# PlanetPhysics/Quantum Operator Algebras

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

## Quantum operator algebras(QOA)

in quantum field theories are defined as the algebras of observable operators, and as such, they are also related to the von Neumann algebra; quantum operators are usually defined on Hilbert spaces, or in some QFTs on Hilbert space bundles or other similar families of spaces.
Note: representations of Banach *-algebras, that are also defined on Hilbert spaces, are related to $C^{*}$-algebra representations which provide a useful approach to defining quantum space-times.
Quantum Operator Algebras in Quantum Field Theories: \htmladdnormallink{QOAs {http://planetphysics.us/encyclopedia/QAT.html} in QFTs} Examples of quantum operators are: the Hamiltonian operator (or Schr\"odinger operator), the position and momentum operators, Casimir operators, Unitary operators, spin operators, and so on. The observable operators are also self-adjoint . More general operators were recently defined, such as Progogine's superoperators. Another development in quantum theories is the introduction of Frech\'et nuclear spaces or `rigged' Hilbert spaces (Hilbert bundles ). The following sections define several types of quantum operator algebras that provide the foundation of modern quantum field theories in mathematical physics.

### Quantum Groups, Quantum Operator Algebras and Related Symmetries.

Quantum theories adopted a new lease of life post 1955 when von Neumann beautifully re-formulated quantum mechanics (QM) and Quantum theories (QT) in the mathematically rigorous context of Hilbert spaces and operator algebras defined over such spaces. From a current physics perspective, von Neumann' s approach to quantum mechanics has however done much more: it has not only paved the way to expanding the role of symmetry in physics, as for example with the Wigner-Eckhart theorem and its applications, but also revealed the fundamental importance in Quantum physics of the state space geometry of quantum operator algebras- Mathematical definitions
Definitions:
* Von Neumann Algebra
* Hopf Algebra
* Groupoids
* Haar \htmladdnormallink{systems {http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} associated to Measured Groupoids or Locally Compact Groupoids.}

### Von Neumann Algebra

Let $\mathbb {H}$ denote a complex (separable) Hilbert space. A \emph{von Neumann algebra} Failed to parse (unknown function "\A"): {\displaystyle \A} acting on $\mathbb {H}$ is a subset of the algebra of all bounded operators Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} such that:
* (i) Failed to parse (unknown function "\A"): {\displaystyle \A} is closed under the adjoint operation (with the
adjoint of an element $T$ denoted by $T^{*}$).
* (ii) Failed to parse (unknown function "\A"): {\displaystyle \A} equals its bicommutant, namely:
   Failed to parse (unknown function "\A"): {\displaystyle \A= \{A \in \cL(\mathbb{H}) : \forall B \in \cL(\mathbb{H}), \forall C\in \A,~ (BC=CB)\Rightarrow (AB=BA)\}~. }
If one calls a commutant of a set Failed to parse (unknown function "\A"): {\displaystyle \A} the special set of bounded operators on Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} which commute with all elements in Failed to parse (unknown function "\A"): {\displaystyle \A} , then this second condition implies that the commutant of the commutant of Failed to parse (unknown function "\A"): {\displaystyle \A} is again the set Failed to parse (unknown function "\A"): {\displaystyle \A} .
\med On the other hand, a von Neumann algebra Failed to parse (unknown function "\A"): {\displaystyle \A} inherits a unital subalgebra from Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} , and according to the first condition in its definition Failed to parse (unknown function "\A"): {\displaystyle \A} does indeed inherit a *-subalgebra structure, as further explained in the next section on C*-algebras. Furthermore, we have notable Bicommutant Theorem which states that Failed to parse (unknown function "\A"): {\displaystyle \A} \emph{is a von Neumann algebra if and only if Failed to parse (unknown function "\A"): {\displaystyle \A} is a *-subalgebra of Failed to parse (unknown function "\cL"): {\displaystyle \cL(\mathbb{H})} , closed for the smallest topology defined by continuous maps $(\xi ,\eta )\longmapsto (A\xi ,\eta )$ for all $<A\xi ,\eta )>$ where $<.,.>$ denotes the inner product defined on $\mathbb {H}$}~. For a well-presented treatment of the geometry of the state spaces of quantum operator algebras, see e.g. Aflsen and Schultz (2003).

