Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
The Jacobson radical of semicrossed
products of the disk algebra
Anchalee Khemphet
Justin Peters
Department of Mathematics, Iowa State University
April 14th , 2012
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Outline
1
Introduction
Semicrossed Products of the Disk Algebra
Previous Results
2
Finite Blaschke Products
Complex Dynamics of Finite Blaschke Products
The Description of the Recurrent Set
3
The Jacobson Radical and Quasinilpotent Elements
The Jacobson Radical
Quasinilpotent Elements
4
References
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Semicrossed Products of the Disk Algebra
Disk Algebra
What is the disk algebra?
Let D be the open unit disc and T the unit circle.
The disk algebra is an algebra
A(D) = {f : f is analytic on D and continuous on D}.
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Semicrossed Products of the Disk Algebra
Finite Blaschke Products
A finite Blaschke product is a function ϕ of the form:
ϕ(z) = u
N
Y
z − ai
,
1 − āi z
i=1
where |u| = 1, and ai ∈ D, i = 1, 2, . . . , N.
Define an endomorphism α of A(D) by
α(f ) = f ◦ ϕ,
where f ∈ A(D) and ϕ is a finite Blaschke product.
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Semicrossed Products of the Disk Algebra
Semicrossed products
Let P be the algebra of formal polynomials
X
U k fk
with fk ∈ A(D) where U satisfies
fU = Uα(f )
for any f ∈ A(D).
The semicrossed product A(D) ×α Z+ is the completion of
P with respect to the following norm.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Semicrossed Products of the Disk Algebra
Semicrossed products
Consider a pair (A, T ) of contractions satisfying
AT = T ϕ(A).
There is a contractive representation ρ of A(D) given by
ρ(f ) = f (A).
Define a representation of P
X
X
ρ × T(
U k fk ) =
T k ρ(fk ).
Then the norm on P is defined by
X
X
||
U k fk || =
sup
||
T k ρ(fk )||.
(ρ,T ) covariant
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Semicrossed Products of the Disk Algebra
Fourier Coefficients
For t ∈ [0, 1], define γt acting on P by
X
X
γt (
U k fk ) =
U k e2πikt fk
which extends to an automorphism of A(D) ×α Z+ .
Define the k th Fourier coefficient πk (F ) of F by
Z
k
U πk (F ) =
e−2πikt γt (F )dm(t)
T
where dm is normalized Lebesgue measure on the unit
circle.
If F is in the polynomial P, then πk (F ) = fk and
||πk (F )|| ≤ ||F ||.
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Semicrossed Products of the Disk Algebra
Quasinilpotent Elements and Jacobson Radical
Definition
1
F is a quasinilpotent element if lim ||F n || n = 0.
n→∞
Definition
The Jacobson radical of an operator algebra A is the maximal
left ideal of A which is contained in the set of quasinilpotent
elements, denoted by Rad(A).
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Previous Results
Previous Results
1
T. Hoover, J. Peters, and W. Wogen, Spectral properties of
semicrossed products, Houston J. Math., 19 (1993),
649–660.
2
A. P. Donsig, A. Katavolos, and A. Manoussos, The
Jacobson radical for analytic crossed products, J. Funct.
Anal., 187 (2001), 129–145.
3
K. A. Davidson and E. Katsoulis, Semicrossed products of
the disk algebra, preprint, arXiv:1104.1398v1, 2011.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Previous Results
Previous Results
Theorem (Donsig et al., 2001)
Rad(C(T) ×α Z+ ) is the set {F ∈ C(T) ×α Z+ : π0 (F ) = 0, and
πk (F ) vanishes on the recurrent set}.
Theorem (Davidson and Katsoulis, 2011)
Let ϕ be a finite Blaschke product, and let α(f ) = f ◦ ϕ. Then
A(D) ×α Z+ is a subalgebra of C(T) ×α Z+ .
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Complex Dynamics of Finite Blaschke Products
Classification of Finite Blaschke Products
Definition
A finite Blaschke product ϕ is said to be
1
elliptic if there exists a fixed point in D;
2
hyperbolic if there exists the Denjoy-Wolff point z0 ∈ T
such that ϕ0 (z0 ) < 1;
3
parabolic if there exists the Denjoy-Wolff point z0 ∈ T such
that ϕ0 (z0 ) = 1.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Complex Dynamics of Finite Blaschke Products
Julia Sets
Definition
The Fatou set F of ϕ is the set of points z in C such that {ϕn } is
a normal family in some neighborhood of z. The Julia set is the
complement of the Fatou set.
