Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Given a compact and regular Hausdorff measure space ( ) , X µ , with µ a Radon measure, it is known that the generalised space ( ) X  of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions ( ) , L X µ ∞ on X. We confirm that the commutative von Neumann algebras ( ) B ⊂  M , with ( ) 2 , L X µ =  , are unitary equivariant to the maximal ideals of the commutative algebra ( ) C X . Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra ( ) C X following its action on ( ) X  and define its representation and ergodic dynamical system on the commutative von Neumann algebras M of ( ) B  .


Preliminaries
Assuming the preliminary materials of [1] which form the background of this work, we are motivated to explore an alternative approach to the representation of the dynamical system of the commutative algebra ( ) C X on a commutative von Neumann algebra using groupoid framework.relations existing between the von Neumann algebra and its commutants.Hence, the groupoid representations relate the predual of the algebra to the Hilbert space constituting its domain.
The starting point for the analytical method is the relationship between the closure of a space or subspace with the boundedness of the functions defined on it.Because the measurability of functions imposes a very little restriction on the space, according to Connes [3], which translates to closure of the space or subspace supporting the function; the space must be a closed interval [ ] , I a b = or a standard Borel space X for measurability to be granted.Hence, measurability of X is a structure defined on X by measurable functions.This measure structure is invariant under Aut X the transformations of X. (cf.[3]).The definition of a Borel measure as a positive operator valued set map by [4] connects these structures to operator algebra.The connection is based on the positivity and completeness of the Hilbert space of square integrable functions gives rise to the Banach algebra ( ) . This good interaction between the two structures is established using the Borel measures on X, whose mutual derivatives define measurable functions, and define linear forms ( ) C X and bilinear/sesquilinear forms or inner product ( ) Spectral consideration is used to identify the algebra of these measures as von Neumann algebra of operators on  .For the spectrum of a self-adjoint operator ( ) T ∈   can be analysed using polynomials ( ) p x which which are cons- titutive of the maximal ideals { } and the geometric structure of the algebra ( ) C X .Every polynomial ( ) p x on X defines an operator as a measurable function, and there is always a sequence of polynomials uniformly approximating a measurable function by Weierstrass approximation theorem.Thus, the continuous extensions of any continuous function vanishing at a point x X ∈ to a function vanishing in some closed subsets containing x define the spectrum of the resulting partially ordered operators.Cf. [4].
These continuous extensions are captured by the uniformly convergent sequences or nets, such that the sequence n ∈ ; which also gives rise to weak convergence This interval is related to the spectral radius of the operator T in a von Neumann algebra M which can be defined as follows.
is the algebra of operators on  of the form ( ) f T for some bounded Borel function f.This is called the von Neumann algebra generated by the operator T.
Hence, M is made up of operators with the same symmetries as T.This means that they commute with all unitary operators ( ( ) Because proper actions relate directly to slice theorem used in the cohomogeneity-one G-space analysis as in [5] [6] [7], some of the main results of the paper also relate to slice theorem.

The Algebra and the Generalized Space
According to [8], the time evolution of dynamical systems modelled by measurepreserving actions of integers  or real numbers IR which represent passage of time are generalized by measure-preserving actions of lattices which are usually "subgroup" of Lie groups.The two basic constituents of the commutative algebra ( ) C X : the Borel group of units ( )   convergence of closed sets to x has associated nets of polynomials or measurable functions converging to a function f defined on x.Given a net f α of contractions in the complete metric space X, as in [9], it follows that f f α → such that ( ) All these are represented on the generalized space ( ) The idea of a generalized space ( ) of Radon measures on X which is conceived as the state space (cf.[8]) is a direct extension of Mackey's conception of a measure class C as a generalized subset.At the centre of this extension is the focus on i) measure preserving transformations of the compact metric space X, and ii) the Dirac measures x δ as generalized or geometric points embedding the points of X in the generalized space.The role assigned to the ergodic transformations by Mackey, is to translate along time in such a way as to ensure the invariance of measure or state.
Every measure preserving continuous linear transformation : X X ϕ → de-composes into ergodic components.This is based on the fact that the "noncommutative spaces" replacing the "phase space" are basically quotient spaces determined by ergodic actions of a Borel group.Hence, X is embedded in the generalized space as its geometric or generalized points; that is, the ( ) According to [8], ergodic theorems express a relationship between averages taken along the orbit of a point under the iteration of a measure-preserving map or transformation.
invariance over states.The ergodicity represented by average over space or states (averages taken over the classes of measures) is given by nets of invariant nonergodic measures converging to an ergodic limit.It is also the convergence of operators to an ergodic operator in a von Neumann algebra.Cf. [4].These two averages are given by the invariance or stability of  and the measure classes { } Remark 2.2 From the proposition, we see that the restriction of * ϕ to the generalized points { } ( ) This shows that the set { } ( ) Proof.The coincidence of zero sets of ( ) . Since the elements of x m vanish at x, its ( ) 1 G -action is transferred to fibre of measure classes (or tangent measures to x δ ) via * ϕ (see [1]).

