C*-algebra
CstarAlgebra
2009-01-10 19:27:37
2009-01-10 19:40:37
Definition
quantum operator algebra
groupoid
von Neumann algebra
C*-algebra
quantum operator algebra
groupoid
von Neumann algebra
Haar system
Hopf algebra
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\section{Von Neumann and C*-algebras: Quantum Operator Algebra and Quantum Theories}
C*-algebra has evolved as a key concept in Quantum Operator Algebra after the introduction of the
von Neumann algebra for the mathematical foundation of Quantum Mechanics. The von Neumann algebra classification is simpler and studied in greater depth than that of general C*-algebra classification theory. The importance of
C*-algebras for understanding the geometry of quantum state spaces (Alfsen and Schultz, 2003 \cite{AS}) cannot be overestimated. Moreover, the introduction of non-commutative C*-algebras in Noncommutative Geometry has already played important roles in expanding the Hilbert space perspective of Quantum Mechanics developed by von Neumann. Furthermore, extended quantum symmetries are currently being approached in terms of groupoid C*- convolution algebra and their representations; the latter also enter into the construction of compact quantum groupoids as developed in the Bibliography cited, and also briefly outlined here in the second section.
The fundamental connections that exist between categories of $C^*$-algebras and those of von Neumann and other
quantum operator algebras, such as JB- or JBL- algebras are yet to be completed and are the subject
of in depth studies \cite{AS}.
A \textbf{C*-algebra} is simultaneously a $*$--algebra and a Banach space -with additional conditions- as defined next.
Let us consider first the definition of an \emph{involution} on a complex algebra $\mathfrak A$.
\begin{definition}
An \emph{involution} on a complex algebra $\mathfrak A$ is a \emph{real--linear map} $T \mapsto T^*$
such that for all
$S, T \in \mathfrak A$ and $\lambda \in \bC$, we have $ T^{**} = T~,~ (ST)^* = T^* S^*~,~ (\lambda T)^* = \bar{\lambda} T^*~. $
\end{definition}
A \emph{*-algebra} is said to be a complex associative algebra together with an involution $*$~.
\begin{definition}
A \emph{C*-algebra} is simultaneously a *-algebra and a Banach space $\mathfrak A$,
satisfying for all $S, T \in \mathfrak A$~ the following conditions:
$ \begin{aligned} \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. \end{aligned}$
\end{definition}
One can easily verify that $\Vert A^* \Vert = \Vert A \Vert$~.
By the above axioms a C*--algebra is a special case of a Banach algebra where the latter requires the above C*-norm property, but not the involution (*) property.
Given Banach spaces $E, F$ the space $\mathcal L(E, F)$ of (bounded) linear operators from $E$ to $F$ forms a Banach space, where for $E=F$, the space $\mathcal L(E) = \mathcal L(E, E)$ is a Banach algebra with respect to the norm
\bigbreak
$\Vert T \Vert := \sup\{ \Vert Tu \Vert : u \in E~,~ \Vert u \Vert= 1 \}~. $
\bigbreak
In quantum field theory one may start with a Hilbert space $H$, and consider the Banach
algebra of bounded linear operators $\mathcal L(H)$ which given to be closed under the usual
algebraic operations and taking adjoints, forms a $*$--algebra of bounded operators, where the
adjoint operation functions as the involution, and for $T \in \mathcal L(H)$ we have~:
$ \Vert T \Vert := \sup\{ ( Tu , Tu): u \in H~,~ (u,u) = 1 \}~,$ and $ \Vert Tu \Vert^2 = (Tu,
Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.$
By a \emph{morphism between C*-algebras} $\mathfrak A,\mathfrak B$ we mean a linear map $\phi :
\mathfrak A \lra \mathfrak B$, such that for all $S, T \in \mathfrak A$, the following hold~:
\bigbreak
$\phi(ST) = \phi(S) \phi(T)~,~ \phi(T^*) = \phi(T)^*~, $
\bigbreak
where a bijective morphism is said to be an isomorphism (in which case it is then an
isometry). A fundamental relation is that any norm-closed $*$-algebra $\mathcal A$ in
$\mathcal L(H)$ is a \PMlinkname{C*-algebra}{CAlgebra3}, and conversely, any \PMlinkname{C*-algebra}{CAlgebra3} is isomorphic to a norm--closed $*$-algebra in $\mathcal L(H)$ for some Hilbert space $H$~.