Remark
1
The Julia set is the closure of the repelling periodic points.
2
The Julia set of a finite Blaschke product is either T or a
Cantor set of T.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Complex Dynamics of Finite Blaschke Products
Hyperbolic Distance
Definition
The hyperbolic distance d in D is defined by
1+
d(z, w) = log
1−
|z−w|
|1−zw|
|z−w|
|1−zw|
where z, w ∈ D.
Definition
A finite Blaschke product ϕ is said to be
1
2
of zero hyperbolic step if lim d(ϕn (z), ϕn+1 (z)) = 0 for
n→∞
some z ∈ D.
of positive hyperbolic step otherwise.
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
The Description of the Recurrent Set
Density of Recurrent Points
Theorem (Basallote et al., 2009, Contreras et al., 2007 and
Hamilton, 1996)
ϕ is of zero hyperbolic step if and only if its Julia set is T if and
only if it is ergodic.
Remark
Here we assume ϕ is ergodic without assuming that ϕ is
measure-preserving.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
The Description of the Recurrent Set
Hyperbolic Step
Theorem (Basallote et al., 2009)
Let ϕ be a finite Blaschke product.
1
If ϕ is elliptic, then it is of zero hyperbolic step.
2
If ϕ is hyperbolic, then it is of positive hyperbolic step.
Theorem (Contreras et al., 2007)
If ϕ is a parabolic finite Blaschke product with Denjoy-Wolff
point 1, then ϕ is of zero hyperbolic step if and only if ϕ00 (1) = 0
N
X
1 − |ai |2
if and only if
Im(ai ) = 0 where ai ’s are as in (2).
|1 − ai |2
i=1
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
The Description of the Recurrent Set
Topological Transitivity
Lemma
If ϕ is elliptic or parabolic with zero hyperbolic step, then the
recurrent points of ϕ are dense in T.
Theorem
If ϕ is elliptic or parabolic with zero hyperbolic step, then ϕ is
topologically transitive.
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
The Description of the Recurrent Set
The Recurrent Set
Theorem (Contreras et al., 2007)
If ϕ is hyperbolic or parabolic with positive hyperbolic step, then
(ϕn (z)) converges to the Denjoy-Wolff point of ϕ, for almost
every z ∈ T.
Theorem
If ϕ is hyperbolic or parabolic with positive hyperbolic step, then
the closure of the set of recurrent points of ϕ is the union of the
Julia set and the Denjoy-Wolff point.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
The Jacobson Radical
Quasinilpotent Elements
Lemma
1
2
Let F ∈ A(D) ×α Z+ . If π0 (F ) is not identically zero or
πk (F ) does not vanish on the fixed point set of ϕ for some
k > 0, then F is not quasinilpotent.
Let f ∈ A(D). If f does not vanish on the set of recurrent
points of ϕ, then there is an n ∈ N such that U n f is not
quasinilpotent.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
The Jacobson Radical
The Jacobson Radical of the Semicrossed Product
Theorem
Rad(A(D) ×α Z+ ) is the set {F ∈ A(D) ×α Z+ : π0 (F ) = 0, and
πk (F ) vanishes on the set of recurrent points of ϕ, for k > 0}.
Proof.
From Donsig’s result, if F is such that π0 (F ) = 0 and πk (F )
vanishes on the recurrent set for j > 0, then F is in
Rad(C(T) ×α Z+ ).
I = Rad(C(T) ×α Z+ ) ∩ A(D) ×α Z+ is contained in
Rad(A(D) ×α Z+ ).
Let F ∈ Rad(A(D) ×α Z+ ).
Then π0 (F ) = 0 and πk (F )(z0 ) = 0 ∀k > 0 where z0 is the
Denjoy-Wolff point.
Suppose that not all fi ’s vanish on J .
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
The Jacobson Radical
Proof (continue).
Then ∃k so that fk (x0 ) 6= 0 and fi ’s vanish on J ∀i < k for
some x0 where ϕn (x0 ) = x0 for some n.
Choose j so that U j (U k fk ) = U m fk where m is a multiple of
n.
For l > 0,
||(U j F )l || ≥ ||πml ((U j F )l )||
≥ |πml ((U j F )l )(x0 )|
= |fk (x0 )fk ◦ ϕm (x0 ) . . . fk ◦ ϕ(l−1)m (x0 )|
= |fk (x0 )|l .
