Twisted dynamical systems, Schrodinger representations

MYKOLA YAREMENKO

Department of Partial Differential Equations,

The National Technical University of Ukraine,

“Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, UKRAINE

Abstract: - Assume

G

is a locally compact Hausdorff group,

A

is a

C

-algebra, and

 

,G



A,

is a

dynamical system, we consider a Takai theorem that states the isomorphism

   

 

ˆˆ

:G G LK L G



     2

AA

is equivariant for

 

ˆ

ˆˆ

ˆ:G G G





  A

and for

   

 

ˆ

ˆ:Ad G LK L G



   2

A

. Also, we show that



-surjective mapping

   

: , ,

CC

C G C G



A A,

can be extended to quotient mapping

 

:/G G G G



   

      A A A AI

for the twisted dynamical system

 

,,G



A,

.

We establish that there exists an isomorphism of the Schrodinger

C

-algebra

 

Sch G





A

to the

reduced crossed product

red

G











A

; and show the representation

 

Sch G LB L G

  





   2

AA

is faithful for each amenable group

G

.

Key-Words: - Takai Duality,



-duality, Wigner function,

C

-algebra, Pontryagin duality, induced

representation, cross product.

1 Introduction (dynamic systems)

Let

G

be a locally compact Hausdorff

group and let



be a Radon measure on

G

.

Assume that

N

is a locally compact subgroup of

G

then

/GN

is locally compact Hausdorff group.

Definition. The group

G

is called an

extension of the group

N

by the group

H

if the

short group sequence

jk

e N G H e   

(1)

is exact, where

j

is a continuous homeomorphism

onto the range of

j

and

k

is a continuous open

surjective mapping.

If we assume that

N

a normal subgroup

of

G

then

H

is a quotient group

/GN

. So,

symbolically we can write a short exact sequence

of groups

//

jk

e N N G N G N e    

. (2)

Definition 2. Let

A

be a

C

-algebra and

G

locally compact Hausdorff group then the

triplet

 

,G



A,

is called a dynamical system

where

 

:G Aut



A

is a strongly continuous

representation.

.

Received: September 15, 2022. Revised: May 16, 2023. Accepted: Jane 11, 2023. Published: July 20, 2023.

PROOF

DOI: 10.37394/232020.2023.3.3

Mykola Yaremenko

E-ISSN: 2732-9941

14

Volume 3, 2023

Definition 3. Let

 

,G



A,

be a dynamic

system on the Hilbert space

H

. Let



be unitary

representation

 

:G U H





and

 

:LB H



A

representation on the Hilbert

space

H

, such that

 

     

,g a g a g

    





. (3)

Then, the pair

 

,



is called a

covariant representation of

 

,G



A,

.

Definition 4. Let

 

,



be a covariant

representation of the dynamic system

 

,G



A,

on the Hilbert space

H

. The

L1

-norm-decreasing



-representation

 

,

C

CGA

on

H

is given by

    

 

   

G

h h d h

      





(4)

for all

 

C

CG





.

We denote a continuous homomorphism

:GR





such that the equality

         

GG

g hg d h h d h

   





(5)

holds for all

 

C

CG





.

To show that mapping





is



-

homomorphism, we compute

   

 

   



 

   

 

,

.

G

h h d h

h h h h d h

      

   

    

  











  



  





1

11

(6)

Applying the Fubini theorem, we write

 

     

 

   

,

GG

h h h g

g d h d g

h h h g

h g d g d h

   

   

  

    

  

     



  



  



1

(7)

so





is



-homomorphism.

Definition 5. Let

 

,G



A,

be a dynamic

system

 

,G



A,

. The norm on

 

,

C

CGA

, of

the function

 

,

C

CG



A

given by

   

 

:,

sup cov ,

is

ariant representation of G



    

















A,

(8)

is called the universal norm.

Definition 6. The completion of the set

 

,

C

CGA

in the universal norm is called the

cross product

G



A

of

A

by

G

and present a

Banach

C

-algebra. The cross-product

G



A

is said to be associated with the dynamic system

 

,G



A,

.

Definition 7. Let

N

and

H

be a pair of

locally compact groups and let mapping

 

:H Aut N





be continuous homomorphism

maps as

    

,h n h n



for all

,n N h H

, and let the short sequence

e N G H e



   

(9)

be exact, where

      

 

h n h n h

    



11

,

homomorphism

:HG





satisfies

 

id H





(identity map on

H

), then the

PROOF

DOI: 10.37394/232020.2023.3.3

Mykola Yaremenko

E-ISSN: 2732-9941

15

Volume 3, 2023

group

G

is called a semi-direct product

NH





of pair of groups

N

and

H

.

