Added basics of C*-algebras.

refs.bib

@@ -151,3 +151,24 @@
   year={2001},
   publisher={Springer Berlin Heidelberg}
 }
+
+
+
+@techreport{aronszajn1964interpolation,
+  title={Interpolation spaces and interpolation methods},
+  author={Aronszajn, Nachman and Gagliardo, Emilio},
+  year={1964}
+}
+@book{brudnyi1991interpolation,
+  title={Interpolation functors and interpolation spaces},
+  author={Brudnyi, Yu A and Krugljak, N Ya},
+  volume={47},
+  year={1991},
+  publisher={Elsevier}
+}
+@book{PietschHistory,
+  title={History of Banach spaces and linear operators},
+  author={Pietsch, Albrecht},
+  year={2007},
+  publisher={Springer}
+}

src/fa/interpolation/functors.tex

@@ -0,0 +1,10 @@
+\section{Interpolation Functors}
+\label{section:interpolation-functors}
+
+\textit{"In the presence of so many different interpolation methods it seemed timely to study the general structure of all possible methods: to determine all of them and to analyze the properties which are common to all."} — \cite[Page 51]{aronszajn1964interpolation}. 
+
+\textit{"This is how things appeared in 1965. Fifteen years later, it was found that the number
+of interpolation methods at our disposal is not large."} — \cite[Page vi, Footnote 3]{brudnyi1991interpolation}. 
+
+The above quotes are taken from \cite[Page 427]{PietschHistory}.
+

src/fa/interpolation/index.tex

@@ -0,0 +1,6 @@
+\chapter{Interpolation Spaces}
+\label{chap:interpolation}
+
+\input{./functors.tex}
+
+

src/fa/tvs/complexify.tex

@@ -58,8 +58,8 @@
 \label{definition:complex-conjugation}
     Let $E$ be a vector space over $\complex$ and $*: E \to E$ be a $\real$-linear map, then $*$ is a \textbf{complex conjugation} if:
     \begin{enumerate}
-        \item For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. 
-        \item For each $x \in E$, $x^{**} = x$. 
+        \item[(C1)] For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. 
+        \item[(C2)] For each $x \in E$, $x^{**} = x$. 
     \end{enumerate}
 
     In which case, $\text{Re}(E) = \bracs{x \in E| x^* = x}$ is the \textbf{real part} of $E$. 
@@ -74,6 +74,7 @@
         \[
         \text{Re}(x) = \frac{x + x^*}{2} \quad \text{Im}(x) = \frac{x - x^*}{2i}
         \]
+        \item For each $x \in E$, $x^* = \text{Re}(x) - i\text{Im}(x)$. 
     \end{enumerate}
 \end{proposition}
 \begin{proof}

src/op/banach/definitions.tex

@@ -27,5 +27,44 @@
     \end{enumerate}
 \end{definition}
 
+\begin{definition}[Unital Homomorphism]
+\label{definition:banach-algebra-unital-homomorphism}
+    Let $A, B$ be unital Banach algebras and $\phi: A \to B$ be a homomorphism, then $\phi$ is a \textbf{unital homomorphism} if $\phi(1) = 1$. 
+\end{definition}
+
+\begin{definition}[Unitisation]
+\label{definition:unitisation}
+    Let $A$ be a Banach algebra over $\complex$, and $\tilde A = \complex \oplus A$ with
+    \[
+    \iota: A \to \complex \oplus A \quad x \mapsto 0 + x
+    \]
+    
+    For each $\lambda + x, \mu + y \in \tilde A$, define
+    \[
+    (\lambda + x)(\mu + y) = \lambda \mu + (\lambda x + \mu x + xy)
+    \]
+    
+    and
+    \[
+    \norm{\lambda + x}_{\tilde A} = |\lambda| \norm{x}_A
+    \]
+
+    then
+    \begin{enumerate}
+        \item $\tilde A$ is a unital associative algebra over $\complex$.
+        \item $\iota: A \to \tilde A$ is a homomorphism. 
+        \item[(U)] For any pair $(B, \phi)$ satisfying (1) and (2), there exists a unique continuous unital homomorphism $\tilde \phi: \tilde A \to B$ such that $\phi(1) = 1$ and the following diagram commutes:
+        \xymatrix{
+        \tilde A \ar@{->}[r]^{\tilde \phi } & B \\
+        A \ar@{->}[u]^{\iota} \ar@{->}[ru]_{\phi} & 
+        }
+        \item $\iota(A)$ is a closed two-sided ideal of $\tilde A$. 
+    \end{enumerate}
+
+    The algebra $\tilde A$ is the \textbf{unitisation} of $A$. 
+\end{definition}
+\begin{proof}
+    (U): For each $\lambda + x \in \tilde A$, let $\tilde \phi(\lambda + x) = \lamdba + \phi(x)$. 
+\end{proof}
 
