THEORY OF FILTRATRIONS OF AF-ALGEBRAS AND ......Filtrations in measure theory We discuss rstly commutative case 1. A ltration is the decreasing sequence of the sigma- elds of measurable

THE MATHEMATICAL LEGACY OF UFFE HAAGERUP

SYMPOSIUM, June 24-26, 2016, Copenhagen

THEORY OF FILTRATRIONS OFAF-ALGEBRAS AND STANDARDNESS.

”Non-commutative version of the theory ofmeasure-theoretical filtrations”.

ANATOLY M. VERSHIK (ST. PETERSBURG DEPT. OFTHE MATHEMATICAL INSTITUTE OF RUSSIAN

ACADEMY OF SCIENCES, MATHEMATICALDEPARTMENT OF ST.PETERSBURG UNIVERSITY,

MOSCOW INSTITUTE OF THE PROBLEMS OFTRANSMISSION INFORMATION)[email protected]

June 25, 2016

Filtrations in measure theory

We discuss firstly commutative case1. A filtration is the decreasing sequence of the sigma-fields ofmeasurable sets of the space (X , µ):

A0 ⊃ A1 ⊃ A2 . . . .

The sigma field A0 = A.Each sigma field canonically corresponds to the measurablepartition of the space (X , µ): each elements of sigma-field A arethe sets which are consist with the blocks of that partition. Sofiltration uniquely mod 0 generates the decreasing sequence ofthe measurable partitions {ξn}n:

ξ0 � ξ1 � . . . ; ξ0 −−− partition on the separate points.

Another type of description of the filtration as decreasing sequenceof subalgebras of functions (or, equivalently — sequence of theoperators of mathematical expectation on sigma-fields Ai ):

L∞(X , µ) ⊃ L∞(Xξ1 , µξ1) . . . or Id > Pξ1 > Pξ2 . . .

Filtrations in measure theoryWe discuss firstly commutative case1. A filtration is the decreasing sequence of the sigma-fields ofmeasurable sets of the space (X , µ):

A0 ⊃ A1 ⊃ A2 . . . .

The sigma field A0 = A.

Each sigma field canonically corresponds to the measurablepartition of the space (X , µ): each elements of sigma-field A arethe sets which are consist with the blocks of that partition. Sofiltration uniquely mod 0 generates the decreasing sequence ofthe measurable partitions {ξn}n:





A0 ⊃ A1 ⊃ A2 . . . .






A0 ⊃ A1 ⊃ A2 . . . .





Tail filtrations on the space of paths of Bratteli diagram ofAF -algebras

We consider the filtrations (decreasing sequences) of commutativesubalgebras of measurable functions and filtration of subalgebras ofAF -algebras.Let A is an AF -algebra and Γ =

∐n Γn its Bratteli diagram

(N-graded locally finite graph) Γn is n− th (finite) level of graph Γ.Consider the space T (Γ) of paths of graph Γ. It is a Cantor(topological) space (inverse limit of finite spaces) with tailequivalence relation τ(Γ) and with tail filtration An, n ≥ 0 inthe space of continuous functions C (T (Γ))The space of paths T (Γ) is the same as Markov compact(notstationary in general) and we can use terminology of the theory ofMarkov processes.If we choose a trace χ (central measure on A) then we obtain afiltration of the Past of markov measure with maximal entropy.


We consider the filtrations (decreasing sequences) of commutativesubalgebras of measurable functions and filtration of subalgebras ofAF -algebras.

Let A is an AF -algebra and Γ =∐

n Γn its Bratteli diagram(N-graded locally finite graph) Γn is n− th (finite) level of graph Γ.Consider the space T (Γ) of paths of graph Γ. It is a Cantor(topological) space (inverse limit of finite spaces) with tailequivalence relation τ(Γ) and with tail filtration An, n ≥ 0 inthe space of continuous functions C (T (Γ))The space of paths T (Γ) is the same as Markov compact(notstationary in general) and we can use terminology of the theory ofMarkov processes.If we choose a trace χ (central measure on A) then we obtain afiltration of the Past of markov measure with maximal entropy.




(N-graded locally finite graph) Γn is n− th (finite) level of graph Γ.

Consider the space T (Γ) of paths of graph Γ. It is a Cantor(topological) space (inverse limit of finite spaces) with tailequivalence relation τ(Γ) and with tail filtration An, n ≥ 0 inthe space of continuous functions C (T (Γ))The space of paths T (Γ) is the same as Markov compact(notstationary in general) and we can use terminology of the theory ofMarkov processes.If we choose a trace χ (central measure on A) then we obtain afiltration of the Past of markov measure with maximal entropy.




(N-graded locally finite graph) Γn is n− th (finite) level of graph Γ.Consider the space T (Γ) of paths of graph Γ. It is a Cantor(topological) space (inverse limit of finite spaces) with tailequivalence relation τ(Γ) and with tail filtration An, n ≥ 0 inthe space of continuous functions C (T (Γ))

The space of paths T (Γ) is the same as Markov compact(notstationary in general) and we can use terminology of the theory ofMarkov processes.If we choose a trace χ (central measure on A) then we obtain afiltration of the Past of markov measure with maximal entropy.




(N-graded locally finite graph) Γn is n− th (finite) level of graph Γ.Consider the space T (Γ) of paths of graph Γ. It is a Cantor(topological) space (inverse limit of finite spaces) with tailequivalence relation τ(Γ) and with tail filtration An, n ≥ 0 inthe space of continuous functions C (T (Γ))The space of paths T (Γ) is the same as Markov compact(notstationary in general) and we can use terminology of the theory ofMarkov processes.

If we choose a trace χ (central measure on A) then we obtain afiltration of the Past of markov measure with maximal entropy.




