $(a^-(f))^*$ also.

**Example 1.1.** If $\mathbf{H}^\# = L_2^\#(M, d^\#\mu)$, then $\otimes_{i=1}^n L_2^\#(M, d^\#\mu) = L_2^\#(\times_{i=1}^n M, \otimes_{i=1}^n d^\#\mu)$ and that $S \otimes_{i=1}^n L_2^\#(M, d^\#\mu) = L_{2,s}^\#(\times_{i=1}^n M, \otimes_{i=1}^n d^\#\mu)$, where $L_{2,s}^\#$ is the set of functions in $L_2^\#$ which are invariant under permutations of the coordinates. The operators $a^-(f)$ and $(a^-(f))^*$ are given by

$$\begin{aligned} a^-(f)\psi^{(n)}(m_1, \dots, m_n) &= \sqrt{n+1} \left( \text{Ext}-\int_M \tilde{f}(m)\psi^{(n+1)}(m, m_1, \dots, m_n)d^\#\mu \right) \\ a^-(f)^*\psi^{(n)}(m_1, \dots, m_n) &= \frac{1}{\sqrt{n}} \sum_{i=1}^n f(m_i)\psi^{(n-1)}(m_1, \dots, \tilde{m}_i, \dots, m_n) \end{aligned} \quad (7)$$

where $\tilde{m}_i$ means that $m_i$ is omitted. If $A$ operates on $L_2^\#(M, d^\#\mu)$ by multiplication by the $\#\mathbb{R}_c^\#$-valued function $\omega(m)$, then

$$(d\Gamma^\#(A)\psi)^{(n)}(m_1, \dots, m_n) = \left(\sum_{i=1}^n \omega(m_i)\right)\psi^{(n)}(m_1, \dots, m_n) \quad (8)$$

Eq.(6) implies that the Segal field operator $\Phi_S^\#(f)$ on $F_0$ defined by

$$\Phi_S^\#(f) = \frac{1}{\sqrt{2}} [a^-(f) + a^-(f)^*] \quad (9)$$

is symmetric and essentially self-$\#$-adjoint. The mapping from $\mathbf{H}^\#$ to the self-$\#$-adjoint operators on $\mathcal{F}_s(\mathbf{H}^\#)$ given by

$$f \mapsto \Phi_S^\#(f) \quad (10)$$

is called the Segal quantization over $\mathbf{H}^\#$. Notice that the Segal quantization is a real (but not complex) linear map since $f \mapsto a^-(f)$ is antilinear and $f \mapsto a^-(f)^*$ is linear. The following theorem gives the properties of the Segal quantization.
**Theorem 1.1.** Let $\mathbf{H}^{\#}$ be hyper infinite dimensional Hilbert space over field $\mathbb{C}_c = \mathbb{R}_c^{\#} + i\mathbb{R}_c^{\#}$ and $\Phi_S^{\#}(f)$ the corresponding Segal quantization. Then:

(a) (self-adjointness) For each $f \in \mathbf{H}^{\#}$ the operator $\Phi_S^{\#}(f)$ is essentially self-adjoint on $F_0$,
the hyperfinite particle vectors.

(b) (cyclicity of the vacuum) $\Omega_0$ is in the domain of all hyperfinite products $\prod_{i=1}^n \Phi_S^{\#}(f_i), n \in \mathbb{N}^{\#}$

and the set $\{\prod_{i=1}^n \Phi_S^{\#}(f_i)\Omega_0 | f_i \text{ and } n \text{ arbitrary}\}$ is #-total in $\mathcal{F}_s(\mathbf{H}^{\#})$.