#### Hopf algebra

First, a unital associative algebra consists of a linear space $A$ together with two linear maps
   Failed to parse (syntax error): {\displaystyle m &: A \otimes A \lra A~,~(multiplication) \\ \eta &: \bC \lra A~,~ (unity) }
satisfying the conditions
   Failed to parse (syntax error): {\displaystyle m(m \otimes \mathbf 1) &= m (\mathbf 1 \otimes m) \\ m(\mathbf 1 \otimes \eta) &= m (\eta \otimes \mathbf 1) = \ID~. }
This first condition can be seen in terms of a commuting diagram~:
   Failed to parse (unknown function "\begin{CD}"): {\displaystyle \begin{CD} A \otimes A \otimes A @> m \otimes \ID>> A \otimes A \\ @V \ID \otimes mVV @VV m V \\ A \otimes A @ > m >> A \end{CD} }
Next suppose we consider `reversing the arrows', and take an algebra $A$ equipped with a linear homorphisms Failed to parse (unknown function "\lra"): {\displaystyle \Delta : A \lra A \otimes A<math>, satisfying, for } a,b \in A</math> :
   Failed to parse (syntax error): {\displaystyle \Delta(ab) &= \Delta(a) \Delta(b) \\ (\Delta \otimes \ID) \Delta &= (\ID \otimes \Delta) \Delta~. }
We call $\Delta$ a comultiplication , which is said to be coasociative in so far that the following diagram commutes
   Failed to parse (unknown function "\begin{CD}"): {\displaystyle \begin{CD} A \otimes A \otimes A @< \Delta\otimes \ID<< A \otimes A \\ @A \ID \otimes \Delta AA @AA \Delta A \\ A \otimes A @ < \Delta << A \end{CD} }
There is also a counterpart to $\eta$, the counity map Failed to parse (unknown function "\vep"): {\displaystyle \vep : A \lra \bC} satisfying
   Failed to parse (unknown function "\ID"): {\displaystyle (\ID \otimes \vep) \circ \Delta = (\vep \otimes \ID) \circ \Delta = \ID~. }
A bialgebra Failed to parse (unknown function "\vep"): {\displaystyle (A, m, \Delta, \eta, \vep)<math> is a linear space } A$withmaps$m, \Delta, \eta, \vep</math> satisfying the above properties.
\med Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism Failed to parse (unknown function "\lra"): {\displaystyle S : A \lra A} , satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$~. This map is defined implicitly via the property~:
   Failed to parse (unknown function "\ID"): {\displaystyle m(S \otimes \ID) \circ \Delta = m(\ID \otimes S) \circ \Delta = \eta \circ \vep~~. }
We call $S$ the antipode map . A Hopf algebra is then a bialgebra Failed to parse (unknown function "\vep"): {\displaystyle (A,m, \eta, \Delta, \vep)} equipped with an antipode map $S$~.
\med Commutative and noncommutative Hopf algebras form the backbone of quantum `groups' and are essential to the generalizations of symmetry. Indeed, in most respects a quantum `group' is identifiable with a Hopf algebra. When such algebras are actually associated with proper groups of matrices there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.

