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\begin{document}

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\section*{unitization of a C-star-algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{operator_algebra}{}\paragraph*{{Operator algebra}}\label{operator_algebra}

[[!include AQFT and operator algebra contents]]

\hypertarget{noncommutative_geometry}{}\paragraph*{{Noncommutative geometry}}\label{noncommutative_geometry}

[[!include noncommutative geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_point_at_infinity}{The point at infinity}\dotfill \pageref*{the_point_at_infinity} \linebreak
\noindent\hyperlink{ktheory_with_compact_support_on_nonunital_algebras}{K-theory with compact support on non-unital $C^\ast$-algebras}\dotfill \pageref*{ktheory_with_compact_support_on_nonunital_algebras} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The generalization of the process of [[one-point compactification]] of possibly non-[[compact topological space|compact]] [[topological spaces]] from [[topology]] to [[non-commutative topology]] is the process of adding units to possibly [[non-unital ring|non-unital]] [[C\emph{-algebras]], thought of as formal duals of non-commutative spaces in [[noncommutative topology]].}

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
For $(A, {\Vert \cdot \Vert_{A}})$ a non-[[unital algebra|unital]] [[C\emph{-algebra]], its \textbf{[[unitisation]]} is the $C^{\ast}$-algebra whose underlying vector space is the [[direct sum]]}

\begin{displaymath}
A^{+} \coloneqq A \oplus \mathbb{C}
\end{displaymath}
of $A$ with the field of [[complex numbers]], equipped with the multiplication law

\begin{displaymath}
(a_{1} \oplus z_{1}) \cdot (a_{2} \oplus z_{2})
\coloneqq (a_{1} a_{2} + z_{2} a_{1} + z_{1} a_{2}) \oplus z_{1} z_{2},
\end{displaymath}
and the [[involution]]

\begin{displaymath}
(a \oplus z)^\ast \coloneqq a^{\ast} \oplus \overline{z}
\end{displaymath}
([[complex conjugation]] is taking place on the right). This really is a $C^\ast$-algebra. The [[norm]] can be characterised as

\begin{displaymath}
{\Vert a \oplus z \Vert}_{A^{+}}
= {\Vert L_{a} + z \cdot \operatorname{Id}_{A} \Vert}_{\mathcal{B}(A)}
\coloneqq \sup_{b \in A, \Vert b \Vert_{A} \leq 1} \Vert a b + z b \Vert_{A}.
\end{displaymath}
if $A$ does not have a multiplicative unit. (Note that the operatornorm vanishes on $1_A\oplus -1_{\mathbb{C}}$ if $A$ does have a multiplicative unit). For $C^\ast$-algebras with a multiplicative unit one can show that the norm equals

\begin{displaymath}
\Vert a\oplus z \Vert_{A^+} = \max\{\|a+z1_A\|,|z|\}
\end{displaymath}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{UnitisationGivesOnePointCompactification}\hypertarget{UnitisationGivesOnePointCompactification}{}
For $X$ a [[locally compact topological space|locally compact]] [[Hausdorff topological space]] and $C(X)$ its possibly non-unital [[C\emph{-algebra]] [[algebra of functions|of functions]], then}

\begin{displaymath}
(C_0(X))^+ \simeq C_0(X) \oplus \mathbb{C} \simeq C(X^+)
\end{displaymath}
this is equivalently the unital $C^\ast$-algebra of functions on the [[one-point compactification]] of $X$.

There is hence a canonical projection

\begin{displaymath}
i^\ast \colon C(X^+) \to \mathbb{C}
\end{displaymath}
The [[topological K-theory]] of $X$ is the [[kernel]] of the induced map in [[operator K-theory]]

\begin{displaymath}
K(X) \simeq ker(i^\ast) \simeq K(C_0(X))
  \,.
\end{displaymath}
\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{the_point_at_infinity}{}\subsubsection*{{The point at infinity}}\label{the_point_at_infinity}

\begin{remark}
\label{DualPointInclusion}\hypertarget{DualPointInclusion}{}
The unitisation of $A$ comes with a canonical [[projection]] [[homomorphism]] of [[C\emph{-algebras]]}

\begin{displaymath}
i^\ast_A \colon A^+ \to \mathbb{C}
\end{displaymath}
given by

\begin{displaymath}
(a + z) \mapsto z
  \,.
\end{displaymath}
Dually this corresponds to the inclusion of the ``point at infinity''.

\end{remark}
\hypertarget{ktheory_with_compact_support_on_nonunital_algebras}{}\subsubsection*{{K-theory with compact support on non-unital $C^\ast$-algebras}}\label{ktheory_with_compact_support_on_nonunital_algebras}

The map $i^\ast_A \colon A^+ \to \mathbb{C}$ of remark \ref{DualPointInclusion} induces a [[homomorphism]] of [[operator K-theory]] [[groups]] of the form

\begin{displaymath}
K(i^\ast_A) \colon K(A^+) \to K(\mathbb{C}) \simeq \mathbb{Z}
  \,.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
The [[kernel]] of this map is the operator K-theory of the original possibly non-unital $C^\ast$-algebra $A$:

\begin{displaymath}
K(A) \coloneqq ker(K(i^\ast_A))
 \,.
\end{displaymath}
\end{defn}
Heuristically it is clear that this is the ``compactly suppported'' K-theory of the possibly non-compact non-commutative space given by the algebra $A$. This statement has been made precise for instance in (\hyperlink{Emerson07}{Emerson 07}, theorem 3.8)):

\begin{prop}
\label{}\hypertarget{}{}
For $X$ a [[locally compact topological space|locally compact]] [[Hausdorff topological space]], there is a [[natural isomorphism]]

\begin{displaymath}
K(X) \simeq [X,Fred]_{cs}
  \,,
\end{displaymath}
where on the right we [[homotopy classes]] of maps of [[compact support]] into the [[classifying space]] for K-theory (space of Fredholm operators).

\end{prop}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Heath Emerson]], \emph{Equivariant representable K-theory} (\href{http://arxiv.org/abs/0710.1410}{arXiv:0710.1410})

\end{itemize}
[[!redirects unitization of a C\emph{-algebra]] [[!redirects unitization of C}-algebras]] [[!redirects unitizations of C\emph{-algebras]] [[!redirects unitisation of a C}-algebra]] [[!redirects unitisation of C\emph{-algebras]] [[!redirects unitisations of C}-algebras]] [[!redirects unitization of a C-star algebra]] [[!redirects unitization of C-star algebras]] [[!redirects unitizations of C-star algebras]] [[!redirects unitisation of a C-star algebra]] [[!redirects unitisation of C-star algebras]] [[!redirects unitisations of C-star algebras]] [[!redirects unitization of a C-star-algebra]] [[!redirects unitization of C-star-algebras]] [[!redirects unitizations of C-star-algebras]] [[!redirects unitisation of a C-star-algebra]] [[!redirects unitisation of C-star-algebras]] [[!redirects unitisations of C-star-algebras]]



\end{document}