Dynamical systems dual to interactions
and graph
C ∗-algebras
Bartosz Kosma Kwa±niewski
University of Biaªystok
14 March 2013, Sde Boker, Israel
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
The general problem of faithfulness
G - discrete
abelian group (with neutral element 0)
L
max
B = t ∈G Bt - G -graded C ∗ -algebra (Bt Bs ⊂ Bt +s , Bt∗ = B−t )
Prop. Let Ψ : B → C be ∗-homomorphism, faithful on B0 . TCAE:
(faithfulness)
Ψ is faithful
(gauge-invariance) there exists a gauge action γ of Gb on Ψ(B ) s.t.
Ψ(Bt ) are spectral subspaces for γ
P
(condit. expect.) we have kb0 k ≤ k t ∈F Ψ(bt )k
for every nite F ⊂ G , bt ∈ Bt , t ∈ F .
Examples



⇒
B = A oα G crossed prod.
where α : G → Aut(A)
the conditions
in Prop. are automatic
is topologically free
B=C∗ (E ) graph C ∗ -algebra
the conditions
⇔
E satises condition (L)
in Prop. are automatic
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Topological freeness for Hilbert bimodules (Israel J. Math.)
Let A be a C ∗ -algebra and X be a Hilbert bimodule over A
with inner products h·, ·iA and A h·, ·i s.t. x · hy , z iA = A hx , y i · z .
X -Ind maps rep. π : A → B(H ) to X -Ind(π) : A → B(X ⊗π H )
X -Ind(π)(b)(x ⊗π h) = (bx ) ⊗π h.
hX , X iA := span{hx , y iA : x , y ∈ X }
=⇒ ideals in A
A hX , X i := span{A hx , y i : x , y ∈ X }
X is a A hX , X i-hX , X iA Morita equivalence module and hence
b.
X -Ind : hX\
, X iA → A \
hX , X i is a partial homeomorphism of A
Thm. (BKK)
i) If X -Ind is topologically free, then every faithful representation
(πA , πX ) of (A, X ) yields faithful representation of A oX Z.
ii) If X -Ind is free, then J 7→ J[
∩ A is a lattice isomorphism
b.
between ideals in A oX Z and open invariant sets in A
iii) If X -Ind is topologically free and minimal A oX Z is simple.
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Interactions and transfer operators
Let A be a unital C ∗ -algebra and α : A → A ∗-endomorphism,
Transfer operator is a linear positive map L : A → A s.t.
L(α(a)b) = aL(b),
a , b ∈ A.
We also assume E = α ◦ L is a conditional expectat. onto α(A).
We will call the pair (α, L) a C ∗ -dynamical
system.
Def. (R. Exel 2007)
Pair (V, H) of positive linear maps V, H : A → A is interaction if
V◦H ◦ V = V,
H ◦ V ◦ H = H,
V(ab) = V(a)V(b)
if a or b is in H(A),
H(ab) = H(a)H(b) if a or b is in V(A).
Any C ∗ -dynamical system (α, L) is an interaction.
Interaction (V, H) is a C ∗ -dynamical system i H(A) is ideal in A.
Prop.
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Complete interactions
Def. Let (V, H) be an interaction.
(V, H) is complete if H(A), V(A) are hereditary subalgebras of A
Prop.
If (V, H) is complete H is uniquely determined by V .
Def. Let (V, H) be a complete interaction.
Covariant representation of (V, H) is (π, S ) where π : A → B(H )
is unital representation and S ∈ B(H ) are s.t.
S π(a)S ∗ = π(V(a)),
S ∗ π(a)S = π(H(a))
a ∈ A.
The crossed product C ∗ (A, V, H) is generated by iA (A) and s
where (iA , s ) is a universal covariant representation of (V, H).
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Def. Let (V, H) be an interaction.
(V, H) is complete if H(A), V(A) are hereditary subalgebras of A
Prop.
If (V, H) is complete H is uniquely determined by V .
Def. Let (V, H) be a complete interaction.
Covariant representation of (V, H) is (π, S ) where π : A → B(H )
is unital representation and S ∈ B(H ) are s.t.
S π(a)S ∗ = π(V(a)),
S ∗ π(a)S = π(H(a))
a ∈ A.
The crossed product C ∗ (A, V, H) is generated by iA (A) and s
where (iA , s ) is a universal covariant representation of (V, H).
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Def. Let (V, H) be an interaction.
(V, H) is complete if H(A), V(A) are hereditary subalgebras of A
Prop.
If (V, H) is complete H is uniquely determined by V .
Def. Let (V, H) be a complete interaction.
Covariant representation of (V, H) is (π, S ) where π : A → B(H )
is unital representation and S ∈ B(H ) are s.t.
S π(a)S ∗ = π(V(a)),
S ∗ π(a)S = π(H(a))
a ∈ A.
The crossed product C ∗ (A, V, H) is generated by iA (A) and s
where (iA , s ) is a universal covariant representation of (V, H).
Prop. Let (V, H) be a complete interaction.