(N-graded locally finite graph) Γn is n− th (finite) level of graph Γ.Consider the space T (Γ) of paths of graph Γ. It is a Cantor(topological) space (inverse limit of finite spaces) with tailequivalence relation τ(Γ) and with tail filtration An, n ≥ 0 inthe space of continuous functions C (T (Γ))The space of paths T (Γ) is the same as Markov compact(notstationary in general) and we can use terminology of the theory ofMarkov processes.If we choose a trace χ (central measure on A) then we obtain afiltration of the Past of markov measure with maximal entropy.

Filtrations arise

• in the theory of random processes (stationary or not), as thesequences of “pasts” (or “futures”);• in the theory of dynamical systems, as filtrations generated byorbits of periodic approximations of group actions;• in statistical physics, as filtrations of families of configurationscoinciding outside some volume;• in the theory of C ∗-algebras and combinatorics, as tail filtrationsof the path spaces of equipped N-graded locally finite graphs(Bratteli diagrams).Many problems about filtrations have appeared in ergodic theory,(decreasing sequences of measurable partitions = filtrations of ofsigma-fields in the standard measure space), theory of stochasticprocesses (martingale theory), boundaries (Martin, exit,Poisson-Furstenberg etc.), theory of approximation of the groupactions; VWB-Ornstein criteria etc.

Filtrations arise

• in the theory of random processes (stationary or not), as thesequences of “pasts” (or “futures”);

• in the theory of dynamical systems, as filtrations generated byorbits of periodic approximations of group actions;• in statistical physics, as filtrations of families of configurationscoinciding outside some volume;• in the theory of C ∗-algebras and combinatorics, as tail filtrationsof the path spaces of equipped N-graded locally finite graphs(Bratteli diagrams).Many problems about filtrations have appeared in ergodic theory,(decreasing sequences of measurable partitions = filtrations of ofsigma-fields in the standard measure space), theory of stochasticprocesses (martingale theory), boundaries (Martin, exit,Poisson-Furstenberg etc.), theory of approximation of the groupactions; VWB-Ornstein criteria etc.

Filtrations arise

• in the theory of random processes (stationary or not), as thesequences of “pasts” (or “futures”);• in the theory of dynamical systems, as filtrations generated byorbits of periodic approximations of group actions;

• in statistical physics, as filtrations of families of configurationscoinciding outside some volume;• in the theory of C ∗-algebras and combinatorics, as tail filtrationsof the path spaces of equipped N-graded locally finite graphs(Bratteli diagrams).Many problems about filtrations have appeared in ergodic theory,(decreasing sequences of measurable partitions = filtrations of ofsigma-fields in the standard measure space), theory of stochasticprocesses (martingale theory), boundaries (Martin, exit,Poisson-Furstenberg etc.), theory of approximation of the groupactions; VWB-Ornstein criteria etc.

Filtrations arise

• in the theory of random processes (stationary or not), as thesequences of “pasts” (or “futures”);• in the theory of dynamical systems, as filtrations generated byorbits of periodic approximations of group actions;• in statistical physics, as filtrations of families of configurationscoinciding outside some volume;

• in the theory of C ∗-algebras and combinatorics, as tail filtrationsof the path spaces of equipped N-graded locally finite graphs(Bratteli diagrams).Many problems about filtrations have appeared in ergodic theory,(decreasing sequences of measurable partitions = filtrations of ofsigma-fields in the standard measure space), theory of stochasticprocesses (martingale theory), boundaries (Martin, exit,Poisson-Furstenberg etc.), theory of approximation of the groupactions; VWB-Ornstein criteria etc.

Filtrations arise

• in the theory of random processes (stationary or not), as thesequences of “pasts” (or “futures”);• in the theory of dynamical systems, as filtrations generated byorbits of periodic approximations of group actions;• in statistical physics, as filtrations of families of configurationscoinciding outside some volume;• in the theory of C ∗-algebras and combinatorics, as tail filtrationsof the path spaces of equipped N-graded locally finite graphs(Bratteli diagrams).

Many problems about filtrations have appeared in ergodic theory,(decreasing sequences of measurable partitions = filtrations of ofsigma-fields in the standard measure space), theory of stochasticprocesses (martingale theory), boundaries (Martin, exit,Poisson-Furstenberg etc.), theory of approximation of the groupactions; VWB-Ornstein criteria etc.

Filtrations arise

• in the theory of random processes (stationary or not), as thesequences of “pasts” (or “futures”);• in the theory of dynamical systems, as filtrations generated byorbits of periodic approximations of group actions;• in statistical physics, as filtrations of families of configurationscoinciding outside some volume;• in the theory of C ∗-algebras and combinatorics, as tail filtrationsof the path spaces of equipped N-graded locally finite graphs(Bratteli diagrams).Many problems about filtrations have appeared in ergodic theory,(decreasing sequences of measurable partitions = filtrations of ofsigma-fields in the standard measure space), theory of stochasticprocesses (martingale theory), boundaries (Martin, exit,Poisson-Furstenberg etc.), theory of approximation of the groupactions; VWB-Ornstein criteria etc.

Tail filtration of AF -algebras and its commutativecounterpart

The general notion of filtrations means a sequence (decreasing inour case of subalgebras, or sigma-fields.Now define tail filtration of algebra A — {An}, correspondingto the diagram Γ:

A = A0 ⊃ A1 ⊃ A2 ⊃ . . .where

An =∑u∈Γn

⊕1λ(u)

⊗An,u,

and algebras An,u are uniquely defined from the decompositions:∀n = 0, 1 . . . :

A ∼=∑u∈Γn

⊕Mλ(u)(C)

⊗An,u,

and λ(u) is dimension of u = number of paths form ∅ to u. Thefiltration {An} is the sequences of the commutants.


The general notion of filtrations means a sequence (decreasing inour case of subalgebras, or sigma-fields.

