Download KL_2009 - Maths, NUS

Extending Pure States on C*-Algebras
and Feichtinger’s Conjecture
Special Program on Operator Algebras
5th Asian Mathematical Conference
Putra World Trade Centre, Kuala Lumpur
MALAYSIA 22-26 June 2009
Wayne Lawton
Department of Mathematics
National University of Singapore
[email protected]
http://www.math.nus.edu.sg/~matwml
Basic Notation
N  Z Q RC
denote the natural, integer, rational, real, complex numbers.
R / Z circle group
S  Borel ( R / Z ), | S | Haar measure
2
h, g  L ( R / Z ), h, g   h g
R/Z
  L ( R / Z ),


h, g


h
g


R/Z
Riesz Pairs
(S , Z1 ), S  Borel ( R / Z ), Z1  Z
satisfying any of the following equivalent conditions
1.   0  h  L2 ( R / Z )
|| h || 1, sup hˆ  Z1   | h |2  
2. 1  0  P  Q  1I
Pg   S g ,
S
(Qg ) ^   Z \ Z1 gˆ
Problem: characterize Riesz pairs
Synthesis Operator
(H , , )
denotes a complex Hilbert space
and the Hermitian form
is linear in
A subset
h
H  H  (h, g )  h , g
and congugate-linear in
g.
B  H defines a Synthesis Operator
Tfin : Cfin ( B)  H
Tfin f  bB f (b)b
Bessel Sets
B is a Bessel Set if Tfin : Cfin ( B)  H
admits an extension
T :  ( B)  H
2
Then its adjoint, the Analysis Operator,
T : H   ( B)

2
(T h)(b)  h, b and the

Frame Operator S  TT : H  H satisfies
exists and
SI

where
  || S || || T ||
2
Frames
B is a Frame for G  span ( B)  H
if it is a Bessel set
that satisfies any of the following equivalent conditions:
1.
2.
3.
T :  ( B)  G is surjective,

2
T |G : G   ( B) is injective,
2
  0  I G  S |G
Proof of Equivalence: [Chr03], pages 102-103.
Example.
B  {ek  ek 1 , k  N } is a Bessel set,
but not a frame for
H   ( N )  span ( B).
Proof: [Chr03], 98-99.
2
Riesz Sets
B is a Riesz Set if it is a Bessel set that satisfies
any of the following equivalent conditions:
T :  ( B)  G  span ( B)

2
2. T |G : G   ( B ) is bijective,
1.
3.
2
is bijective,
  0  I 2 ( B )  T T

Proof of Equivalence: [Chr03], 66-68, 123-125.
Example: Union of n > 1 Riesz bases for
H
H,
is always a frame for
but never a Riesz set.


Remark: T T is the grammian, and S  TT is the
dual- grammian used by Amos Ron and Zuowei Shen
http://www.math.nus.edu.sg/~matzuows/publist.html
Stationary Sets
B is a Stationary Set if there exists h H and a unitary
k
U : H  H such that B  {U h : k  Z }.
Then the function
g : Z  C , g (k )  h,U h
k
is positive definite so by a theorem of Bochner [Boc57]
there exists a positive Borel measure
g (k )   e
such that
2ix
dv
on
R/Z
dv.
R/Z
Example
H  L ( R / Z , dv), h  1, (U h)( x)  e
2
k
2ikx
Stationary Sets
If
B
is stationary set then
B
is a
1. Bessel set iff there exists a symbol function
  L ( R / Z )  dv   ( x)dx

and then
B
2. Frame iff
is a
  0  x  R / Z ,  ( x )  0     ( x )
3. Tight Frame iff
4. Riesz set iff

is constant on its support
  0  x  R / Z ,    ( x )
Proof. [Chr], 143-145.
Stationary Bessel Sets

with symbol   L ( R / Z )
Representation as Exponentials
H  L ( R / Z , ( x )dx )
2
B {e
(e
2imx
,e
2ikx
2ikx
: k Z }
)  ˆ (k  m)
Representation as Translates
H   ( Z ), h  gˆ , g  
2
B  {h(m  k ) : k  Z }
(h(  m), h(  k ))  ˆ (k  m)
Two Conjectures
Definition A Fechtinger set is a finite union of Riesz sets
Feichtinger Conjecture: Every Bessel set is Feichtinger set
Definition Let
  0. An  - Riesz set is one satisfying
(1   ) I  T T  (1   ) I

