# PlanetPhysics/Quantum Operator Algebras

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## Quantum operator algebras(QOA)Edit

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 -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.Edit

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 AlgebraEdit

Let denote a complex (separable) Hilbert space. A \emph{von
Neumann algebra} **Failed to parse (unknown function "\A"): {\displaystyle \A}**
acting on 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 denoted by ).

- (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 for all
where denotes the inner product defined on }~. For a well-presented
treatment of the geometry of the state spaces of quantum operator algebras, see e.g. Aflsen and Schultz (2003).

#### Hopf algebraEdit

First, a unital associative algebra consists of a linear space 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 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 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 , 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 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 , for ~. 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 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 ~.

\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.

#### GroupoidsEdit

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
to ~. 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)] (\gamma_1, \gamma_2) \in \grp^{(2)}</math>~.

\med
\item[(2)]
~, for all **Failed to parse (unknown function "\grp"): {\displaystyle x \in \grp^{(0)}}**
~.

\med \item[(3)] \gamma \in \grp</math>~.

\med \item[(4)] ~.

\med
\item[(5)]
Each has a two--sided inverse with ~.
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 * .

\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:
. Here, **Failed to parse (unknown function "\grp"): {\displaystyle \grp^0 = X }**
, (the diagonal of ) and .
\med
So = .
When , *R* is called a *trivial* groupoid. A special case of a trivial groupoid is
. (So every *i* is equivalent to every *j* ). Identify with the matrix unit . Then the groupoid is just matrix multiplication except that we only multiply when , and . We do not really lose anything by restricting the multiplication, since the pairs 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp_{lc}}**
. What replaces the left Haar measure on **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}**
is a system of measures (**Failed to parse (unknown function "\grp"): {\displaystyle u \in \grp_{lc}^0}**
), where is a positive regular Borel measure on **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u}**
with dense support. In addition, the \^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 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
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 groupoidsEdit

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 , the *costar of * denoted
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 ~. 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 on **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
, ii) a
measure on and iii) a measure on
such that for every Baire set of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \grp}**
, the following hold on
setting ~:

\item[(1)] is measurable. \med \item[(2)] ~. \med \item[(3)] , for allFailed to parse (unknown function "\grp"): {\displaystyle t \in \grp(x,z)}andFailed to parse (unknown function "\grp"): {\displaystyle x, z \in \grp}~.

\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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
, with
f*(x) .
\med
One has **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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
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
'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 of square integrable sections of ,
it is required that the function \nu \Phi</math> of
**Failed to parse (unknown function "\grp"): {\displaystyle C_c(\grp_{lc})}**
is then given by:\\ .

The triple is called a \textit{measurable
**Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}**
--Hilbert bundle}.

## All SourcesEdit

^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}^{[16]}^{[17]}^{[18]}^{[19]}^{[20]}^{[21]}^{[22]}^{[23]}^{[24]}^{[25]}^{[26]}^{[27]}^{[28]}

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