#### Groupoids

Recall that a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} is, loosely speaking, a small category with inverses over its set of objects Failed to parse (unknown function "\grp"): {\displaystyle X = Ob(\grp)} ~. One often writes Failed to parse (unknown function "\grp"): {\displaystyle \grp^y_x} for the set of morphisms in Failed to parse (unknown function "\grp"): {\displaystyle \grp} from $x$ to $y$~. A topological groupoid consists of a space Failed to parse (unknown function "\grp"): {\displaystyle \grp} , a distinguished subspace Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = \obg \subset \grp} , called {\it the space of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} , together with maps
   Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} } }
called the {\it range} and {\it source maps} respectively,
together with a law of composition
   Failed to parse (unknown function "\grp"): {\displaystyle \circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{ ~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) = r(\gamma_2)~ \}~ \lra ~\grp~, }
such that the following hold~:~
\item[(1)] $s(\gamma _{1}\circ \gamma _{2})=r(\gamma _{2})~,~r(\gamma _{1}\circ \gamma _{2})=r(\gamma _{1})<math>~,forall$(\gamma_1, \gamma_2) \in \grp^{(2)}</math>~.
\med \item[(2)] $s(x)=r(x)=x$~, for all Failed to parse (unknown function "\grp"): {\displaystyle x \in \grp^{(0)}} ~.
\med \item[(3)] $\gamma \circ s(\gamma )=\gamma ~,~r(\gamma )\circ \gamma =\gamma <math>~,forall$\gamma \in \grp</math>~.
\med \item[(4)] $(\gamma _{1}\circ \gamma _{2})\circ \gamma _{3}=\gamma _{1}\circ (\gamma _{2}\circ \gamma _{3})$~.
\med \item[(5)] Each $\gamma$ has a two--sided inverse $\gamma ^{-1}$ with $\gamma \gamma ^{-1}=r(\gamma )~,~\gamma ^{-1}\gamma =s(\gamma )$~. Furthermore, only for topological groupoids the inverse map needs be continuous. \med It is usual to call Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = Ob(\grp)} {\it the set of objects} of Failed to parse (unknown function "\grp"): {\displaystyle \grp} ~. For Failed to parse (unknown function "\grp"): {\displaystyle u \in Ob(\grp)} , the set of arrows Failed to parse (unknown function "\lra"): {\displaystyle u \lra u} forms a group Failed to parse (unknown function "\grp"): {\displaystyle \grp_u} , called the isotropy group of Failed to parse (unknown function "\grp"): {\displaystyle \grp} at $u$ .
\med Thus, as is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalise bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to Brown (2006). \med
$>>$ Several examples of groupoids are: (a) locally compact groups, transformation groups , and any group in general (e.g. [59] (b) equivalence relations (c) tangent bundles (d) the tangent groupoid (e.g. [4]) (e) holonomy groupoids for foliations (e.g. [4]) (f) Poisson groupoids (e.g. [81]) (g) graph groupoids (e.g. [47, 64]).
\med As a simple, helpful example of a groupoid, consider (b) above. Thus, let R be an equivalence relation on a set X. Then R is a groupoid under the following operations: $(x,y)(y,z)=(x,z),(x,y)^{-1}=(y,x)$. Here, Failed to parse (unknown function "\grp"): {\displaystyle \grp^0 = X } , (the diagonal of $X\times X$ ) and $r((x,y))=x,s((x,y))=y$. \med So $R^{2}$ = $\left\{((x,y),(y,z)):(x,y),(y,z)\in R\right\}$. When $R=X\times X$, R is called a trivial groupoid. A special case of a trivial groupoid is $R=R_{n}=\left\{1,2,...,n\right\}$ $\times$ $\left\{1,2,...,n\right\}$. (So every i is equivalent to every j ). Identify $(i,j)\in R_{n}$ with the matrix unit $e_{ij}$. Then the groupoid $R_{n}$ is just matrix multiplication except that we only multiply $e_{ij},e_{kl}$ when $k=j$, and $(e_{ij})^{-1}=e_{ji}$. We do not really lose anything by restricting the multiplication, since the pairs $e_{ij},{e_{kl}}$ excluded from groupoid multiplication just give the 0 product in normal algebra anyway. For a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} to be a locally compact groupoid means that Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is required to be a (second countable) locally compact Hausdorff space, and the product and also inversion maps are required to be continuous. Each Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} as well as the unit space Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^0} is closed in Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . What replaces the left Haar measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} is a system of measures $\lambda ^{u}$ (Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0} ), where $\lambda ^{u}$ is a positive regular Borel measure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u} with dense support. In addition, the $\lambda ^{u}$ \^a~@~Ys are required to vary continuously (when integrated against Failed to parse (unknown function "\grp"): {\displaystyle f \in C_c(\grp_{lc}))} and to form an invariant family in the sense that for each x, the map $y\mapsto xy$ is a measure preserving homeomorphism from Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^s(x)} onto Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^r(x)} . Such a system $\left\{\lambda ^{u}\right\}$ is called a left Haar system for the locally compact groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . \med
This is defined more precisely next.