One can thus also define \emph{the category $\mathcal{C}^*$ of C*-algebras and morphisms between C*-algebras}.
For a \PMlinkname{C*-algebra}{CAlgebra3} $\mathfrak A$, we say that $T \in \mathfrak A$ is \emph{self--adjoint} if $T
= T^*$~. Accordingly, the self--adjoint part $\mathfrak A^{sa}$ of $\mathfrak A$ is a real
vector space since we can decompose $T \in \mathfrak A^{sa}$ as ~:
$ T = T' + T^{''} := \frac{1}{2} (T + T^*) + \iota (\frac{-\iota}{2})(T - T^*)~.$
A \emph{commutative} C*--algebra is one for which the associative multiplication is
commutative. Given a commutative C*--algebra $\mathfrak A$, we have $\mathfrak A \cong C(Y)$,
the algebra of continuous functions on a compact Hausdorff space $Y~$.
The classification of {$C^*$-algebras} is far more complex than that of von Neumann algebras that provide
the fundamental algebraic content of quantum state and operator spaces in quantum theories.
\subsection{Quantum Compact Groupoids}
Compact quantum groupoids were introduced in Landsman (1998) as a
simultaneous generalization of a compact groupoid and a quantum group. Since this construction is relevant to the definition of locally compact quantum groupoids and their representations investigated here, its exposition is required before we can step up to the next level of generality. Firstly, let $\mathfrak A$ and $\mathfrak B$ denote C*--algebras equipped with a *--homomorphism $\eta_s : \mathfrak B \lra \mathfrak A$, and a *--antihomomorphism $\eta_t : \mathfrak B \lra \mathfrak A$ whose images in $\mathfrak A$
commute. A non--commutative Haar measure is defined as a completely
positive map $P: \mathfrak A \lra \mathfrak B$ which satisfies
$P(A \eta_s (B)) = P(A) B$~. Alternatively, the composition $\E = \eta_s \circ P : \mathfrak A \lra \eta_s (B) \subset \mathfrak A$ is a faithful conditional expectation.
Next consider $\mathsf{G}$ to be a (topological) groupoid as defined in the Appendix. We denote by $C_c(\mathsf{G})$ the space of smooth complex--valued functions with compact support on $\mathsf{G}$~. In particular, for all $f,g \in C_c(\mathsf{G})$, the
function defined via convolution
\begin{equation} (f ~*~g)(\gamma)
= \int_{\gamma_1 \circ \gamma_2 = \gamma} f(\gamma_1) g
(\gamma_2)~,
\end{equation}
is again an element of $C_c(\mathsf{G})$, where the convolution product
defines the composition law on $C_c(\mathsf{G})$~. We can turn
$C_c(\mathsf{G})$ into a *--algebra once we have defined the involution
$*$, and this is done by specifying $f^*(\gamma) = \overline{f(\gamma^{-1})}$~.
We recall that following Landsman (1998) a \emph{representation} of a groupoid $\grp$, consists of a
family (or field) of Hilbert spaces $\{\mathcal H_x \}_{x \in X}$
indexed by $X = \ob~ \grp$, along with a collection of maps $\{
U(\gamma)\}_{\gamma \in \grp}$, satisfying:
\begin{itemize}
\item[1.]
$U(\gamma) : \mathcal H_{s(\gamma)} \lra \mathcal H_{r(\gamma)}$,
is unitary.
\item[2.]
$U(\gamma_1 \gamma_2) = U(\gamma_1) U( \gamma_2)$, whenever
$(\gamma_1, \gamma_2) \in \grp^{(2)}$~ (the set of arrows).
\item[3.]
$U(\gamma^{-1}) = U(\gamma)^*$, for all $\gamma \in \grp$~.