Now define tail filtration of algebra A — {An}, correspondingto the diagram Γ:

A = A0 ⊃ A1 ⊃ A2 ⊃ . . .where

An =∑u∈Γn

⊕1λ(u)

⊗An,u,


A ∼=∑u∈Γn

⊕Mλ(u)(C)

⊗An,u,




A = A0 ⊃ A1 ⊃ A2 ⊃ . . .

whereAn =

∑u∈Γn

⊕1λ(u)

⊗An,u,


A ∼=∑u∈Γn

⊕Mλ(u)(C)

⊗An,u,




A = A0 ⊃ A1 ⊃ A2 ⊃ . . .where

An =∑u∈Γn

⊕1λ(u)

⊗An,u,


A ∼=∑u∈Γn

⊕Mλ(u)(C)

⊗An,u,




A = A0 ⊃ A1 ⊃ A2 ⊃ . . .where

An =∑u∈Γn

⊕1λ(u)

⊗An,u,


A ∼=∑u∈Γn

⊕Mλ(u)(C)

⊗An,u,




A = A0 ⊃ A1 ⊃ A2 ⊃ . . .where

An =∑u∈Γn

⊕1λ(u)

⊗An,u,


A ∼=∑u∈Γn

⊕Mλ(u)(C)

⊗An,u,




A = A0 ⊃ A1 ⊃ A2 ⊃ . . .where

An =∑u∈Γn

⊕1λ(u)

⊗An,u,


A ∼=∑u∈Γn

⊕Mλ(u)(C)

⊗An,u,


Comments

The commutative subalgebra GZ(=Gelfand-Zetlin) of AF -algebraA inherit filtration on A and if we fix a central measure χ weobtain the filtration in L∞χ (T (Γ)).This is the same filtration whichwe define in the space of paths.Correspondence between classification of tail filtration ofAF -algebras and classification of the tail filtrations ofGZ -algebra (with respect to the traces or central measures)or in topological sense?Tail filtrations can be considered as ”co-approximation” of AFalgebras versus to approximation of it with finite-dimensionalsub-algebras, corresponding to the structure of graph Γ. (Cocycleof cotransition probability)Important: Tail filtration does not define uniquely theapproximation and Bratteli diagram, because various graphs Γ candefine the same C ∗-algebra and the same tail filtration in it.

Comments

The commutative subalgebra GZ(=Gelfand-Zetlin) of AF -algebraA inherit filtration on A and if we fix a central measure χ weobtain the filtration in L∞χ (T (Γ)).This is the same filtration whichwe define in the space of paths.

Correspondence between classification of tail filtration ofAF -algebras and classification of the tail filtrations ofGZ -algebra (with respect to the traces or central measures)or in topological sense?Tail filtrations can be considered as ”co-approximation” of AFalgebras versus to approximation of it with finite-dimensionalsub-algebras, corresponding to the structure of graph Γ. (Cocycleof cotransition probability)Important: Tail filtration does not define uniquely theapproximation and Bratteli diagram, because various graphs Γ candefine the same C ∗-algebra and the same tail filtration in it.

Comments

The commutative subalgebra GZ(=Gelfand-Zetlin) of AF -algebraA inherit filtration on A and if we fix a central measure χ weobtain the filtration in L∞χ (T (Γ)).This is the same filtration whichwe define in the space of paths.Correspondence between classification of tail filtration ofAF -algebras and classification of the tail filtrations ofGZ -algebra (with respect to the traces or central measures)or in topological sense?

Tail filtrations can be considered as ”co-approximation” of AFalgebras versus to approximation of it with finite-dimensionalsub-algebras, corresponding to the structure of graph Γ. (Cocycleof cotransition probability)Important: Tail filtration does not define uniquely theapproximation and Bratteli diagram, because various graphs Γ candefine the same C ∗-algebra and the same tail filtration in it.

Comments

The commutative subalgebra GZ(=Gelfand-Zetlin) of AF -algebraA inherit filtration on A and if we fix a central measure χ weobtain the filtration in L∞χ (T (Γ)).This is the same filtration whichwe define in the space of paths.Correspondence between classification of tail filtration ofAF -algebras and classification of the tail filtrations ofGZ -algebra (with respect to the traces or central measures)or in topological sense?Tail filtrations can be considered as ”co-approximation” of AFalgebras versus to approximation of it with finite-dimensionalsub-algebras, corresponding to the structure of graph Γ. (Cocycleof cotransition probability)

Important: Tail filtration does not define uniquely theapproximation and Bratteli diagram, because various graphs Γ candefine the same C ∗-algebra and the same tail filtration in it.

Comments


Comments


Main example: dyadic AF -algebras

Consider the main partial case, so called dyadic filtrations. Inmeasure-theoretic case was studied from the end of 60-th. Dyadicfiltration. For AF -algebras:Definition A filtration {An} of AF -algebra A called dyadic if⋂

nAn = {const1} and for all n ∈ N:

A ∼= M2n(C)⊗An ∀n ∈ N, and

⋂n

An = {const} (∗)

If a Bratteli diagram Γ generates a dyadic filtration then for eachu ∈ Γn, λ(u) = 2n.Example: The standard dyadic filtration of AF -algebra is a dyadicfiltration which is isomorphic to the filtration of infinite tensorproduct: A =

⊗∞1 M2(C) = (M2(C))⊗∞ :

{An}∞n=0 : An =∞⊗k=n

M2(C), n = 1, . . . .

(Dyadic filtration of AF -algebra (when exists) is an analogue of thenotion of countable tensor product of algebra M2(C))

Main example: dyadic AF -algebrasConsider the main partial case, so called dyadic filtrations. Inmeasure-theoretic case was studied from the end of 60-th. Dyadicfiltration. For AF -algebras:Definition A filtration {An} of AF -algebra A called dyadic if⋂



⋂n

An = {const} (∗)

If a Bratteli diagram Γ generates a dyadic filtration then for eachu ∈ Γn, λ(u) = 2n.

Example: The standard dyadic filtration of AF -algebra is a dyadicfiltration which is isomorphic to the filtration of infinite tensorproduct: A =

⊗∞1 M2(C) = (M2(C))⊗∞ :

{An}∞n=0 : An =∞⊗k=n

M2(C), n = 1, . . . .