 

src/op/banach/gelfand.tex

@@ -3,7 +3,7 @@
 
 \begin{definition}[Gelfand Transform]
 \label{definition:gelfand-transform}
-    Let $A$ be a unital Banach algebra, then the \textbf{Gelfand transform} is the homomorphism
+    Let $A$ be a unital Banach algebra, then the \textbf{Gelfand transform} is the contractive homomorphism
     \[
     \Gamma = \Gamma_A: A \to C(\Omega(A); \complex) \quad (\Gamma_Ax)(\varphi) = \varphi(x)
     \]
@@ -19,7 +19,6 @@
 \label{proposition:gelfand-transform-gymnastics}
     Let $A$ be a commutative unital Banach algebra and $x \in A$, then:
     \begin{enumerate}
-        \item $\Gamma_A$ is a contractive homomorphism. 
         \item $\Gamma_A(1) = 1$. 
         \item $x \in G(A)$ if and only if $\Gamma_A x \in G(C(\Omega(A); \complex))$. 
         \item $(\Gamma_Ax)(\Omega(A)) = \sigma_A(x)$. 
@@ -27,11 +26,11 @@
     \end{enumerate}
 \end{proposition}
 \begin{proof}[Proof, {{\cite[Theorem 1.1.13]{FollandHarmonic}}}. ]
-    (2): For each $\phi \in \Omega(A)$, $\phi(1) = 1$, so $\Gamma_A(1) = 1$. 
+    (1): For each $\phi \in \Omega(A)$, $\phi(1) = 1$, so $\Gamma_A(1) = 1$. 
 
-    (3): Since $A$ is commutative, $x \not\in G(A)$ if and only if the ideal generated by $x$ is proper, if and only if there exists a maximal ideal containing $x$, if and only if there exists $\phi \in \Omega(A)$ with $\phi(x) = 0$. 
+    (2): Since $A$ is commutative, $x \not\in G(A)$ if and only if the ideal generated by $x$ is proper, if and only if there exists a maximal ideal containing $x$, if and only if there exists $\phi \in \Omega(A)$ with $\phi(x) = 0$. 
 
-    (4): By (1) and (3), 
+    (3): By (2), 
     \[
     (\Gamma_Ax)(\Omega(A)) = \sigma_{C(\Omega(A); \complex)}(\Gamma x) = \sigma_A(x)
     \]
@@ -46,12 +45,12 @@
     \end{enumerate}
 \end{proposition}
 \begin{proof}
-    (1) $\Rightarrow$ (2): For each $x \in A$, by the \hyperref[spectral radius formula]{proposition:spectral-radius-hadamard} and (5) of \autoref{proposition:gelfand-transform-gymnastics},
+    (1) $\Rightarrow$ (2): For each $x \in A$, by the \hyperref[spectral radius formula]{proposition:spectral-radius-hadamard} and (4) of \autoref{proposition:gelfand-transform-gymnastics},
     \[
     \norm{\Gamma_A x}_u = [x]_{sp} = \norm{x}_A
     \]
 