\end{itemize}
Suppose now $\mathsf{G}_{lc}$ is a Lie groupoid. Then the isotropy group
$\mathsf{G}_x$ is a Lie group, and for a (left or right) Haar
measure $\mu_x$ on $\mathsf{G}_x$, we can consider the Hilbert
spaces $\mathcal H_x = L^2(\mathsf{G}_x, \mu_x)$ as exemplifying the
above sense of a representation. Putting aside some technical
details which can be found in Connes (1994) and Landsman (2006), the
overall idea is to define an operator of Hilbert spaces
\begin{equation}\pi_x(f) : L^2(\mathsf{G_x},\mu_x) \lra L^2(\mathsf{G}_x, \mu_x)~,
\end{equation}
given by
\begin{equation}
(\pi_x(f) \xi)(\gamma) = \int f(\gamma_1) \xi (\gamma_1^{-1}
\gamma)~ d\mu_x~,
\end{equation}
for all $\gamma \in \mathsf{G}_x$, and
$\xi \in \mathcal H_x$~. For each $x \in X =\ob ~\mathsf{G}$, $\pi_x$
defines an involutive representation $\pi_x : C_c(\mathsf{G}) \lra
\mathcal H_x$~. We can define a norm on $C_c(\mathsf{G})$ given by
\begin{equation}
\Vert f \Vert = \sup_{x \in X} \Vert \pi_x(f) \Vert~,
\end{equation}
whereby the completion of $C_c(\mathsf{G})$ in this norm, defines
\emph{the reduced C*--algebra $C^*_r(\mathsf{G})$ of $\mathsf{G}_{lc}$}. It is
perhaps the most commonly used C*--algebra for Lie groupoids
(groups) in noncommutative geometry.
The next step requires a little familiarity with the theory of
Hilbert modules (see e.g. Lance, 1995). We define a left
$\mathfrak B$--action $\lambda$ and a right $\mathfrak B$--action
$\rho$ on $\mathfrak A$ by $\lambda(B)A = A \eta_t (B)$ and
$\rho(B)A = A \eta_s(B)$~. For the sake of localization of the
intended Hilbert module, we implant a $\mathfrak B$--valued inner
product on $\mathfrak A$ given by $\langle A, C \rangle_{\mathfrak
B} = P(A^* C)$ ~. Let us recall that $P$ is defined as a \emph{completely positive map}.
Since $P$ is faithful, we fit a new norm on $\mathfrak A$ given by $\Vert A \Vert^2 = \Vert P(A^* A)
\Vert_{\mathfrak B}$~. The completion of $\mathfrak A$ in this new
norm is denoted by $\mathfrak A^{-}$ leading then to a Hilbert
module over $\mathfrak B$~.
The tensor product $\mathfrak A^{-} \otimes_{\mathfrak B}\mathfrak
A^{-}$ can be shown to be a Hilbert bimodule over $\mathfrak B$,
which for $i=1,2$, leads to *--homorphisms $\vp^{i} : \mathfrak A
\lra \mathcal L_{\mathfrak B}(\mathfrak A^{-} \otimes \mathfrak
A^{-})$~. Next is to define the (unital) C*--algebra $\mathfrak A
\otimes_{\mathfrak B} \mathfrak A$ as the C*--algebra contained in
$ \mathcal L_{\mathfrak B}(\mathfrak A^{-} \otimes \mathfrak
A^{-})$ that is generated by $\vp^1(\mathfrak A)$ and
$\vp^2(\mathfrak A)$~. The last stage of the recipe for defining a
compact quantum groupoid entails considering a certain coproduct
operation $\Delta : \mathfrak A \lra \mathfrak A
\otimes_{\mathfrak B} \mathfrak A$, together with a coinverse $Q :
\mathfrak A \lra \mathfrak A$ that it is both an algebra and
bimodule antihomomorphism. Finally, the following axiomatic
relationships are observed~:
\begin{equation}
\begin{aligned}
(\ID \otimes_{\mathfrak B} \Delta) \circ \Delta &= (\Delta
\otimes_{\mathfrak B} \ID) \circ \Delta \\ (\ID \otimes_{\mathfrak
B} P) \circ \Delta &= P \\ \tau \circ (\Delta \otimes_{\mathfrak
B} Q) \circ \Delta &= \Delta \circ Q
\end{aligned}
\end{equation}
where $\tau$ is a flip map : $\tau(a \otimes b) = (b \otimes a)$~. \\
There is a natural extension of the above definition of quantum compact groupoids
to \textit{locally compact} quantum groupoids by taking $\mathsf{G}_{lc}$ to be a locally compact groupoid (instead of a compact groupoid), and then following the steps in the above construction with the topological groupoid $\mathsf{G}$ being replaced by $\mathsf{G}_{lc}$. Additional integrability and Haar measure system conditions need however be also satisfied as in the general case of locally compact groupoid \textit{representations} (for further details, see for example the monograph by Buneci (2003).
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