⋂n

An = {const} (∗)


⊗∞1 M2(C) = (M2(C))⊗∞ :

{An}∞n=0 : An =∞⊗k=n

M2(C), n = 1, . . . .





⋂n

An = {const} (∗)


⊗∞1 M2(C) = (M2(C))⊗∞ :

{An}∞n=0 : An =∞⊗k=n

M2(C), n = 1, . . . .


Concrete examples of (dyadic) filtrations

1)Bernoulii (tensor product)

2)Random walk (RS);(V,Hoffmann-Rudolf,Parry, etc.)3)Action of

∑Z2, Entropy of filtration.

4)Graph of Ordered and Unordered pairs.(Corresponding Bratteli diagrams, Tower of measures.


1)Bernoulii (tensor product)2)Random walk (RS);(V,Hoffmann-Rudolf,Parry, etc.)

3)Action of∑

Z2, Entropy of filtration.4)Graph of Ordered and Unordered pairs.(Corresponding Bratteli diagrams, Tower of measures.


1)Bernoulii (tensor product)2)Random walk (RS);(V,Hoffmann-Rudolf,Parry, etc.)3)Action of

∑Z2, Entropy of filtration.

4)Graph of Ordered and Unordered pairs.(Corresponding Bratteli diagrams, Tower of measures.

Main questions

Questions:Is it true that each dyadic ergodic filtration of measure space isBernoulli (or product filtration?The same question for for AF -algebra:Is it true that all filtration in form (*) are isomorphic to thefiltration of (M2(C))⊗∞?Answer V70:There exist a continuum of the non-isomorphic ergodicdyadic filtrations of measure space and correspondingly —consequently, a continuum of the non-isomorphic dyadicAF -algebrasThe same is true for dyadic filtration of AF -algebra.General Problem To classify the tail filtrations, in particularydyadic filtrations, in commutative and non-commutative cases.

Main questions

Questions:Is it true that each dyadic ergodic filtration of measure space isBernoulli (or product filtration?The same question for for AF -algebra:Is it true that all filtration in form (*) are isomorphic to thefiltration of (M2(C))⊗∞?

Answer V70:There exist a continuum of the non-isomorphic ergodicdyadic filtrations of measure space and correspondingly —consequently, a continuum of the non-isomorphic dyadicAF -algebrasThe same is true for dyadic filtration of AF -algebra.General Problem To classify the tail filtrations, in particularydyadic filtrations, in commutative and non-commutative cases.

Main questions

Questions:Is it true that each dyadic ergodic filtration of measure space isBernoulli (or product filtration?The same question for for AF -algebra:Is it true that all filtration in form (*) are isomorphic to thefiltration of (M2(C))⊗∞?Answer V70:There exist a continuum of the non-isomorphic ergodicdyadic filtrations of measure space and correspondingly —consequently, a continuum of the non-isomorphic dyadicAF -algebras

The same is true for dyadic filtration of AF -algebra.General Problem To classify the tail filtrations, in particularydyadic filtrations, in commutative and non-commutative cases.

Main questions


Main questions


Continuation

We will consider filtrations either in a standard separable Borelspace, as the filtration of the “pasts” of a discrete time randomprocess {ξn}, −n ∈ N, in the space of realizations of thisprocess, more generally — as a filtration in the standard separablemeasure space (Lebesgue space);oras the tail filtration in the path space T (Γ) of an equipped gradedgraph Γ;more generally — filtration in the Cantor space (without measure).The filtration called ”discrete” if the conditional filtration{Ai/An; i = 0, 1, . . . n − 1} over sigma-field An for all n arefiltration (hierarchy) of the finite space with measure.

Continuation

We will consider filtrations either in a standard separable Borelspace, as the filtration of the “pasts” of a discrete time randomprocess {ξn}, −n ∈ N, in the space of realizations of thisprocess, more generally — as a filtration in the standard separablemeasure space (Lebesgue space);or

as the tail filtration in the path space T (Γ) of an equipped gradedgraph Γ;more generally — filtration in the Cantor space (without measure).The filtration called ”discrete” if the conditional filtration{Ai/An; i = 0, 1, . . . n − 1} over sigma-field An for all n arefiltration (hierarchy) of the finite space with measure.

Continuation

We will consider filtrations either in a standard separable Borelspace, as the filtration of the “pasts” of a discrete time randomprocess {ξn}, −n ∈ N, in the space of realizations of thisprocess, more generally — as a filtration in the standard separablemeasure space (Lebesgue space);oras the tail filtration in the path space T (Γ) of an equipped gradedgraph Γ;more generally — filtration in the Cantor space (without measure).

The filtration called ”discrete” if the conditional filtration{Ai/An; i = 0, 1, . . . n − 1} over sigma-field An for all n arefiltration (hierarchy) of the finite space with measure.

Continuation

We will consider filtrations either in a standard separable Borelspace, as the filtration of the “pasts” of a discrete time randomprocess {ξn}, −n ∈ N, in the space of realizations of thisprocess, more generally — as a filtration in the standard separablemeasure space (Lebesgue space);oras the tail filtration in the path space T (Γ) of an equipped gradedgraph Γ;more generally — filtration in the Cantor space (without measure).The filtration called ”discrete” if the conditional filtration{Ai/An; i = 0, 1, . . . n − 1} over sigma-field An for all n arefiltration (hierarchy) of the finite space with measure.

Limit invariants: Kolmogorov 0-1 Law

Let ξn, n < 0 sequence of random variables. The sigma-algebraAn =<< ξk , k ≤ −n >> - sigma-algebra of the past:

A0 ⊃ A1 ⊃ A2 . . . ,

this is tail filtration of the process ξn, n < 0,Kolmogoroff ”0− 1” Law: if ξn, n ≤ 0, is Bernoulli process then⋂

n An = N (trivial sigma-field).Filtration in the measure space called ergodic, or regular, orKolmogoroff, or has zero-one-law — if (N is trivial sigma-field):⋂

n

An = N,

is trivial sigma-fields.