-    (2) $\Rightarrow$ (1): For each $x \in A$, by (5) of \autoref{proposition:gelfand-transform-gymnastics}, 
+    (2) $\Rightarrow$ (1): For each $x \in A$, by (4) of \autoref{proposition:gelfand-transform-gymnastics}, 
     \[
     \normn{x^2}_A \ge [x^2]_{sp} = \normn{\Gamma_A x^2}_u = \normn{\Gamma_A x}_u^2 = \normn{x}_A^2
     \]

src/op/banach/multiplicative.tex

@@ -14,7 +14,6 @@
         \item $\norm{\phi}_{A^*} = 1$. 
         \item $\phi(G(A)) \subset \complex \setminus \bracs{0}$. 
     \end{enumerate}
-    
 \end{proposition}
 \begin{proof}
     (3): For each $x \in G(A)$, $1 = \phi(xx^{-1}) = \phi(x)\phi(x^{-1}) \ne 0$. 
@@ -24,12 +23,27 @@
     (2): For each $\lambda \in \complex$, $\phi(\lambda 1) = \lambda$, so $\norm{\phi}_{A^*} \le 1$. 
 \end{proof}
 
+\begin{proposition}
+\label{proposition:multiplicative-less-unit}
+    Let $A$ be a Banach algebra and $\phi \in A^*$ be a multiplicative functional, then $\norm{\phi}_{A^*} \le 1$. 
+\end{proposition}
+\begin{proof}
+    Let $\tilde A$ be the unitisation of $A$, then by (U) of the \hyperref[unitisation]{definition:unitisation}, $\phi$ extends to a multiplicative functional $\tilde \phi$ on $\tilde A$. Therefore $\norm{\phi}_{A^*} \le \normn{\tilde \phi}_{{\tilde A}^*} = 1$. 
+\end{proof}
+
+
 \begin{definition}[Space of Multiplicative Linear Functionals]
 \label{definition:multiplicative-linear-functional-space}
-    Let $A$ be a unital Banach algebra, then $\Omega(A)$ is the \textbf{space of multiplicative linear functionals}, which is a compact Hausdorff space under the weak-* topology. 
+    Let $A$ be a Banach algebra, then $\Omega(A)$ is the \textbf{space of multiplicative linear functionals}, and with respect to the weak-* topology, 
+    \begin{enumerate}
+        \item If $A$ is unital, then $\Omega(A)$ is a compact Hausdorff space. 
+        \item $\Omega(A) \cup \bracs{0}$ is a compact Hausdorff space, and $\Omega(A)$ is a LCH space.
+    \end{enumerate}
 \end{definition}
 \begin{proof}
-    By the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}.
+    (1): By \autoref{proposition:multiplicative-unit}, $\Omega(A)$ is a weak-* closed subset of $\bracsn{\phi \in A^*:\norm{\phi}_{A^*} = 1}$, so it is compact by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}.
+
+    (2): By \autoref{proposition:multiplicative-less-unit}, $\Omega(A) \cup \bracs{0}$ is a weak-* closed subset of $\ol{B_{A^*}(0, 1)}$, so it is compact by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}.
 \end{proof}
 
 \begin{proposition}

src/op/c-star/index.tex

@@ -0,0 +1,6 @@
+\chapter{$C^*$-Algebras}
+\label{chap:c-star-algebras}
+
+\input{./involution.tex}
+\input{./sa.tex}
+\input{./order.tex}