R- Conjecture: For every   0, every Riesz set is a
finite union of
- Riesz sets
Pave-able Operators
b  B( ( Z )) is pave-able if   0, n  N
and a partition Z  Z    Z 
1
n
2
(1)
|| Pj (b  diag (b)) Pj ||   || b  diag (b) ||
where
Pj :  ( Z )   ( Z )
2
Pj

2

is the diagonal projection
c
e

c
e

k
k
k
k
kZ
kZ
j
Observation This holds iff for every
are a finite union of
-Riesz sets

b
Theorem 1.2 in [BT87] There exists
density such that
satisfies (1)
P1
  0 the columns of
Z1  Z with positive
States on C*-Algebras
A State on a unital
C

- algebra
A
is a linear functional
 : A  C that satisfies any of the following equiv. cond.
1. ||  ||  1 and  ( I )  1

2. ||  ||  1 and a  A,  ( a a )  0
  {states on A} is convex and weakly compact
A Pure State is an extremal state
Krein-Milman 
Examples
  conv {pure states }
 ( Z )  C ( Z ), p  Z ,  (a )  a( p)

B ( Z ), v   ( Z ),  (a)  av, v
2
2
The Kadison-Singer Problem


A   (Z )
~ on B(2 ( Z )) ?
have a unique extension to a state 
Does every pure state
on
Remarks
Problem arose from Dirac quantization
Hahn-Banach extensions always exist
~
~
p  Z  Z  !    |A (a )  ae p , e p
YES answer to KS is equivalent to:
- combination of the Feichtinger and
R conjectures
2
- Paving Conjecture: every b  B(  ( Z )) is pave-able
- other conjectures in mathematics and engineering
Two Conjectures for Stationary Sets
B be a Bessel set with symbol   L ( R / Z )
Observation B satisfies Feichtinger’d conjecture iff   0

Let
Z  Z1    Z n , ( S , Zi ) is a Riesz pair where
S  {x  R / Z :  ( x )   }
[HKW86,86] If  is Riemann integrable then B satisfies
both the Feichtinger and
conjectures.
R
Theorem 4.1 in [BT91] If
then
  0  kZ | ˆ (k ) | | k |  
2
  0, B is a finite union of  -Riesz bases
Corollary 4.2 in [BT91] There exist dense open subsets of
R/Z whose complements have positive measure and whose
characteristic functions satisfy the hypothesis above.
Observation The characteristic functions of their
complementary ‘fat Cantor sets’ satisfy both conjectures
Feichtinger Conjecture for Stationary Sets
We consider a stationary Bessel set B with symbol 
Then
B  e
2ikx
:k Z
 H  span ( B)
where the closure is wrt the hermitian product
h, g
Definition If
Z1  Z

h
g


R/Z
then we call
B1   e2ikx : k  Z1
Theorem If

( , Z1 ) a Riesz pair if
 is a Riesz set.
m  Z , n  N , Z1  m  nZ then ( , Z1 )
is a Riesz pair iff
Corollary Never for
  0   j 0  ( x  )  
n 1
  C where C
j
n
is a fat Cantor set
New Results
  M ( R / Z )  ˆ  L ( Z )
Pseudomeasure
S  Borel ( R / Z ), Z1  Z
,  M ( R / Z ), sup(   v)  Z1
Y  S  sup(  )  R / Z
Theorem 1. If
then
1. h  L ( R / Z ), || h ||  1, sup( h)  Y  
2
2
2. f  (    )  h   | f |  ||ˆ ||
2
S
3. inf  | f | / || f ||  0  ( S , Z1 ) not RB
2
S
4.
this happens if
2
 