A0 ⊃ A1 ⊃ A2 . . . ,

this is tail filtration of the process ξn, n < 0,

Kolmogoroff ”0− 1” Law: if ξn, n ≤ 0, is Bernoulli process then⋂n An = N (trivial sigma-field).

Filtration in the measure space called ergodic, or regular, orKolmogoroff, or has zero-one-law — if (N is trivial sigma-field):⋂

n

An = N,




A0 ⊃ A1 ⊃ A2 . . . ,


n An = N (trivial sigma-field).

Filtration in the measure space called ergodic, or regular, orKolmogoroff, or has zero-one-law — if (N is trivial sigma-field):⋂

n

An = N,




A0 ⊃ A1 ⊃ A2 . . . ,


n An = N (trivial sigma-field).Filtration in the measure space called ergodic, or regular, orKolmogoroff, or has zero-one-law — if (N is trivial sigma-field):⋂

n

An = N,


Bernoulli filtration in commutative case

DefinitionHomogeneous standard filtrations of the measure space is afiltration which is isomorphic in measure theoretic sense toBernoulli filtration (or filtration of product type) with arbitrarycomponents: filtration on the space

∏∞n−1(rn,mrn), where rn is

finite space with rn ∈ Nr 0 points, and mk a uniform measure onrn. Dyadic filtration: rn ≡ 2

We will give a general definition of standard filtration later.

Bernoulli filtration in commutative case

DefinitionHomogeneous standard filtrations of the measure space is afiltration which is isomorphic in measure theoretic sense toBernoulli filtration (or filtration of product type) with arbitrarycomponents: filtration on the space

∏∞n−1(rn,mrn), where rn is

finite space with rn ∈ Nr 0 points, and mk a uniform measure onrn. Dyadic filtration: rn ≡ 2

We will give a general definition of standard filtration later.

Homogeneous and semi-homogeneous filtrations -traces,”central measures” on the space opaths

Filtration called homogeneous if for each n almost all elements ofthe partition ξn are finite measure- space with the uniformconditional measure, and number of point are the same for given n.Filtration called semi-homogeneous if conditional measure ofalmost all elements of partition ξn is uniform.Homogeneous filtration whose number of points in one elements ofpartition ξn equal to rn called r -adic filtration (dyadic for r = 2).Each discrete ergodic filtration correctly define an ergodicequivalence relation: two points x , y belongs to the same class ifthere exists such n that they belongs to the same element ofpartition ξn.A concrete discrete filtration called Markov filtration if is the pastof a one-sided Markov chain with discrete time, with finite list oftransition probabilities and arbitrary state space. Each discretefiltration can be realized as Markov one.


Filtration called homogeneous if for each n almost all elements ofthe partition ξn are finite measure- space with the uniformconditional measure, and number of point are the same for given n.

Filtration called semi-homogeneous if conditional measure ofalmost all elements of partition ξn is uniform.Homogeneous filtration whose number of points in one elements ofpartition ξn equal to rn called r -adic filtration (dyadic for r = 2).Each discrete ergodic filtration correctly define an ergodicequivalence relation: two points x , y belongs to the same class ifthere exists such n that they belongs to the same element ofpartition ξn.A concrete discrete filtration called Markov filtration if is the pastof a one-sided Markov chain with discrete time, with finite list oftransition probabilities and arbitrary state space. Each discretefiltration can be realized as Markov one.


Filtration called homogeneous if for each n almost all elements ofthe partition ξn are finite measure- space with the uniformconditional measure, and number of point are the same for given n.Filtration called semi-homogeneous if conditional measure ofalmost all elements of partition ξn is uniform.

Homogeneous filtration whose number of points in one elements ofpartition ξn equal to rn called r -adic filtration (dyadic for r = 2).Each discrete ergodic filtration correctly define an ergodicequivalence relation: two points x , y belongs to the same class ifthere exists such n that they belongs to the same element ofpartition ξn.A concrete discrete filtration called Markov filtration if is the pastof a one-sided Markov chain with discrete time, with finite list oftransition probabilities and arbitrary state space. Each discretefiltration can be realized as Markov one.


Filtration called homogeneous if for each n almost all elements ofthe partition ξn are finite measure- space with the uniformconditional measure, and number of point are the same for given n.Filtration called semi-homogeneous if conditional measure ofalmost all elements of partition ξn is uniform.Homogeneous filtration whose number of points in one elements ofpartition ξn equal to rn called r -adic filtration (dyadic for r = 2).

Each discrete ergodic filtration correctly define an ergodicequivalence relation: two points x , y belongs to the same class ifthere exists such n that they belongs to the same element ofpartition ξn.A concrete discrete filtration called Markov filtration if is the pastof a one-sided Markov chain with discrete time, with finite list oftransition probabilities and arbitrary state space. Each discretefiltration can be realized as Markov one.


Filtration called homogeneous if for each n almost all elements ofthe partition ξn are finite measure- space with the uniformconditional measure, and number of point are the same for given n.Filtration called semi-homogeneous if conditional measure ofalmost all elements of partition ξn is uniform.Homogeneous filtration whose number of points in one elements ofpartition ξn equal to rn called r -adic filtration (dyadic for r = 2).Each discrete ergodic filtration correctly define an ergodicequivalence relation: two points x , y belongs to the same class ifthere exists such n that they belongs to the same element ofpartition ξn.

A concrete discrete filtration called Markov filtration if is the pastof a one-sided Markov chain with discrete time, with finite list oftransition probabilities and arbitrary state space. Each discretefiltration can be realized as Markov one.


Filtration called homogeneous if for each n almost all elements ofthe partition ξn are finite measure- space with the uniformconditional measure, and number of point are the same for given n.Filtration called semi-homogeneous if conditional measure ofalmost all elements of partition ξn is uniform.Homogeneous filtration whose number of points in one elements ofpartition ξn equal to rn called r -adic filtration (dyadic for r = 2).Each discrete ergodic filtration correctly define an ergodicequivalence relation: two points x , y belongs to the same class ifthere exists such n that they belongs to the same element ofpartition ξn.A concrete discrete filtration called Markov filtration if is the pastof a one-sided Markov chain with discrete time, with finite list oftransition probabilities and arbitrary state space. Each discretefiltration can be realized as Markov one.