src/op/c-star/involution.tex

@@ -0,0 +1,57 @@
+\section{Involutions}
+\label{section:involutions}
+
+\begin{definition}[Involution]
+\label{definition:involution}
+    Let $A$ be an associative algebra over $\complex$, and $*: A \to A$ be a $\real$-linear map, then $*$ is an \textbf{involution} if:
+    \begin{enumerate}
+        \item[(C1)] For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. 
+        \item[(C2)] For each $x \in A$, $x^{**} = x$. 
+        \item[(I)] For every $x, y \in A$, $(xy)^* = y^*x^*$.  
+    \end{enumerate}
+
+    The space $A$ equipped with an involution is an \textbf{involutive algebra} over $\complex$. 
+\end{definition}
+
+\begin{definition}[$C^*$-Algebra]
+\label{definition:c-star-algebra}
+    Let $A$ be an involutive Banach algebra over $\complex$, then $A$ is a \textbf{$C^*$-algebra} if for every $x \in A$, $\normn{x^*x}_A = \norm{x}_A^2$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:c-star-algebra-gymnastics}
+    Let $A$ be a $C^*$ algebra, then:
+    \begin{enumerate}
+        \item For each $x \in A$, $\norm{x}_A = \normn{x^*}_A\norm{x}_A$. 
+    \end{enumerate}
+
+    If $A$ is unital, then
+    \begin{enumerate}[start=1]
+        \item For each $\lambda \in \complex$, $\lambda^* = \ol \lambda$. 
+        \item For any $x \in A$, $x \in G(A)$ if and only if $x^* \in G(A)$.  
+        \item For every $x \in A$, $\sigma_A(x^*) = \bracsn{\ol \lambda| \lambda \in \sigma_A(x)}$.
+        \item For each $x \in A$, $[x]_{sp} = [x^*]_{sp}$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): For each $x \in A$, $\norm{x}_A^2 = \normn{x^*x}_A \le \norm{x}_A \normn{x^*}_A$. 
+
+    (2): For every $x \in A$, $1^*x^* = (x1)^* = x^* = (1x)^* = x^*1^*$, so $1^* = 1$ by uniqueness of the inverse. 
+
+    (3): For any $x \in A$, $(x^{-1})^*x^* = (x^{-1}x)^* = 1 = (xx^{-1})^* = x^*(x^{-1})^*$.
+\end{proof}
+
+\begin{definition}[*-Homomorphism]
+\label{definition:star-homomorphism}
+    Let $A, B$ be $C^*$-algebras and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if:
+    \begin{enumerate}
+        \item $\phi$ is a homomorphism of Banach algebras.
+        \item For every $x \in A$, $\phi(x^*) = \phi(x)^*$. 
+    \end{enumerate}
+
+    If in addition, $\phi(1) = 1$, then $\phi$ is a \textbf{unital *-homomorphism}. 
+\end{definition}
+
+
+
+

src/op/c-star/order.tex

@@ -0,0 +1,10 @@
+\section{Order Structures of $C^*$-Algebras}
+\label{section:order-c-star-algebra}
+
+\begin{definition}[Positive]
+\label{definition:positive-c-star-algebra}
+    Let $A$ be a $C^*$-algebra and $x \in A$, then $x$ is \textbf{positive} if there exists $y \in A$ such that $x = y^*y$. 
+\end{definition}
+
+
+

src/op/c-star/sa.tex

@@ -0,0 +1,36 @@
+\section{Self-Adjoint Elements}
+\label{section:c-star-self-adjoint}
+
+\begin{definition}[Self-Adjoint]
+\label{definition:self-adjoint}
+    Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then $x$ is \textbf{self-adjoint} if $x = x^*$. The space $A_{sa} = \bracs{x \in A| x = x^*}$ is the \textbf{self-adjoint part} of $A$, and:
+    \begin{enumerate}
+        \item $A_{sa}$ is a $\real$ subspace of $A$. 
+        \item $A = \complex(A_{sa})$ as a vector space.
+        \item For each $x \in A$, let
+        \[
+        \text{Re}(x) = \frac{x + x^*}{2} \quad \text{Im}(x) = \frac{x - x^*}{2i}
+        \]
+        
+        then $\text{Re}(x), \text{Im}(x) \in A_{sa}^2$ and $x = \text{Re}(x) + i\text{Im}(x)$.
+        \item For each $x \in A$, $x^* = \text{Re}(x) - i\text{Im}(x)$. 
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    By \autoref{proposition:complex-conjugation-properties}. 
+\end{proof}
+
+\begin{definition}[Normal]
+\label{definition:c-star-normal}
+    Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then the following are equivalent:
+    \begin{enumerate}
+        \item $\text{Re}(x)\text{Im}(x) = \text{Im}(x)\text{Re}(x)$. 
+        \item $x^*x = xx^*$. 
+    \end{enumerate}
+    
+    If the above holds, then $x$ is \textbf{normal}. 
+\end{definition}
+
+
+
+