There is a Hilbert A-bimodule X = span{a ⊗ b : a, b ∈ A} where
ha ⊗ b, c ⊗ d iA = b∗ H(a∗ c )d ,
B.K.K.
∗ ∗
A ha ⊗ b, c ⊗ d i = aV(bd )c .
Dual to an interaction and graph
C ∗ -algebras
Prop.
If (V, H) is complete H is uniquely determined by V .
Def. Let (V, H) be a complete interaction.
Covariant representation of (V, H) is (π, S ) where π : A → B(H )
is unital representation and S ∈ B(H ) are s.t.
S π(a)S ∗ = π(V(a)),
S ∗ π(a)S = π(H(a))
a ∈ A.
The crossed product C ∗ (A, V, H) is generated by iA (A) and s
where (iA , s ) is a universal covariant representation of (V, H).
Prop. Let (V, H) be a complete interaction.
There is a Hilbert A-bimodule X = span{a ⊗ b : a, b ∈ A} where
ha ⊗ b, c ⊗ d iA = b∗ H(a∗ c )d ,
∗ ∗
A ha ⊗ b, c ⊗ d i = aV(bd )c .
We have a one-to-one correspondence between the cov. rep. (π, S )
of (V, H) and representations (π, πX ) of (A, X )
πX (a ⊗ b) = π(a)S π(b),
B.K.K.
x ∈ X,
S = πX (1 ⊗ 1).
Dual to an interaction and graph
C ∗ -algebras
Def. Let (V, H) be a complete interaction.
Covariant representation of (V, H) is (π, S ) where π : A → B(H )
is unital representation and S ∈ B(H ) are s.t.
S π(a)S ∗ = π(V(a)),
S ∗ π(a)S = π(H(a))
a ∈ A.
The crossed product C ∗ (A, V, H) is generated by iA (A) and s
where (iA , s ) is a universal covariant representation of (V, H).
Prop. Let (V, H) be a complete interaction.
There is a Hilbert A-bimodule X = span{a ⊗ b : a, b ∈ A} where
ha ⊗ b, c ⊗ d iA = b∗ H(a∗ c )d ,
∗ ∗
A ha ⊗ b, c ⊗ d i = aV(bd )c .
We have a one-to-one correspondence between the cov. rep. (π, S )
of (V, H) and representations (π, πX ) of (A, X )
πX (a ⊗ b) = π(a)S π(b),
B.K.K.
x ∈ X,
S = πX (1 ⊗ 1).
Dual to an interaction and graph
C ∗ -algebras
Prop. Let (V, H) be a complete interaction.
There is a Hilbert A-bimodule X = span{a ⊗ b : a, b ∈ A} where
ha ⊗ b, c ⊗ d iA = b∗ H(a∗ c )d ,
∗ ∗
A ha ⊗ b, c ⊗ d i = aV(bd )c .
We have a one-to-one correspondence between the cov. rep. (π, S )
of (V, H) and representations (π, πX ) of (A, X )
πX (a ⊗ b) = π(a)S π(b),
B.K.K.
x ∈ X,
S = πX (1 ⊗ 1).
Dual to an interaction and graph
C ∗ -algebras
Prop. Let (V, H) be a complete interaction.
There is a Hilbert A-bimodule X = span{a ⊗ b : a, b ∈ A} where
ha ⊗ b, c ⊗ d iA = b∗ H(a∗ c )d ,
∗ ∗
A ha ⊗ b, c ⊗ d i = aV(bd )c .
We have a one-to-one correspondence between the cov. rep. (π, S )
of (V, H) and representations (π, πX ) of (A, X )
πX (a ⊗ b) = π(a)S π(b),
x ∈ X,
S = πX (1 ⊗ 1).
In particular, C ∗ (A, V, H) ∼
= A oX Z.
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Prop. Let (V, H) be a complete interaction.
There is a Hilbert A-bimodule X = span{a ⊗ b : a, b ∈ A} where
ha ⊗ b, c ⊗ d iA = b∗ H(a∗ c )d ,
∗ ∗
A ha ⊗ b, c ⊗ d i = aV(bd )c .
We have a one-to-one correspondence between the cov. rep. (π, S )
of (V, H) and representations (π, πX ) of (A, X )
πX (a ⊗ b) = π(a)S π(b),
x ∈ X,
S = πX (1 ⊗ 1).
In particular, C ∗ (A, V, H) ∼
= A oX Z.
Let (V, H) be a complete interaction.
[
[
The dual Vb : V(
A) → H(
A) to ∗-isomorphism V : H(A) → V(A)
b
could be naturally considered as a partial homeomorphism of A
Prop.
X -Ind = Hb = Vb −1
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Let (V, H) be a complete interaction.
[
[
The dual Vb : V(
A) → H(
A) to ∗-isomorphism V : H(A) → V(A)
b
could be naturally considered as a partial homeomorphism of A
Prop.
X -Ind = Hb = Vb −1
Thm.
i) If Vb is topologically free, then every faithful cov. represen.