Criteria of the standardness for dyadic filtrations

Consider the ergodic dyadic filtration ξn (in the form of measurablepartitions of [0, 1] with Lebesgue measure).The group Dn of all automorphisms of n-level dyadic tree with 2n

points acts on each block C : |C | = 2n of partition ξn, 1, . . . andconsequently acts on the functions on C .Let η = {B1,B2, . . .Bk} -an arbitrary finite measurable partition of[0, 1] with k blocks. On each block C ∈ ξn define a functionfn : C → k : fn(c) = i ∈ k : c = C ∩ Bi . Denote the orbit of actionof the group Dn on the vectors {fn(c)}c∈C as Orbn(C ). Finallydefine on the set of orbits of the group Dn the metric rn:rn(O1,O2) = minx∈O1,y∈O2 ρn(x , y), where ρn is Hamming metricon the vectors with value in kCriteria of standardness Dyadic filtration {ξn} is Bernoulli (orproduct-type or standard) iff ∀ finite measurable partition η

limn

∫[0,1]×[0,1]

rn(Orbn(C ),Orbn(C ′))dCdC ′ = 0.


Consider the ergodic dyadic filtration ξn (in the form of measurablepartitions of [0, 1] with Lebesgue measure).

The group Dn of all automorphisms of n-level dyadic tree with 2n


limn

∫[0,1]×[0,1]




points acts on each block C : |C | = 2n of partition ξn, 1, . . . andconsequently acts on the functions on C .Let η = {B1,B2, . . .Bk} -an arbitrary finite measurable partition of[0, 1] with k blocks. On each block C ∈ ξn define a functionfn : C → k : fn(c) = i ∈ k : c = C ∩ Bi . Denote the orbit of actionof the group Dn on the vectors {fn(c)}c∈C as Orbn(C ). Finallydefine on the set of orbits of the group Dn the metric rn:rn(O1,O2) = minx∈O1,y∈O2 ρn(x , y), where ρn is Hamming metricon the vectors with value in k

Criteria of standardness Dyadic filtration {ξn} is Bernoulli (orproduct-type or standard) iff ∀ finite measurable partition η

limn

∫[0,1]×[0,1]





limn

∫[0,1]×[0,1]


Criteria of standardness, continuation

It is natural to fix a trace of algebra and discuss about the imageof AF -algebra in the corresponding II1 representation. It is possibleto have different answer for different traces.AFC ∗-algebra is standard if for any indecomposable tracecorresponding tail filtration of the paths is standard.To describe standard AFC ∗-algebras.Criteria for homogeneous for general homogeneousfiltrations, iteration of Kantorovich metric (intrinsic metric)It is make sense in the case of AF -algebras to distinguish weak andstrong standardness: weak means that in all II1 factorrepresentations the image of algebra is standard in the sense whichis described below.


It is natural to fix a trace of algebra and discuss about the imageof AF -algebra in the corresponding II1 representation. It is possibleto have different answer for different traces.AFC ∗-algebra is standard if for any indecomposable tracecorresponding tail filtration of the paths is standard.To describe standard AFC ∗-algebras.

Criteria for homogeneous for general homogeneousfiltrations, iteration of Kantorovich metric (intrinsic metric)It is make sense in the case of AF -algebras to distinguish weak andstrong standardness: weak means that in all II1 factorrepresentations the image of algebra is standard in the sense whichis described below.


It is natural to fix a trace of algebra and discuss about the imageof AF -algebra in the corresponding II1 representation. It is possibleto have different answer for different traces.AFC ∗-algebra is standard if for any indecomposable tracecorresponding tail filtration of the paths is standard.To describe standard AFC ∗-algebras.Criteria for homogeneous for general homogeneousfiltrations, iteration of Kantorovich metric (intrinsic metric)

It is make sense in the case of AF -algebras to distinguish weak andstrong standardness: weak means that in all II1 factorrepresentations the image of algebra is standard in the sense whichis described below.


It is natural to fix a trace of algebra and discuss about the imageof AF -algebra in the corresponding II1 representation. It is possibleto have different answer for different traces.AFC ∗-algebra is standard if for any indecomposable tracecorresponding tail filtration of the paths is standard.To describe standard AFC ∗-algebras.Criteria for homogeneous for general homogeneousfiltrations, iteration of Kantorovich metric (intrinsic metric)It is make sense in the case of AF -algebras to distinguish weak andstrong standardness: weak means that in all II1 factorrepresentations the image of algebra is standard in the sense whichis described below.

Finite isomorphism, isomorphism, main problem

Two filtrations {An}∞n=0 and {A′n}∞n=0 called finitely isomorphic iffor each N the finite fragments for n = 0, 1 . . .N of its aremetrically isomorphic.(For each n there exists mp automorphismTn for each k < n TnAk = A′k}.The sets of all conditional measures of almost all elements of theall partitions ξn, n = 1, . . . are invariants of the finite isomorphism.Are there other invariants of ergodic filtrations besides the finiteinvariants?The problem of classification of discrete Markov filtrations in thecategory of measure spaces or in other categories is deep and quitetopical.


Two filtrations {An}∞n=0 and {A′n}∞n=0 called finitely isomorphic iffor each N the finite fragments for n = 0, 1 . . .N of its aremetrically isomorphic.(For each n there exists mp automorphismTn for each k < n TnAk = A′k}.

The sets of all conditional measures of almost all elements of theall partitions ξn, n = 1, . . . are invariants of the finite isomorphism.Are there other invariants of ergodic filtrations besides the finiteinvariants?The problem of classification of discrete Markov filtrations in thecategory of measure spaces or in other categories is deep and quitetopical.


