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\section*{E-theory}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

[[!include AQFT and operator algebra contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

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

[[!include noncommutative geometry - contents]]

\hypertarget{theory}{}\section*{{$E$-theory}}\label{theory}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{universal_characterization}{Universal characterization}\dotfill \pageref*{universal_characterization} \linebreak
\noindent\hyperlink{RelationToKKTheory}{Relation to KK-theory}\dotfill \pageref*{RelationToKKTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_shape_theory}{Relation to shape theory}\dotfill \pageref*{relation_to_shape_theory} \linebreak


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

\emph{$E$-Theory} is the name of a [[category]] whose [[objects]] are [[C\emph{-algebras]] and whose [[hom-sets]] are homotopy classes of slightly generalized $C*$-homomorphisms, called \emph{[[asymptotic C\emph{-homomorphisms]]\_. These hom-sets have the structure of an [[abelian group]] and are also called the \emph{E-groups} of their arguments. Since under [[Gelfand duality]] [[C}-algebras]] may be thought of as exhibiting [[noncommutative topology]], one also speaks of}[[noncommutative stable homotopy theory]]\_.}

This construction may be understood as the universal improvement of [[KK-theory]] under [[excision]] (\hyperlink{Higson90}{Higson 90}). Accordingly, the $E$-groups behave like groups of a [[K-theory]]-like [[generalized cohomology theory]].

In terms of [[noncommutative topology]] (regarding, in view of [[Gelfand duality]], noncommutative [[C\emph{-algebras]] as [[algebras of functions]] on ``noncommutative topological spaces'') one may understand this as dealing with ``locally badly behaved space'' such as certain [[quotients]] of [[foliations]] (\href{ConnesHigson90}{Connes-Higson 90}) in a way that resembles a noncommutative version of [[shape theory]] (\hyperlink{Dadarlat94}{Ddrlat 94}).}

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

First some notation and terminology.

For $A \in$ [[C\emph{Alg]], we write}

\begin{displaymath}
\Sigma A \coloneqq C_0((0,1),A)
\end{displaymath}
for the $C^\ast$-algebra of continuous $A$-valued functions on the open inverval [[vanishing at infinity]]. This is also called the \emph{suspension} of $A$.

For $A,B \in$ [[C\emph{Alg]], write $[A,B]$ for the set of [[homotopy]]-[[equivalence classes]] of [[asymptotic C}-homomorphisms]] $A \to B$. As discussed there

\begin{enumerate}%
\item there is a natural [[composition operation]] $[A,B] \times [B,C] \to [A,C]$;


\item $[A,\Sigma B]$ is naturally an [[abelian group]].



\end{enumerate}
Finally, write $\mathcal{K} \in$ [[C\emph{Alg]] for the $C^\ast$-algebra of [[compact operators]] on an infinite-dimensional [[separable Hilbert space]]. For $A \in C^\ast Alg$ the [[tensor product of C}-algebras]] $A \otimes \mathcal{K}$ is also called the \emph{stabilization} of $A$.

\begin{defn}
\label{}\hypertarget{}{}
For $A,B \in$ [[C\emph{Alg]], the \textbf{E-group} of $A$ with [[coefficients]] in $B$ is}

\begin{displaymath}
E(A,B) \coloneqq [(\Sigma A )\otimes \mathcal{K}, (\Sigma B) \otimes \mathcal{K}]
  \in Ab
  \,.
\end{displaymath}
Under the induced [[composition]] operation this yields an [[additive category]] $E$ whose [[objects]] are [[C\emph{-algebras]], and whose [[hom-objects]] are $E(-,-)$.}

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{universal_characterization}{}\subsubsection*{{Universal characterization}}\label{universal_characterization}

E-theory is the [[universal construction|universal]] localization [[C\emph{Alg]] $\to E$ which is homotopy-invariant, stable and preserves [[exact sequences]] in the middle.}

(\ldots{})

\hypertarget{RelationToKKTheory}{}\subsubsection*{{Relation to KK-theory}}\label{RelationToKKTheory}

Because KK-theory is the universal \emph{split exact} (stable and homotopy-invariant) localization of [[C\emph{Alg]], and E-theory the universal half-exact localization, and since every [[split exact sequence]] is in particular exact, there is a universal [[functor]]}

\begin{displaymath}
KK \to E
\end{displaymath}
from the [[KK-theory]] [[homotopy category]] to that of $E$-theory.