2. The Takai dynamical system

Let

G

be an Abelian locally compact group

and let

 

,G



A,

be a dynamical system. A

homomorphism

 

ˆ

ˆ:G Aut G



A

is given by

extending of the mapping

     

ˆ: , ,

CC

C G C G



AA

,

       

ˆg g g

    

.

We denote the space of all linear compact

operators on the Hilbert space

 

LG

2

by

 

LK L G

2

.

The Takai duality theorem states that if

assume

 

,G



A,

is a dynamical system then the

isomorphism

   

 

ˆˆ

:G G LK L G



     2

AA

is

equivariant for

 

ˆ

ˆˆ

ˆ:G G G





  A

and for

   

 

ˆ

ˆ:Ad G LK L G



   2

A

, where the

mapping

 

:G LK L G



 2

A

is the right

regular representation.

We remind our reader two statements of the

Peter-Weyl theorem: the first statement, the

collection of matrix coefficients of the group

G

is

dense in

 

CG

relevant to the uniform topology;

the second statement, assume

:GH





is a

unitary representation of

G

in Hilbert space

 

H L G2

, then

 

H L G2

can be presented in

the form of the direct sum of irreducible finite-

dimensional unitary representation of

G

. Let

G

be

compact, the Peter-Weyl theorem implies that each

ˆ

ˆG





equals a subrepresentation of the left-regular

representation

 

:G U L G



2

.

The proof of the Takai theorem is based on

the following sequence of isomorphisms

 

   

 

ˆˆˆ

,

.

id

G G G G

C G G

C G G C G G

LK L G























     

  

    

 

12

1

3

2

34

5

0

00

2

AA

A

AA

A

(10)

The subalgebras

 

ˆ,

C

C G GA

and

 

ˆ,

C

C G GA

are dense in the

 

ˆˆ

GG



A

and

 

ˆ

id GG







1

A

, respectively. The

isomorphism

   

ˆˆ

: , ,

CC

C G G C G G   

1AA

is given by

      

,,f g g f g

  



1

for all

 

ˆ,

C

f C G GA

next extends to

 

ˆˆˆ

:id

G G G G





      1

1AA

. The

mapping

1

is continuous in the inductive limit

topology.

The second isomorphism

 

ˆ

: , , ,

CC

C G G C G C G  

20

AA

is a

Fourier transform given by

        

ˆ

,,

G

f g h f g h d

   





2

(11)

for all

 

ˆ,

C

f C G GA

.

The third isomorphism

3

is defined as

      

 

,,f g h h f g h







1

3

for all

 

,,

C

f C G C G0A

. Next, we need the

following lemma.

Lemma (Raeburn) 1. Let

 

,,G





be a

dynamical system and let



be a

C

-algebra,

then

 

max maxid GG





      

(12)

is equal in the isomorphic sense.

The proof is based on the Raeburn theorem.

PROOF

DOI: 10.37394/232020.2023.3.3

Mykola Yaremenko

E-ISSN: 2732-9941

16

Volume 3, 2023

We define an isomorphism

   

 

:C G G LK L G



   2

0

by

    

 

,,

G

f g h f k h k g d k







1

(13)

for

   

C

f C G G C G G



  

0

and

   

C

C G L G





2

. Thus, by Raeburn lemma,

there exists an equivariant isomorphism

   

 

,id

C G G LK L G







  

42

0AA

.

The necessary isomorphism

   

 

ˆˆ

:G G LK L G



     2

AA

can be

written as a combination

     

4 3 2 1

.

3. Twisted dynamical system

Let

 

,G



A,

be a dynamic system. Let

N

be a normal subgroup of

G

. Let

 

UM A

be a

unitary group of multiplier algebra of

A

.

Definition 8. A continuous homomorphism

 

:N UM



A

such that

     

,n a n n a

  



and

 

,g n gng

  



1

for all

nN

,

gG

,

aA

, is called a twisting map. The covariant

representation

 

,



of the dynamic system

 

,G



A,

is called preserving

 

:N UM



A

if equality

 

nn

  



holds for all

nN

.

The quartet

 

,,G



A,

is called a twisted

dynamical system.

Definition 9. Let

 

,G

ii

A

be a canonical

covariant homomorphism from

 

,G



A,

to

 

MG



A

. Let

I

be the ideal of multiplier

algebra

 

MG



A

of the cross product

G



A

generated by a set consisting of

   

 

 

:

G

i n i n n N





A

.