(π, S ) of (V, H) yields faithful representation of C ∗ (A, V, H).
ii) If Vb is free, then J 7→ J[
∩ A is a lattice isomorphism between
∗
b.
ideals in C (A, V, H) and open invariant sets in A
iii) If Vb is topologically free and minimal C ∗ (A, V, H) is simple.
B.K.K.
Dual to an interaction and graph
C ∗ -algebras
Graph
C ∗-algebras
Let E = (E 0 , E 1 , r , s ) be a nite graph (both E 0 , E 1 nite)
E -family consists of {Pv }v ∈E and {Se }e ∈E where Pv are
mutually orthogonal projections s.t.
0
Se∗ Se = Pr (e ) ,
Pv
X
=
e ∈s −1 (v )
1
Se Se∗ ,
e ∈ E 1 , v ∈ s (E 1 ).
C ∗ -algebra C ∗ (E ) is the C ∗ -algebra generated by the
universal E -family {pv }v ∈E , {se }e ∈E .
C ∗ (E ) = span {sµ sν∗ : µ ∈ E n , ν ∈ E m , n, m ∈ N}
FE = span {sµ sν∗ : µ, ν ∈ E n , n ∈ N} the AF core of C ∗ (E )
DE = span sµ sµ∗ : µ ∈ E n , n ∈ N the canonical MASA in FE
Graph
0
B.K.K.
1
Dual to an interaction and graph
C ∗ -algebras
C ∗ (E ) = span {sµ sν∗ : µ ∈ E n , ν ∈ E m , n, m ∈ N}
FE = span {sµ sν∗ : µ, ν ∈ E n , n ∈ N} the AF core of C ∗ (E )
DE = span sµ sµ∗ : µ ∈ E n , n ∈ N the canonical MASA in FE
Def.
Let nv := |r −1 (v )|, v ∈ E 0 .
X
1
s :=
e ∈E1
√
nr (e )
se
− universal partial isometry
Prop. C ∗ (E ) ∼
= DE o(φE ,L) N (Exel's crossed product) where
X 1
X
∗
∗
φE (a) =
e ∈E
se ase ,
L(a) =
1
P
For a ∈ DE we have L(a) = e ,f ∈E 1
√
e ∈E
1
nr (e )
se ase .
1
∗
∗
nr (e ) nr (f ) se asf = s as .
Prop. C ∗ (E ) ∼
= C ∗ (FE , V, H) (interaction crossed product) where
V(a) = sas ∗ ,
B.K.K.
H(a) = s ∗ as .
Dual to an interaction and graph
C ∗ -algebras
Glimm product states and the dual to
V
on
FbE
Let µ = (µ1 , µ2 , ...) be an innite path, and denote by πµ the
GNS-representation associated to the pure state ωµ : FE → C
ωµ (s
s
∗
ν η)
1 ν = η = (µ1 , ..., µn )
0 otherwise
(
=
for ν, η ∈ E n
Prop.
πµ ∼
= πν
⇐⇒ ∃N (µN , µN +1 , ...) = (νN , νN +1 , ...).
Thm. (Description of the dual to interaction (V, H) of graph E )
The set Fb∞ := {πµ ∈ FcE : µ = (µ1 , µ2 , ...)} is Vb -invariant,
cE \ Fb∞ does not contain periodic points and
F
b (µ ,µ ,µ ,...) ) = π(µ ,µ ,...) ,
V(π
1
2
3
2
3
B.K.K.
for any (µ1 , µ2 , µ3 , ...)
Dual to an interaction and graph
C ∗ -algebras
Prop.
πµ ∼
= πν
⇐⇒ ∃N (µN , µN +1 , ...) = (νN , νN +1 , ...).
Thm. (Description of the dual to interaction (V, H) of graph E )
The set Fb∞ := {πµ ∈ FcE : µ = (µ1 , µ2 , ...)} is Vb -invariant,
cE \ Fb∞ does not contain periodic points and
F
b (µ ,µ ,µ ,...) ) = π(µ ,µ ,...) ,
V(π
1
2
3
2
3
B.K.K.
for any (µ1 , µ2 , µ3 , ...)
Dual to an interaction and graph
C ∗ -algebras
Prop.
πµ ∼
= πν
⇐⇒ ∃N (µN , µN +1 , ...) = (νN , νN +1 , ...).
Thm. (Description of the dual to interaction (V, H) of graph E )
The set Fb∞ := {πµ ∈ FcE : µ = (µ1 , µ2 , ...)} is Vb -invariant,
cE \ Fb∞ does not contain periodic points and
F
b (µ ,µ ,µ ,...) ) = π(µ ,µ ,...) ,
V(π
1
2
3
2
3
for any (µ1 , µ2 , µ3 , ...)
We have the following dynamical dichotomy:
a) either there are Vb -periodic orbits consisting of isolated points
in FcE (they correspond to loops without exits in E ), or
b) every nonempty open set in Fb∞ contains uncountable number
of non-periodic points for Vb
(this holds if every loop in E has an exit)
X -Ind = Vb −1 is topologically free ⇔ every loop in E has an exit
B.K.K.
Dual to an interaction and graph
C ∗ -algebras