□
From this result, the dynamism defined by the transformations ϕ is encoded in the symmetry of the measures classes contributing to the convergent nets.
Hence, the connection between ergodic theory and the dynamics defined by continuous transformations on compact metric spaces is encoded by the closure of the resulting convex set of non-ergodic ϕ -invariant measures with ergodic measures as boundary.Cf. [8].

The Principal Groupoid and its Action
In what follows, we present the action of the commutative algebra ( ) employment also associates a z-ultrafilter related to a maximal ideal x m to the dynamical system.Thus, given the zero map ( ) is a closed cover for X.Cf. [10].An open cover for X can be constructed from their complements, a countable number of ( ) is an open covering for X and each inverse image ( ) forming the transition functions ( ) ( ) These transition functions also form unitary group if the Borel measures defined on the closed subsets are given the following characterisation.Definition 3.1 [4] A positive operator valued measure is a triple ( ) where X is a set,  is a ring (or σ-algebra) of subsets of X, and µ is an operator valued set function on  with the following properties 1) µ is positive, i.e. ( ) 2) µ is additive, i.e. ( ) ( ) ( ) 3) µ is continuous in the sense that ( ) These conditions are satisfied by the complements of ultra filters of zero sets ( ) x Z m of the maximal ideals x m of ( ) C X [10].Given a maximal filter x m , we have µ is an increasing sequence of Hermitian operators, with ( ) ( ) The group of automorphisms or transformations Proof.That µ is positive operator-valued implies the map ( ) Then Borel measures on X define positive operators on  since ( ) . As already noted, they are also linear forms ϕ on ( ) C X by the map ( ) is identified with the unitary operator on ( ) 2 , L X µ given by ( ) , Aut X µ by [11].Thus, ( ) is a subgroup of the unitary group in view of the positive operators defined by the measures associated with define the system of homeomorphisms which are the transition functions of the fibre bundle structure.
Proof.Given the definition of operator valued measures and with the preceding formulations, the group of automorphisms or transformations The action is fibrewise since , where Z is the zero map.□ We now use symmetry groupoid to capture these bundle symmetries.
Theorem 3. 4 The symmetries of the commutative algebra ( ) Proof.We use the trivialized action of the group of units ( ) x m to formulate the bundle structure.The indexed family