We define the twisted crossed product by

 

/

def

G G G



  

   A A AI

. (14)

Definition 10. We denote by

 

,

C

CG



A,

the subclass of continuous functions from

G

to

A

and satisfy the conditions: first, there exists a

compact subset

KG

such that

 

supp KN





; second, for all

 

,

C

CG



A,

the equality

     

ng g n

  





holds for all

nN

and

gG

.

Let

N



and

/GN



be Haar measure on

N

and

/GN

, respectively, then there the equality

         

/

N G N

G G N N

g d g hn d n d h

    



  

(15)

holds for all

 

C

CG





.

The convolution product

 

,

C

CG

  

 A,

is given by

   

 

/

,GN

GN

h k k k h d k

     





1

,

(16)

and function

 

   



 

/ , ,h G N h h h

  



  

 11

,

 

,

C

CG



A,

. (17)

So, we obtain that the set

 

,

C

CG



A,

is a



-algebra with the convolution operations given

by (16) and (17).

Lemma 2. For every function

 

,

C

CG



A

, we define a function

   

,

C

CG



 A,

by

    

 

N

g gn gng d n

   





1

, (18)

PROOF

DOI: 10.37394/232020.2023.3.3

Mykola Yaremenko

E-ISSN: 2732-9941

17

Volume 3, 2023

thus, there exists a structure-preserving



-

surjective mapping

   

: , ,

CC

C G C G



A A,

.

Proof. Let

N

be a normal subgroup of a

locally compact group

G

. A continuous function

:GR



B

is called the Bruhat approximate

cross-section of

G

by

N

if the equality

   

1

N

gn d n





B

holds for all

gG

, and condition that

   

supp satur KB

is compact for any compact

KG

. Let

:GR



B

be Bruhat

approximate cross-section of

G

by

N

so we have

that

       

,

C

g g g C G



B A

for any

fixed function

 

,

C

CG



A,

and the equality

  

   

 

   

N

g

gn gn gng d n g



    







1

holds, and so



is a surjection.

Now, we show that

 





  

so we

write

   

 



 

   

 

   



 

/ , ,

/,

,

/,

,

N

g

G N h h g

G N h

h g n g ng d n

G N h

h g n g g n d n





   

   















  



  

  









  





11

1

11

1

1 1 1

 



 

/,

,N

N

G N h

n h g n n d n

   







  





1

11

   

 

   



 

   

/ , ,

,

.

N

G N h GN h

hn ng gn g d n

G h gn gng d n

g

   

  







   

  



   









11

1 1 1 1

11

Similarly, since

  

 

   

 

   

    

 

    

/

,

N

NG

GN

g

k k k gn gng d k d n

k k k g d k

g



     

   







  

   

   





11

1

we have

     

   

    

.

Assume covariant representation

 

,



preserves



, so that

 

,HH

e nI nI





for all

nN

, we define a norm

   

 

sup : , preserves



      



for all

 

,

C

CG



A,

.

The norm of the quotient product

G





A

is given as

 







. Thus, we obtain the following

theorem.

Theorem 1. Let

 

,,G



A,

be a twisted

dynamical system. Then, the completion of the set

 

,

C

CG



A,

with respect to the norm defined as

   

 

sup : , preserves



      



coincides with the crossed-product

 

/

def

G G G



  

   A A AI

, and



-

surjective mapping

   

: , ,

CC

C G C G



A A,

can be

PROOF

DOI: 10.37394/232020.2023.3.3

Mykola Yaremenko

E-ISSN: 2732-9941

18

Volume 3, 2023

extended to quotient mapping

:GG





   AA

.

4. An example of the Schrodinger

representation

Let

G

be a locally compact group and let

set

 

GA

be a

C

-algebra of all bounded left

translation invariant and left uniformly continuous

functions such that

  

 

,h g h g

  



1

.

Definition. Let

:GG





be a

measurable mapping, then we define a binal

operation given by

 

   

 



 

,

G

px

z x p z

z x z x p z x d z





  

   



  





12

1

1 1 1

2

for all functions

 

,,LG



1

12 A

; and the

involution define by

 



,,p x x x x p x



   



  

1

1 1 1

12

.

We define a crossed product

G





A

as

the enveloping

C

-algebra of the Banach



-

algebra

 

,LG

1A

with the convolution product



and completion in the universal norm

   

 

sup : : , ( , ) .

PP P L G LB H H





1A

The space

 

,

C

CGA

of all continuous

functions

GA

with compact support is a dense

subalgebra of

G





A

, where

:GG





is a

measurable function.