‘contains’ a point measure
New Results
Definition Given a triplet
topological group,
and
( X ,  ,V )
where
X is a compact
 : Z  X is a homomorphism,
V
is an open neighborhood of the the identity in
1
we call Z ( X ,  ,V )   (V ) a Kronecker set.
X,
Remark Characteristic functions of Kronecker sets are
uniformly recurrent points in the Bebutov system [Beb40]
( ,  ),   {0,1} , ( s)( k )  s(k  1)
Z
This notion coincides with almost periodic in [GH55].
Corollary 1. If
then
Z1 is a Kronecker set and S
( S , Z1 )
is not a Riesz pair.
is a fat Cantor set
New Results
Z1  Z is syndetic if there exists
n  N  Z  Z1  {0,1,2,..., n  1}, thick if
n  N m  Z  m  {0,1,2,..., n  1}  Z1,
Definitions A subset
and piecewise syndetic if it is the intersection of a syndetic
and a thick set [F81].
Theorem 1.23 in [F81] page 34. If
is a partition then one of the
Z  Z1    Z n
Z i is piecewise syndetic.
Observation in proof of Theorem 1.24 in [F81] page 35. If Z i
is piecewise syndetic then the orbit closure of  Z i  
contains the characteristic function of a syndetic set.
Theorem 2. B satisfies Feichtinger’s conjecture iff
is a Riesz pair for some syndetic (almost per.) Z1
( , Z1 )
References
J. Anderson, Extreme points in sets of positive linear
maps on B(H), J. Func. Anal. 31(1979), 195-217.
M. V. Bebutov, On dynamical systems in the space of
continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940).
S. Bochner, Lectures on Fourier Integrals, Princeton
University Press, 1959.
H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III.
Acta Math. 45(1925),29-127;46(1925),101-214;47(1926),237-281
J. Bourgain and L. Tzafriri, Invertibility of “large” submatrices
with applications to the geometry of Banach spaces and
harmonic analysis, Israel J. Math., 57#2(1987), 137-224.
J. Bourgain and L. Tzafriri, On a problem of Kadison and
Singer, J. Reine Angew. Math. 420(1991), 1-43
M. Bownik and D. Speegle, The Feichtinger conjecture for
wavelet frames, Gabor frames, and frames of translates,
Canad. J. Math. 58#6 (2006), 1121-2243.
References
P. G. Casazza, O. Christenson, A. Lindner, and R. Vershynin,
Frames and the Feichtinger conjecture, Proc. Amer. Math.
Soc. 133#4 (2005), 1025-1033.
P. G. Casazza, M . Fickus, J. Tremain, and E. Weber, The
Kadison-Singer problem in mathematics and engineering,
Contep. Mat., vol. 414, Amer. Math. Soc., Providence, RI,
2006, pp. 299-355.
P. G. Casazza and E. Weber, The Kadison-Singer problem
and the uncertainty principle, Proc. Amer. Math. Soc. 136
(2008), 4235-4243.
P. G. Casazza and R. Vershynin, Kadison-Singer meets
Bourgain-Tzafriri, preprint
www.math.ucdavis.edu/~vershynin/papers/kadison-singer.pdf
References
O. Christenson, An Introduction to Frames and Riesz
Bases, Birkhauser, 2003.
H. Furstenberg, Recurrence in Ergodic Theory and
Combinatorial Number Theory, Princeton Univ. Press, 1981.
W. H. Gottschalk and G. A. Hedlund,Topological Dynamics,
Amer. Math. Soc., Providence, R. I., 1955.
H. Halpern, V. Kaftal, and G. Weiss,The relative Dixmier
property in discrete crossed products, J. Funct. Anal. 69
(1986), 121-140.
H. Halpern, V. Kaftal, and G. Weiss, Matrix pavings and
Laurent operators, J. Operator Theory 16#2(1986), 355-374.
R. Kadison and I. Singer, Extensions of pure states, American
J. Math. 81(1959), 383-400.
N. Weaver, The Kadison-Singer problem in discrepancy
theory, preprint