(c) (commutation relations) For each $\psi \in F_0$ and $f, g \in \mathbf{H}^{\#}$

$$[\Phi_S^{\#}(f)\Phi_S^{\#}(g) - \Phi_S^{\#}(g)\Phi_S^{\#}(f)]\psi = i \operatorname{Im}(f,g)_{\mathbf{H}^{\#}}\psi. \quad (11)$$

Further, if $W(f)$ denotes the external unitary operator $\operatorname{Ext}\operatorname{exp}(i\Phi_S^{\#}(f))$ then

$$W(f+g) = \left[ \operatorname{Ext}\operatorname{exp}\left( \frac{-i \operatorname{Im}(f,g)_{\mathbf{H}^{\#}}}{2} \right) \right] W(f)W(g) \quad (12)$$

(d) (#-continuity) If $\{f_n\}_{n=1}^{\infty}$ is hyper infinite sequence such as $\#\lim_{n \to \infty} f_n = f$ in $\mathbf{H}^{\#}$,
then: (i) $\#\lim_{n \to \infty} W(f_n)\psi$ exists for all $\psi \in \mathcal{F}_s(\mathbf{H}^{\#})$ and

$$\#\lim_{n \to \infty} W(f_n)\psi = W(f)\psi \quad (13)$$

(ii) $\#\lim_{n \to \infty} \Phi_S^{\#}(f_n)\psi$ exists for all $\psi \in F_0$ and

$$\#\lim_{n \to \infty} \Phi_S^{\#}(f_n)\psi = \Phi_S^{\#}(f)\psi. \quad (14)$$

(e) For every unitary operator $U$ on $\mathbf{H}^{\#}, \Gamma^{\#}(U) : D(\overline{\Phi_S^{\#}(f)}) \to D(\overline{\Phi_S^{\#}(Uf)})$ and for $\psi \in D(\overline{\Phi_S^{\#}(Uf)})$

$$\Gamma^{\#}(U)\overline{\Phi_S^{\#}(f)}\Gamma^{\#}(U)^{-1}\psi = \overline{\Phi_S^{\#}(Uf)}\psi \quad (15)$$

for all $f \in \mathbf{H}^{\#}$.

**Proof.** Let $\psi \in \mathbf{H}_S^{\#(n)}$. Since $\Phi_S^{\#}(f) : F_0 \to F_0$, $\psi$ is in $C^{\infty}(\Phi_S^{\#}(f))$. Further, it follows from Eq.(5)-Eq.(6), and the fact that $\|b^{-}(f)\| = \|f\|$, that

$$\|(a^{\star}(f))^{k}\psi\|_{\#} \le (\operatorname{Ext}-\prod_{i=1}^{k}\sqrt{p+i})\|f\|_{\#}^{k}\|\psi\|_{\#} \quad (16)$$

where $a^{\star}(f)$ represents either $a^{-}(f)$ or $a^{-}(f)^{*}$. Therefore,

$$\|\Phi_S^{\#}(f)^k \psi\|_{\#} \le 2^{k/2}((n+k)!)^{1/2} \|f\|_{\#}^k \|\psi\|_{\#} \quad (17)$$

## References.

[1] J. Foukzon, Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I). Journal of Advances in Mathematics and Computer Science, 36(2), 73–88. https://doi.org/10.9734/jamcs/2021/v36i230339

[2] Foukzon, J. (2021). Set Theory INC# $\infty$# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions. Journal of Advances in Mathematics and Computer Science, 36(4), 90–112. https://doi.org/10.9734/jamcs/2021/v36i430359

[3] J. Foukzon, Set Theory $NC_{\infty}^{\#}$ Based on Bivalent Infinitary Logic with Restricted Modus Ponens Rule. Basic Analysis on External Non-Archimedean Field $\mathbb{R}_c^{\#}$ https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3989960
[2] J. Foukzon, The Solution of the Invariant Subspace Problem. Part I. Complex Hilbert space. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4039068

[3] Foukzon, J. (2022). The Solution of the Invariant Subspace Problem. Complex Hilbert Space. External Countable Dimensional Linear spaces Over Field *ℝ<sup>#</sup>. Part II. Journal of Advances in Mathematics and Computer Science, 37(11), 31-69. https://doi.org/10.9734/jamcs/2022/v37i111721

[4] J. GLIMM and A. JAFFE, Singular Perturbations of Selfadjoint Operators
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL. XXII,
401-414 (1969)