src/op/example/c0.tex

@@ -0,0 +1,35 @@
+\section{$C_0(X)$}
+\label{section:vanishing-infinity-algebra}
+
+\begin{definition}[$C_0(X)$]
+\label{definition:vanishing-infinity-algebra}
+    Let $X$ be a LCH space, then $C_0(X; \complex)$ equipped with pointwise operations and the uniform norm is a $C^*$-algebra. 
+\end{definition}
+
+\begin{theorem}
+\label{theorem:vanishing-infinity-multiplicative-functional}
+    Let $X$ be a LCH space, then the mapping
+    \[
+    E: X \to \Omega(C_0(X)) \quad E(x)(f) = f(x)
+    \]
+
+    is a homeomorphism. Under the identification $X = \Omega(C_0(X))$, the Gelfand transform is the identity. 
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem 7.4]{Zhu}}}. ]
+    Let $X^* = X \sqcup \bracs{\infty}$ be the \hyperref[one-point compactification]{definition:alexandroff-compactification} of $X$. For each $\phi \in \Omega(C_0(X))$, let
+    \[
+    \phi^*: BC(X^*) \to \complex \quad f \mapsto \phi(f - f(\infty)) + f(\infty)
+    \]
+
+    then for each $f, g \in BC(X^*)$, 
+    \begin{align*}
+        fg &= (f - f(\infty))(g - g(\infty)) + f(\infty)(g - g(\infty)) + g(\infty)(f - f(\infty)) + f(\infty)g(\infty) \\
+        \phi^*(fg) &= \phi(f - f(\infty))\phi(g - g(\infty)) + f(\infty)\phi(g - g(\infty)) \\
+        &+ g(\infty)(f - f(\infty)) + f(\infty)g(\infty) \\
+        &= \braksn{\phi(f - f(\infty)) + f(\infty)}\braksn{\phi(g - g(\infty)) + g(\infty)} \\
+        &= \phi^*(f)\phi^*(g)
+    \end{align*}
+
+    so $\phi^* \in \Omega(BC(X^*))$. By \autoref{theorem:multiplicative-functional-bc}, there exists $x \in X^*$ such that $\phi^*(f) = f(x)$ for all $f \in BC(X^*)$. Since $\phi \ne 0$, $x \in X$, and $\phi = E(x)$. 
+\end{proof}
+

src/op/example/convolution.tex

@@ -77,11 +77,11 @@
 \label{proposition:convolution-integer-spectrum}
     Let $\ell^1(\integer)$ be the convolution algebra on $\integer$ and $f \in \ell^1(\integer)$, then
     \[
-    \sigma(f) = \bracs{\sum_{n \in \integer}f(n)z^n \bigg | z \in \partial B_\complex(0, 1)}
+    \sigma_{\ell^1(\integer)}(f) = \bracs{\sum_{n \in \integer}f(n)z^n \bigg | z \in \partial B_\complex(0, 1)}
     \]
 \end{proposition}
 \begin{proof}
-    By \autoref{theorem:convolution-integer-gelfand} and (4) of \autoref{proposition:gelfand-transform-gymnastics}.
+    By \autoref{theorem:convolution-integer-gelfand} and (3) of \autoref{proposition:gelfand-transform-gymnastics}.
 \end{proof}
 
 

src/op/example/index.tex

@@ -3,6 +3,7 @@
 
 \input{./matrix.tex}
 \input{./bounded.tex}
+\input{./c0.tex}
 \input{./hardy.tex}
 \input{./disk.tex}
 \input{./convolution.tex}

src/op/index.tex

@@ -2,5 +2,6 @@
 \label{part:operator-algebras}
 
 \input{./banach/index.tex}
+\input{./c-star/index.tex}
 \input{./example/index.tex}
 \input{./notation.tex}

src/topology/main/c0.tex

@@ -1,6 +1,9 @@
 \section{Continuous Functions Vanishing at Infinity}
 \label{section:vanish-at-infinity}
 
+The following section concerns the properties of spaces of vector valued functions vanishing at infinity. 
+For details regarding the complex-valued cased, in particular its properties as an algebra, see 
+
 \begin{definition}[Vanish at Infinity]
 \label{definition:vanish-at-infinity}
     Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $f$ \textbf{vanishes at infinity} if for every $U \in \cn_E^o(0)$, $\bracs{f \not\in U}$ is compact.