Restricted to [[nuclear C\emph{-algebras]] this is a [[full and faithful functor]]. (\hyperlink{Higson90}{Higson 90}) (\ldots{})}

If in the definition of E-theory by [[asymptotic C\emph{-homomorphisms]] one restricts to those which take values in contractive [[completely positive maps]], then the results is isomorphic to KK-theory again. (K. Thomsen, \hyperlink{Introduction}{Introduction, p. 34}). The above universal functor $KK \to E$ is then just the corresponding [[forgetful functor]].}

It follows that the Kasparov product in [[KK-theory]] is equivalently given by the composition of the corresponding completely positive [[asymptotic C\emph{-homomorphisms]].}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[KK-bootstrap category]]

\end{itemize}
[[!include noncommutative motives - table]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The idea of E-theory was introduced in

\begin{itemize}%
\item [[Alain Connes]], [[Nigel Higson]], \emph{D\'e{}formations, morphismes asymptotiques et $K$-th\'e{}orie bivariante}, C. R. Acad. Sci. Paris S\'e{}r. I Math. \textbf{311} (1990), no. 2, 101--106, \href{http://www.ams.org/mathscinet-getitem?mr=1065438}{MR91m:46114}, \href{ftp://ftp.bnf.fr/578/N5781521_PDF_107_112DM.pdf}{pdf}

\end{itemize}
\begin{itemize}%
\item [[Nigel Higson]], \emph{Categories of fractions and excision in KK-theory} J. Pure Appl. Algebra, 65(2):119--138, (1990) (\href{http://www.math.psu.edu/higson/math/Papers_files/Higson%20-%201990%20-%20Categories%20of%20fractions%20and%20excision%20in%20KK-theory.pdf}{pdf})

\end{itemize}
Reviews and surveys include

\begin{itemize}%
\item \emph{Introduction to KK-theory and E-theory}, Lecture notes (Lisbon 2009) (\href{http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf}{pdf slides})

\end{itemize}
A standard textbook account is in section 25 of

\begin{itemize}%
\item [[Bruce Blackadar]], \emph{[[K-Theory for Operator Algebras]]}

\end{itemize}
The [[stable homotopy theory|stable]] [[homotopy theory]] aspects are further discussed in

\begin{itemize}%
\item Martin Grensing, \emph{Noncommutative stable homotopy theory} (\href{http://arxiv.org/abs/1302.4751}{arXiv:1302.4751})


\item Rasmus Bentmann, \emph{Homotopy-theoretic E-theory and n-order} (\href{http://arxiv.org/abs/1302.6924}{arXiv:1302.6924})



\end{itemize}
See also

\begin{itemize}%
\item web page of a project \href{http://www.math.ku.dk/~jg/papers/etheory.html}{Noncommutative topology - homotopy functors and E-theory}


\item [[Snigdhayan Mahanta]], \emph{Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality} \href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf}



\end{itemize}
\begin{itemize}%
\item \href{http://faculty.tcu.edu/epark/papers/ETheory_JFA.pdf}{pdf}

\end{itemize}
\hypertarget{relation_to_shape_theory}{}\subsubsection*{{Relation to shape theory}}\label{relation_to_shape_theory}

Relation to [[shape theory]] is discussed in

\begin{itemize}%
\item [[Marius Dādārlat]], \emph{Shape theory and asymptotic morphisms for $C^\ast$-algebras}, Duke Math. J. \textbf{73} (3):687-711, 1994, \href{http://www.ams.org/mathscinet-getitem?mr=1262931}{MR95c:46117}, \href{http://www.math.purdue.edu/~mdd/Publications/shape.pdf}{pdf}

\end{itemize}
\begin{itemize}%
\item Vladimir Manuilov, Klaus Thomsen, \emph{Shape theory and extensions of $C^\ast$-algebras}, (\href{http://arxiv.org/abs/1007.1663}{arxiv/1007.1663})

\end{itemize}
[[!redirects E-theory]]

[[!redirects asymptotic morphism]] [[!redirects asymptotic morphisms]]



\end{document}