{ }
x x X ∈ = m  of geometries constitute a bundle  over X, with the projection , such that ( ) . The "symmetries" of these geometric (closed) points of the commutative algebra The smooth bundle symmetry ( )   is a Lie groupoid on X with respect to the following structure.For m , the partial multiplication is the composition of maps; the inverse of ( ) G ξ ∈  is its inverse as an isomorphism.The isotropy groups are the general linear groups ( ) of the fibres which are all isomorphic [12].□ Remark 3.5 The general linear groups ( ) x G m coincide with the unitary group when the bundle is considered a Hilbert bundle.They define the (partial)   symmetries of the system, which the Lie groupoid ( ) X    represents.
Given the Lie groupoid ( ) X    , its symmetries are modelled on the generalized space ( ) . This will be achieved through the formulation of the action of the Lie groupoid ( ) X    on the space of the generalized points { } homeomorphic to  .This will present the generalized space ( ) as a measure groupoid giving a generalized measure-theoretic approach to the dynamical system defined by the action of ( ) C X [13].
From Mackey's definition of generalized subset, there is a correspondence between closed subsets of X and Radon measures in ( ) ; such that the points of X coincide with the Dirac measures ( ) x X δ ∈  which are invariant ergodic measures [8].The Dirac measures define the point functionals (cf.[14]).
Because ( ) δ a probability measure.Hence, ( ) ( ) The action of the Lie groupoid on the set of generalized points { } , which is a continuous open map from the space  onto the unit space X, defines a left action of ( ) where ( )     is the set of composable pair ( ) , x ξ δ .This means ( ) . The action defines a groupoid equivalence on  .Given any pair , x y δ δ ∈ , we say that x y δ δ x y ξ δ ξ δ are composable pairs in ( ) This action of the Lie groupoid ( )   on  is free and proper.Because ξ δ δ ⋅ = implies that ξ is a unit.It is proper also because the map is a proper map; that is, the inverse image of a compact set is compact.The two make  a principal ( )   -space.Hence, the natural projection ( )    means that the groupoid  has a left action on  [15].□ Given that  is a left principal ( ) ⊂ ×      is a space of equivalence classes or pairs in  on which a diagonal action of ( )   is defined as follows: and ( ) Thus, the quotient groupoid H defines a right action on  .We therefore have: Thus, the action is given by composition ( ) , where ξ is unique in ( ) The action is well defined for given , , Proof.The proof follows the above.As we have seen, ( )   and H are locally compact groupoids, and  is a (locally) compact space that is i) a left principal ( )   -space, ii) a right principal H-space; and iii) the two actions commute; iv) the map ( ) X, and v) the map : From the construction, (iv) and (v) follow from the fact that if we have ( ) In [16] it was shown that every action of a Lie groupoid  on the arrows induces an action on the space of objects.So, the partial multiplication defined by  defines a self-action of the arrows which is reflected on the space of objects X and corresponds to elements in

( )
Aut X of the compact set X by homomorphisms.The composition of elements of ( ) Aut X form a unitary group which preserves the nets of Radon measures converging to ergodic measures or operators.Another formulation of the above as a gauge groupoid of a principal G-bundle is given in [12].We will consider the Haar system of measures for the Advances in Pure Mathematics There is always a symmetric quasi-invariant probability in a measure class.
For according to Hahn in Theorem 2.1 of [13], every probability measure in an invariant measure class C is quasi-invariant.Using his conditions, we strengthened the quasi-invariant condition for a Haar measure by modifying it to agree with a maximal ideal ) 3) Given this modification, we now have that for every co-null Borel set is a measure groupoid called an inessential reduction ( ,C   in [13], where the inessential reduction for an open conull From this, we see that each invariant measure class C ∈  form a system of . Since each system is defined on the t-fibre ( ) ( )( ) Thus, for a Borel function F on the groupoid ( ) with t-decomposition ( ) , where f U is conull as in the above; the quasi-invariance of ν implies With these constructions, Hahn showed that every measure groupoid has a measure ν satisfying (1) for ξ in an inessential reduction.
The concept of ergodicity will be delineated next and related to the dynamical system of the measure groupoid.

Convolution Algebra and Dynamical System
According to Hahn [13], the measure groupoid is ergodic if and only if there is a is null.In other words, ergodicity implies the existence of Dirac probability measures { } defined at each point of X.Thus, the existence of a ( )   -action on  makes it ergodic groupoid.This also implies that every Borel function φ on the base X can be expressed in the form of a positive Borel function F on the arrows ( ) , where F satisfies  .This means that the Borel functions on the arrows preserve the equivalence the groupoid ( ) defines on the base space X.Alternatively, as stated above, the ( ) preserves the Borel structure of the generalized space.
Subsequently, a real-valued Borel function F on the measure groupoid ( ) µ -a.e and for µ -almost all ξ , corresponds to a Borel function φ on X such that F s φ =  a.e.Thus, the invariant functions on the equivalence space X X  (or on the space X with ( ) -action) are of the form t φ  or s φ  .This shows the Borel functions are ( ) Therefore, the dynamical system is related to the convergence of the ultrafilters This follows from the involutive map on any Borel ( ) ( ) . Thus, the space ( ) ( ) is made into a normed * -algebra, with the norm of f given as the supremum norm The representation of this convolution algebra , .Thus, .The operator is shown to be an isometry as follows.
( ) ( ) We have shown this to be related to the ergodicity of the measure groupoid