#### Haar systems for locally compact topological groupoids

Let
   Failed to parse (unknown function "\xymatrix"): {\displaystyle \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)}}=X }
be a locally compact, locally trivial topological groupoid with its transposition into transitive (connected) components. Recall that for $x\in X$, the costar of $x$ denoted ${\rm {{CO}^{*}(x)}}$ is defined as the closed set Failed to parse (unknown function "\grp"): {\displaystyle \bigcup\{ \grp(y,x) : y \in \grp \}} , whereby
   Failed to parse (unknown function "\grp"): {\displaystyle \grp(x_0, y_0) \hookrightarrow \rm{CO}^*(x) \lra X~, }
is a principal Failed to parse (unknown function "\grp"): {\displaystyle \grp(x_0, y_0)} --bundle relative to fixed base points $(x_{0},y_{0})$~. Assuming all relevant sets are locally compact, then following Seda (1976), a \emph{(left) Haar system on Failed to parse (unknown function "\grp"): {\displaystyle \grp} } denoted Failed to parse (unknown function "\grp"): {\displaystyle (\grp, \tau)} (for later purposes), is defined to comprise of i) a measure $\kappa$ on Failed to parse (unknown function "\grp"): {\displaystyle \grp} , ii) a measure $\mu$ on $X$ and iii) a measure $\mu _{x}$ on ${\rm {{CO}^{*}(x)}}$ such that for every Baire set $E$ of Failed to parse (unknown function "\grp"): {\displaystyle \grp} , the following hold on setting $E_{x}=E\cap {\rm {{CO}^{*}(x)}}$~:
\med
The presence of a left Haar system on Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} has important topological implications: it requires that the range map Failed to parse (unknown function "\grp"): {\displaystyle r : \grp_{lc} \rightarrow \grp_{lc}^0<math> is open. For such a } \grp_{lc}</math> with a left Haar system, the vector space Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} is a convolution *--algebra , where for Failed to parse (unknown function "\grp"): {\displaystyle f, g \in C_c(\grp_{lc})} : \\ \med $f*g(x)=\int f(t)g(t^{-1}x)d\lambda ^{r(x)}(t)$, with f*(x) $={\overline {f(x^{-1})}}$. \med One has Failed to parse (unknown function "\grp"): {\displaystyle C^*(\grp_{lc})} to be the enveloping C*--algebra of Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} (and also representations are required to be continuous in the inductive limit topology). Equivalently, it is the completion of Failed to parse (unknown function "\grp"): {\displaystyle \pi_{univ}(C_c(\grp_{lc}))} where $\pi _{univ}$ is the universal representation of Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} . For example, if Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc} = R_n} , then Failed to parse (unknown function "\grp"): {\displaystyle C^*(\grp_{lc})} is just the finite dimensional algebra Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc}) = M_n} , the span of the $e_{ij}$'s.
There exists (e.g.[63, p.91]) a measurable Hilbert bundle Failed to parse (unknown function "\grp"): {\displaystyle (\grp_{lc}^0, \mathbb{H}, \mu)} with Failed to parse (unknown function "\grp"): {\displaystyle \mathbb{H} = \left\{ \mathbb{H}^u_{u \in \grp_{lc}^0} \right\}<math> and a G-representation L on } \H</math>. Then, for every pair $\xi ,\eta$ of square integrable sections of $\mathbb {H}$, it is required that the function $x\mapsto (L(x)\xi (s(x)),\eta (r(x)))<math>be$\nu$--measurable.Therepresentation$\Phi</math> of Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})} is then given by:\\ $\left\langle \Phi (f)\xi \vert ,\eta \right\rangle =\int f(x)(L(x)\xi (s(x)),\eta (r(x)))d\nu _{0}(x)$.
The triple $(\mu ,\mathbb {H} ,L)$ is called a \textit{measurable Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}} --Hilbert bundle}.