The Schrodinger representation

 

,,LG



2

is given by

 

z x z x







1

and

 

   



,

for any bounded function



, so that

 



is a

multiplication operator.

The covariant representation is given by the

integral

 

   

 

   

 



   

  

,

G

def

G

z z z d z

xz x xz z d z

Sch x





    

   









  









1

11

defined for

 

,LG



1A

and

 

LG



2

.

Let

 



A

be an enveloping

C

-algebra

of

 

pr pr

id F L G

1

AA

where operation

pr



means the projective tensor product and

F

is

a Fourier transform. Then, there exists an extension

 



FA

of

 

pr

id FA

such that

   

:G

  



  F A A A

.

There are well-known statements that: for

the locally compact group

G

to be amenable it is

necessary and sufficient that the left-regular

representation was faithful representation in

 

CG



; assuming

G

is a locally compact amenable group

then the reduced crossed product coincides with the

universal crossed product.

Theorem 2. The

C

-algebra

 

Sch G





A

is isomorphic the reduced crossed

product

red

G











A

. Assume that the group

G

is

amenable then the representation

 

Sch G LB L G

  





   2

AA

define

by

  

 



   

,

G

Sch x

xz x xz z d z



   









1

11

PROOF

DOI: 10.37394/232020.2023.3.3

Mykola Yaremenko

E-ISSN: 2732-9941

19

Volume 3, 2023

is faithful.

Proof. The Fourier transform is an

isomorphic mapping. The reduced crossed product

red

G











A

can be defined as the range of the left

regular



-representation in

 

LB L G G

2

. The

faithfulness of representation

 

Sch G LB L G

  





   2

AA

follows

from the equality of universal and reduced crossed

products for the amenable locally compact groups.

References

[1]. B. Abadie, Takai duality for crossed

products by Hilbert C ∗ -bimodules, J. Operator

Theory 64 (2010), 19–34.

[2]. S. Albandik and R. Meyer, Product systems

over Ore monodies Doc. Math. 20 (2015) 1331–

1402.

[3]. A. Alldridge, C. Max, M. R. Zirnbauer,

Bulk-Boundary Correspondence for Disordered

Free-Fermion Topological Phases, Commun. Math.

Phys. 377, 1761–1821 (2020).

[4]. E. Bedos, S. Kaliszewski, J. Quigg, and D.

Robertson, A new look at crossed product

correspondences and associated C ∗ –algebras, J.

Math. Anal. Appl. 426 (2015), 1080– 1098.

[5]. N. Dupuis, L. Canet, A. Eichhorn, W.

Metzner, J. Pawlowski, M. Tissier, and N.

Wschebor. The nonperturbative functional

renormalization group and its applications, Physics

Reports 910, 1–114 (2021).

[6]. V. Deaconu, Group actions on graphs and C

∗ -correspondences, Houston J. Math. 44 (2018),

147–168.

[7]. V. Deaconu, A. Kumjian, and J. Quigg,

Group actions on topological graphs, Ergodic

Theory Dynam. Systems 32 (2012),1527–1566.

[8]. A. Carey, G. C. Thiang, The Fermi gerbe of

Weyl semimetals, Letters Math. Phys. 111, 1-16

(2021).

[9]. S. Kaliszewski, J. Quigg and D. Robertson,

Coactions on Cuntz-Pimsner algebras, Math. Scand.

116 (2015), 222–249.

[10]. E. Katsoulis, Non-selfadjoint operator

algebras: dynamics, classification, and C ∗ -

envelopes, Recent advances in operator theory and

operator algebras, 27–81, CRC Press, Boca Raton,

FL, (2018).

[11]. E. Katsoulis, C∗ -envelopes and the Hao-Ng

Isomorphism for discrete groups, International

Mathematics Research Notices, Volume 2017, Issue

18 (2017), 5751–5768.

[12]. I. Raeburn. Dynamical systems and operator

algebras. In National Symposium on Functional

Analysis, Optimization and Applications, 109–119.

Australian National University, Mathematical

Sciences Institute, (1999).

[13]. S. Sundar, C∗-algebras associated to

topological Ore semigroups, M¨unster J. of Math. 9

(2016), no. 1, 155–185.

[14]. M.I. Yaremenko Calderon-Zygmund

Operators and Singular Integrals, Applied

Mathematics & Information Sciences: Vol. 15: Iss.

1, Article 13, (2021).

PROOF

DOI: 10.37394/232020.2023.3.3

Mykola Yaremenko

E-ISSN: 2732-9941

20

Volume 3, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

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