( ) 2 , 2 ,
L X µ , and the fact that the Radon-Nikodym Theorem asserts that the derivatives d d µ ν are measurable functions f on X.It follows that while the Borel structure on X gives rise to the Hilbert space ( ) L X µ and von Neumann algebra, the topological structure defined by continuous functions : f X IR → commuting with T. Thus, given that S ∈ M , then the (lattice) algebra ( ) C X on the generalized space ( ) X  of nonnegative Radon measures on the space X; and the maximal ideals x m are the (projective) modules which characterize and encode the symmetries of measurable functions vanishing on the neighbourhoods of each point of X.These symmetries are represented by z-ultrafilters x  of zero sets (affine algebraic varieties of ( ) C X ) converging to each x X ∈ .The complements of these algebraic sets constitute the open neighbourhood base of points of X.The ultrafilter x

1 G
X using the groupoid equivalence.The ( ) C X -action is determined at each point maximal ideal is a module of the (lattice) algebra ( ) C X and a ( ) 1 G -space at every point x X ∈ .Hence, there is a trivialization of an action groupoid on X which we will now explore in order to describe the ( ) C X dynamical system on X and on the generalized space ( ) aid in the understanding of the dynamics associated to the commutative algebra ( ) C X at each x X ∈ .Their constitutes the structure group of the fibre bundle since it defines an action on the fibres x m given as the groupoid ( )   expresses the smoothly "varying symmetries" of the bundle.
the locally compact and Hausdorff orbit space ( ) \    is an open map, where ( ) \ is the equivalence relation defined by the open map ρ (or ( )   -action) on  .The equivalence classes are defined by having the same image in X.Since ( )   acts by composition on  , we have

Proposition 3 . 7
The groupoid of equivalence o H H  defined by ( )   -action on  defines a right action on the space  .Proof.Given the derived groupoid tinuous open map from the (locally) compact space  onto the unit space

,′Corollary 3 . 8 Theorem 3 . 9
and if the three [ ] [ ] [ ] are same orbit then there must be a unique element of ( )   such that x z δ δ ′  .This is given by The left ( )   -action ρ and right H-action σ commute on  .Proof.Given the right action σ of the equivalence groupoid H on  , it follows that  is a right principal H-space.The left action ρ of ( )   and the right action σ of H commute σ ρ ρ σ =   .The following diagram illustrates this commutativity of left ( )   -action ρ and right H-action σ on  .So, the action ρ induces a homeomorphism of : The space of generalized or geometric points  is a

xm
or the corresponding zero-sets [ ]

a
the system of Haar measures is not unique.Hence, any invariant measure class C determines the measure groupoid be a probability on the base space.The pair ( ) µ such that for some inessential reduction ( ) o   of ( ) Borel function on ( ) can therefore define a net of such positive Borel function F on the arrows ( )

2 µ 2 , 1 G
of the modular function ∆ which Peter Hahn defined and employed in Theorem 3.8 of[13] as ( ) that µ is a system of Haar measures supported on the fibres { }, x x X ∈ m; but given simply as µ because they are same or (groupoid) equivalent measures.Thus, the Hilbert space ( ) made up of the fibres ( )( ), x −  .But because the Borel functions F defined on the arrows are equal to Borel functions t φ  defined on X, having the net convergence t t[13], which is a simplified definition of von Neumann algebra arising from the maximal ideals x m and ergodic action of the Borel group ( ) on a compact measure space X.From the foregoing, the simplification is achieved by considering the system of Haar measures on the principal Lie groupoid ( ) X    , and using them in the Thus, the * -representation is a unitary representation since in conformity with our understanding of the Borel functions on the measure groupoid as probability measures on X or Haar system of measures on the groupoid.von Neuman algebra by representation.Proof.The Borel functions defined on the arrows of ( )   are defined on the fibres x m which contain the polynomials on X.So they are all defined on the operators on  as the representation showed.Hence, they are all of the form ( ) f T , which makes them von Neumann algebra as defined in the opening section.Alternatively, using Connes' characterization of commutative von Neumann algebra in 1.3 of[3] as the algebras of operators on Hilbert space that are invariant under a group (or subgroup) of unitary operators, it follows that the convolution algebra which implies invariance under transformations of X.A third characterization of a commutative von Neumann algebra by Connes[3] as an involutive algebra of operators that is closed under weak limits still reinforces the result.The presence of an ultra-filter the convergence of nets of local bisections α φ φ → of the principal Lie groupoid ( )  .)□