[5] Foukzon, Jaykov, Model $P(\varphi)_4$ Quantum Field Theory. A Nonstandard Approach Based on Nonstandard Pointwise-Defined Quantum Fields (July 14, 2022). Available at SSRN: https://ssrn.com/abstract=4163159 or http://dx.doi.org/10.2139/ssrn.4163159

[6] Jaykov Foukzon, Model $P(\varphi)_4$ Quantum Field Theory. A Nonstandard Approach Based on Nonstandard Pointwise-Defined Quantum Fields
AIP Conf. Proc. 2872, 060028 (2023) https://doi.org/10.1063/5.0162832

[7] Jaykov Foukzon, Set Theory $\mathcal{INC}_{\infty}^{#}$ Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions. Journal of Advances in Mathematics and Computer Science Issue: 2021 - Volume 36 [Issue 4] https://journaljamcs.com/index.php/JAMCS/issue/view/227

[8] A. Robinson, Non-standard analysis. (Reviwed re-edition of the 1st edition of (1966) Princeton: Princeton University Press, 1996.)

[9] Stroyan, K.D., Luxemburg, W.A.J. Introduction to the theory of infinitesimals. New York: Academic Press (1st ed.), 1976

[10] S. Albeverio, J. E. Fenstad, R. Høegh-Krohn, Nonstandard Methods in Stochastic Analysis and Mathematical Physics (Dover Books on Mathematics), February 26, 2009 Paperback: 526 pages ISBN-10: 0486468992, ISBN-13: 978-0486468990

[11] M. Davis, Applied Nonstandard Analysis (Dover Books on Mathematics) ISBN-13: 978-0486442297 ISBN-10: 0486442292

[12] Foukzon, J. (2024). Model $\lambda(\varphi^{2n})_4, n \ge 2$ quantum field theory: A nonstandard approach based on nonstandard pointwise-defined quantum fields.
Jaykov Foukzon 2024 J. Phys.: Conf. Ser. 2701 012113 DOI
10.1088/1742-6596/2701/1/012113

[18] Foukzon, J. (2022). Internal Set Theory IST# Based on Hyper Infinitary Logic with Restricted Modus Ponens Rule: Nonconservative Extension of the Model Theoretical NSA. Journal of Advances in Mathematics and Computer Science, 37(7), 16–43. https://doi.org/10.9734/jamcs/2022/v37i730463

[19] E. Nelson, Internal Set Theory, an axiomatic approach to nonstandard analysis, Bull. Am. Math. Soc., 83:6 (1977) 1165-1198.

[20] Vaught, Robert L. Set theory an introduction. Springer Science & Business Media, 2001.

[21] Raymond M. Smullyan, Set Theory and the Continuum Problem, ISBN-13 978-0486474847

[22] K. Kuratowski, Set Theory, with an Introduction to Descriptive Set Theory, Hardcover Published February 26, 1976 by North-Holland ISBN 9780720404708 (ISBN10: 0720404703)
[23] G. Birkhoff, Lattice Theory, third edition (American Mathematical Society Colloquium Publications, Volume 25) Hardcover – January 1, 1967

[24] Foukzon, J., There Is No Standard Model of ZFC and ZFC2 with Henkin Semantics Advances in Pure Mathematics > Vol.9 No.9, September 2019 https://doi.org/10.4236/apm.2019.99034

[25] Foukzon, J., There is No Standard Model of ZFC and ZFC_2 with Henkin semantics. Generalized Lob's Theorem. Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology. arXiv:1301.5340v15 [math.GM]

[26] Foukzon, J., Relevant First-Order Logic $LP^\#$ and Curry's Paradox Resolution, Pure and Applied Mathematics Journal, Volume 4, Issue 1-1, January 2015
Pages: 6-12 Published: Jan. 19, 2015
https://doi.org/10.11648/j.pamj.s.2015040101.12
arXiv:0804.4818 [math.LO]

[27] Axiomatic Theories of Truth Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/truth-axiomatic/#TypeFreeTrut

[28] Liar Paradox Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/liar-paradox/