Two filtrations {An}∞n=0 and {A′n}∞n=0 called finitely isomorphic iffor each N the finite fragments for n = 0, 1 . . .N of its aremetrically isomorphic.(For each n there exists mp automorphismTn for each k < n TnAk = A′k}.The sets of all conditional measures of almost all elements of theall partitions ξn, n = 1, . . . are invariants of the finite isomorphism.

Are there other invariants of ergodic filtrations besides the finiteinvariants?The problem of classification of discrete Markov filtrations in thecategory of measure spaces or in other categories is deep and quitetopical.


Two filtrations {An}∞n=0 and {A′n}∞n=0 called finitely isomorphic iffor each N the finite fragments for n = 0, 1 . . .N of its aremetrically isomorphic.(For each n there exists mp automorphismTn for each k < n TnAk = A′k}.The sets of all conditional measures of almost all elements of theall partitions ξn, n = 1, . . . are invariants of the finite isomorphism.Are there other invariants of ergodic filtrations besides the finiteinvariants?

The problem of classification of discrete Markov filtrations in thecategory of measure spaces or in other categories is deep and quitetopical.


Two filtrations {An}∞n=0 and {A′n}∞n=0 called finitely isomorphic iffor each N the finite fragments for n = 0, 1 . . .N of its aremetrically isomorphic.(For each n there exists mp automorphismTn for each k < n TnAk = A′k}.The sets of all conditional measures of almost all elements of theall partitions ξn, n = 1, . . . are invariants of the finite isomorphism.Are there other invariants of ergodic filtrations besides the finiteinvariants?The problem of classification of discrete Markov filtrations in thecategory of measure spaces or in other categories is deep and quitetopical.

Standardness for filtrations of the Borel (Cantor),forBratteli diagram

For the space of paths in graded graphs (Bratteli diagrams) ournotion of standardness made a distinguish class of graphs.

DefinitionAF -algebra called standard if for any ergodic central measure tailfiltration if standard in the previous sense.

Examples of standard graphs: Pascal, Young etc. Concentration,Limit shape theorem.

DefinitionThe result of transferring the metric ρX on the space X to theBorel space Y along the equipped map

φ : X → Y

is the metric ρY on Y defined by the formula

ρY (y1, y2) = kρX (νy1 , νy2),

where kρ is the classical Kantorovich metric on Borel probabilitymeasures on (X , ρX ).

This limit is just so called absolute boundary of the directed graph(Bratteli diagram) — for central measures his is the list of tracesof corresponded AF -algebra.






φ : X → Y


ρY (y1, y2) = kρX (νy1 , νy2),








φ : X → Y


ρY (y1, y2) = kρX (νy1 , νy2),








φ : X → Y


ρY (y1, y2) = kρX (νy1 , νy2),



What does it mean standardness

We define a class of ergodic filtrations — Standard Filtrations.Definition. A filtration {An} of the measure space called standardif is any quotient filtration over partition ξ : {An/ξ} which isfinitely isomorphic to it is isomorphic.This class has the following properties:Theorem 1)two standard ergodic filtrations are isomorphic iff theyare finitely isomorphic; e.g. the standard ergodic filtration has nometric invariants except finite.2)each ergodic filtration is finitely isomorphic to a standardfiltrations.Theorem So the class of all ergodic filtration is a fibre bundle overset of standard filtrations.Standardness is generalization of independence (”eventuallyindependence”).


We define a class of ergodic filtrations — Standard Filtrations.Definition. A filtration {An} of the measure space called standardif is any quotient filtration over partition ξ : {An/ξ} which isfinitely isomorphic to it is isomorphic.This class has the following properties:

Theorem 1)two standard ergodic filtrations are isomorphic iff theyare finitely isomorphic; e.g. the standard ergodic filtration has nometric invariants except finite.2)each ergodic filtration is finitely isomorphic to a standardfiltrations.Theorem So the class of all ergodic filtration is a fibre bundle overset of standard filtrations.Standardness is generalization of independence (”eventuallyindependence”).


We define a class of ergodic filtrations — Standard Filtrations.Definition. A filtration {An} of the measure space called standardif is any quotient filtration over partition ξ : {An/ξ} which isfinitely isomorphic to it is isomorphic.This class has the following properties:Theorem 1)two standard ergodic filtrations are isomorphic iff theyare finitely isomorphic; e.g. the standard ergodic filtration has nometric invariants except finite.

2)each ergodic filtration is finitely isomorphic to a standardfiltrations.Theorem So the class of all ergodic filtration is a fibre bundle overset of standard filtrations.Standardness is generalization of independence (”eventuallyindependence”).


We define a class of ergodic filtrations — Standard Filtrations.Definition. A filtration {An} of the measure space called standardif is any quotient filtration over partition ξ : {An/ξ} which isfinitely isomorphic to it is isomorphic.This class has the following properties:Theorem 1)two standard ergodic filtrations are isomorphic iff theyare finitely isomorphic; e.g. the standard ergodic filtration has nometric invariants except finite.2)each ergodic filtration is finitely isomorphic to a standardfiltrations.

Theorem So the class of all ergodic filtration is a fibre bundle overset of standard filtrations.Standardness is generalization of independence (”eventuallyindependence”).


We define a class of ergodic filtrations — Standard Filtrations.Definition. A filtration {An} of the measure space called standardif is any quotient filtration over partition ξ : {An/ξ} which isfinitely isomorphic to it is isomorphic.This class has the following properties:Theorem 1)two standard ergodic filtrations are isomorphic iff theyare finitely isomorphic; e.g. the standard ergodic filtration has nometric invariants except finite.2)each ergodic filtration is finitely isomorphic to a standardfiltrations.Theorem So the class of all ergodic filtration is a fibre bundle overset of standard filtrations.

Standardness is generalization of independence (”eventuallyindependence”).