Thus, U j F is not quasinilpotent, a contradiction.
fi must vanish on J for all i, and so F ∈ I.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
The Jacobson Radical
Theorem
Rad(A(D) ×α Z+ ) is
1
nonzero if ϕ is hyperbolic or parabolic with positive
hyperbolic step;
2
zero if ϕ is elliptic or parabolic with zero hyperbolic step.
References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
The Jacobson Radical
The Jacobson Radical of the Semicrossed Product
Corollary
A(D) ×α Z+ has nonzero quasinilpotent elements if ϕ is
hyperbolic or parabolic with positive hyperbolic step.
Corollary
A(D) ×α Z+ is semi-simple if and only if the recurrent points are
dense in T.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Quasinilpotent Elements
Elliptic case
Lemma
If ϕ is elliptic with the Denjoy-Wolff point 0, then ϕ is
measure-preserving.
Lemma
For any nonzero f ∈ A(D), if ϕ is elliptic with the Denjoy-Wolff
point 0, then Uf is not a quasinilpotent element of A(D) ×α Z+ .
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Quasinilpotent Elements
Proof.
1
1
Note that ||(Uf )n || n = sup |f (x)f ◦ ϕ(x) . . . f ◦ ϕn−1 (x)| n .
x∈T
f ∈ A(D) → log |f | is integrable.
Since ϕ is measure-preserving and the Ergodic Theorem,
1
log |f (x)f ◦ ϕ(x) . . . f ◦ ϕn−1 (x)| n
Z
n−1
1X
=
log |f ◦ ϕk (x)| →
log |f |dm a.e.
n
T
k =0
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Quasinilpotent Elements
Proof (continue).
Let x0 ∈ T be such that the above convergence holds.
Then
1
lim ||(Uf )n || n
n→∞
1
≥ lim |f (x0 )f ◦ ϕ(x0 ) . . . f ◦ ϕn−1 (x0 )| n
n→∞
1
= exp{ lim log |f (x0 )f ◦ ϕ(x0 ) . . . f ◦ ϕn−1 (x0 )| n }
n→∞
Z
= exp{ log |f |dm}
T
> 0.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Quasinilpotent Elements
Isomorphism of Semicrossed Products
Theorem
Let ϕ and ψ be non-trivial finite Blaschke products. If ϕ and ψ
are conjugate, ψ = τ −1 ◦ ϕ ◦ τ for some conformal mapping τ ,
then Z+ ×α A(D) is isomorphic to Z+ ×β A(D), where
α(f ) = f ◦ ϕ and β(f ) = f ◦ ψ, f ∈ A(D).
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Quasinilpotent Elements
Theorem
If ϕ is elliptic, then A(D) ×α Z+ has no nonzero quasinilpotent
elements.
Proof.
Any elliptic map is conjugate to an elliptic map fixes zero.
Let F be a quasinilpotent element, and let πi (F ) = fi .
Then f0 = 0 and suppose that ∃k > 0, fk 6= 0 where
fi = 0 ∀i < k .
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Quasinilpotent Elements
Proof (continue).
Then
1
1
||F n || n ≥ ||πnk (F n )|| n
1
= sup |fk (x)fk ◦ ϕk (x) . . . fk ◦ ϕ(n−1)k (x)| n
x∈T
1
= ||(Uβ fk )n || n ,
where Uβ fk ∈ A(D) ×β Z+ and β(f ) = f ◦ ϕk .
Note ϕk is also elliptic the Denjoy-Wolff point 0.
1
1
By previous lemma, lim ||F n || n ≥ lim ||(Uβ fk )n || n > 0.
n→∞
n→∞
F is not quasinilpotent, a contradiction.
Hence, F = 0.
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Bibliography
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Commuting finite Blaschke products with no fixed points in
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Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Bibliography
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Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Bibliography
A. P. Donsig, A. Katavolos, and A. Manoussos, The
Jacobson radical for analytic crossed products, J. Funct.
Anal., 187 (2001), 129–145.
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References
Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Bibliography
T. Hoover, J. Peters, and W. Wogen, Spectral properties of
semicrossed products, Houston J. Math., 19 (1993),
649–660.
E. T. A. Kakariadis, Semicrossed products of C∗ -algebras
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Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
References
Bibliography
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Introduction
Finite Blaschke Products
The Jacobson Radical and Quasinilpotent Elements
Thank you!!!
References