We define a class of ergodic filtrations — Standard Filtrations.Definition. A filtration {An} of the measure space called standardif is any quotient filtration over partition ξ : {An/ξ} which isfinitely isomorphic to it is isomorphic.This class has the following properties:Theorem 1)two standard ergodic filtrations are isomorphic iff theyare finitely isomorphic; e.g. the standard ergodic filtration has nometric invariants except finite.2)each ergodic filtration is finitely isomorphic to a standardfiltrations.Theorem So the class of all ergodic filtration is a fibre bundle overset of standard filtrations.Standardness is generalization of independence (”eventuallyindependence”).

Martingale interpretation of standardness

We formulate the criteria in terms of past of Markov processes{Xn, n ≤ 0}Theorem(standard criteria). The Markov chain{xn, n ≤ 0, xn ∈ Xn} (Xn is the state space at moment N and itcould be depend on n) the filtration of the past of it called standardif ∀ε > 0,∃N ∈ N ∀n < −Nand An ⊂ XnProb(A) > 1− ε withthe following property

Exn,x ′nDist(Prob(.|xn),Prob(.|x ′n)) < ε, xn, x′n ∈ Xn

where Dist is a Kantorovich-like metric between conditionalmeasures Prob(.|xn) as a measures ”on the trees of the future”(see below).Theorem(V − 71) The standard dyadic (more generally,homogeneous) filtration is isomorphic to Bernoulli filtration (=thefiltration of the past of the classical Bernoulli scheme).


We formulate the criteria in terms of past of Markov processes{Xn, n ≤ 0}

Theorem(standard criteria). The Markov chain{xn, n ≤ 0, xn ∈ Xn} (Xn is the state space at moment N and itcould be depend on n) the filtration of the past of it called standardif ∀ε > 0,∃N ∈ N ∀n < −Nand An ⊂ XnProb(A) > 1− ε withthe following property











Comments

The condition in the definition asserts the convergence inprobability of the conditional measures in the very strong (unform)metric which take care about hierarchy of the future of thetrajectories. This is further strengthen of the martingale theoremwhich asserts the simple convergence of conditional structures andtook place for all ergodic filtrations.

Criteria:

lim

∫X

∫Xρn(x , y)dµ(x)dµ(y) = 0

The convergence in the condition is a strong generalization ofweak convergence of empirical (conditional) distributions to theunconditional distribution.There is no limit distribution but there are very strongconcentration of the many dimensional distributions up to couplingwhich preserve the hierarchy of conditional measures.

Comments

The condition in the definition asserts the convergence inprobability of the conditional measures in the very strong (unform)metric which take care about hierarchy of the future of thetrajectories. This is further strengthen of the martingale theoremwhich asserts the simple convergence of conditional structures andtook place for all ergodic filtrations.Criteria:

lim

∫X



Comments


lim

∫X


The convergence in the condition is a strong generalization ofweak convergence of empirical (conditional) distributions to theunconditional distribution.

There is no limit distribution but there are very strongconcentration of the many dimensional distributions up to couplingwhich preserve the hierarchy of conditional measures.

Comments


lim

∫X



Criteria of Standardness in term of Markov processes

Let X space of the trajectories of the Markov process (n < 0), ξn-is n-th partition of X of the filtration Define a sequence ofsemi=metrics as follows: ρ̄0 = ρ, and

ρ̄n+1(x , y) = k̄ρn(µC(x), µC(y))

where C (x),C (y) — elements of ξn which contain x , y , and k̄ρ isrevised Kantorovich metric for measures on the tree which wasdefined before.

DefinitionA filtration {An}n∈N is called standard if

limn→∞

∫ ∫X×X

ρ̄n(x , y)dµ(x)dµ(y) = 0 (1)

for any initial metric ρ.


Let X space of the trajectories of the Markov process (n < 0), ξn-is n-th partition of X of the filtration

Define a sequence ofsemi=metrics as follows: ρ̄0 = ρ, and

ρ̄n+1(x , y) = k̄ρn(µC(x), µC(y))



limn→∞

∫ ∫X×X

ρ̄n(x , y)dµ(x)dµ(y) = 0 (1)




ρ̄n+1(x , y) = k̄ρn(µC(x), µC(y))



limn→∞

∫ ∫X×X

ρ̄n(x , y)dµ(x)dµ(y) = 0 (1)




ρ̄n+1(x , y) = k̄ρn(µC(x), µC(y))



limn→∞

∫ ∫X×X

ρ̄n(x , y)dµ(x)dµ(y) = 0 (1)


What does it mean NON-standardness and ”highest 0-1Laws”

EVANESCENT (or VIRTUALLY) measure metric spaces andGromov-V. invariants of m-m-spaces.Theorem on classification of the measure-metric spaces.τ = (X , µ, ρ) admissible metric-measure space.Consider a map

F : (X∞, µ∞)→ M∞(R+),

whereF ({xn}n) = {ρ(xi , xk)}i ,k ; n, i , k = 1 . . .

Then random matrixF∗(µ

∞) ≡ Dτ

(”matrix distribution”) is the complete invariant of the triple τ w.r.to measure preserving isometry. The map τ −−− > Dτ iscontinuous in the right sense. What happened if there is asequence of m −m spaces which does not converge?Virtual matrix distributions.



F : (X∞, µ∞)→ M∞(R+),



∞) ≡ Dτ

(”matrix distribution”) is the complete invariant of the triple τ w.r.to measure preserving isometry. The map τ −−− > Dτ iscontinuous in the right sense.

What happened if there is asequence of m −m spaces which does not converge?Virtual matrix distributions.



F : (X∞, µ∞)→ M∞(R+),



∞) ≡ Dτ

(”matrix distribution”) is the complete invariant of the triple τ w.r.to measure preserving isometry. The map τ −−− > Dτ iscontinuous in the right sense. What happened if there is asequence of m −m spaces which does not converge?Virtual matrix distributions.

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THEORY OF FILTRATRIONS OF AF-ALGEBRAS AND ......Filtrations in measure theory We discuss rstly commutative case 1. A ltration is the decreasing sequence of the sigma- elds of measurable