The Numerical Range of Cψ Cφ and Cφ Cψ Clifford, John; Dabkowski, Michael; Wiggins, Alan
Concrete operators (Warsaw, Poland),
01/2021, Volume:
8, Issue:
1
Journal Article
Peer reviewed
Open access
In this paper we investigate the numerical range of
and
on the Hardy space where
is an inner function fixing the origin and
and
are points in the open unit disc. In the case when |
| = |
| = 1 we ...characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when
and
are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.
This paper addresses a conjecture in the work by Kadison and Kastler Kadison RV, Kastler D (1972) Am J Math 94:38–54 that a von Neumann algebra M on a Hilbert space Formula should be unitarily ...equivalent to each sufficiently close von Neumann algebra N , and, moreover, the implementing unitary can be chosen to be close to the identity operator. This conjecture is known to be true for amenable von Neumann algebras, and in this paper, we describe classes of nonamenable factors for which the conjecture is valid. These classes are based on tensor products of the hyperfinite II ₁ factor with crossed products of abelian algebras by suitably chosen discrete groups.
We study the relationship between the dynamics of the action α of a discrete group G on a von Neumann algebra M, and structural properties of the associated crossed product inclusion L(G)⊆M⋊αG, and ...its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion L(G)⊆M⋊αG such as the Haagerup Approximation Property, and is essential to capturing “almost periodic” behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial W⁎-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras.
Strong singularity for subfactors Grossman, Pinhas; Wiggins, Alan
The Bulletin of the London Mathematical Society,
August 2010, Volume:
42, Issue:
4
Journal Article
Peer reviewed
Open access
We examine the notion of α-strong singularity for subfactors of a II1 factor, which is a metric quantity that relates the distance between a unitary in the factor and a subalgebra with the distance ...between that subalgebra and its unitary conjugate. Through planar algebra techniques, we demonstrate the existence of a finite Jones index singular subfactor of the hyperfinite II1 factor that cannot be strongly singular with α = 1, in contrast to the case for masas. Using work of Popa, Sinclair, and Smith, we show that there exists an absolute constant 0 < c < 1 such that all singular subfactors are c-strongly singular. Under the hypothesis of 2-transitivity, we prove that finite index subfactors are α-strongly singular with a constant that tends to 1 as the index tends to infinity and infinite index subfactors are 1-strongly singular. Finally, we give a proof that proper finite index singular subfactors do not have the weak asymptotic homomorphism property relative to the containing factor.
In this paper, we study the product of a composition operator
C
φ
with the adjoint of a composition operator
C
ψ
∗
on the Hardy space
H
2
(
D
)
. The order of the product gives rise to two different ...cases. We completely characterize when the operator
C
φ
C
ψ
∗
is invertible, isometric, and unitary and when the operator
C
ψ
∗
C
φ
is isometric and unitary. If one of the inducing maps
φ
or
ψ
is univalent, we completely characterize when
C
ψ
∗
C
φ
is invertible.
Normalizers of irreducible subfactors Smith, Roger; White, Stuart; Wiggins, Alan
Journal of mathematical analysis and applications,
04/2009, Volume:
352, Issue:
2
Journal Article
Peer reviewed
Open access
We consider normalizers of an infinite index irreducible inclusion
N
⊆
M
of II
1 factors. Unlike the finite index setting, an inclusion
u
N
u
∗
⊆
N
can be strict, forcing us to also investigate the ...semigroup of one-sided normalizers. We relate these one-sided normalizers of
N in
M to projections in the basic construction and show that every trace one projection in the relative commutant
N
′
∩
〈
M
,
e
N
〉
is of the form
u
∗
e
N
u
for some unitary
u
∈
M
with
u
N
u
∗
⊆
N
generalizing the finite index situation considered by Pimsner and Popa. We use this to show that each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of infinite index irreducible subfactors arising from subgroup–group inclusions
H
⊆
G
. Here the one-sided normalizers arise from appropriate group elements modulo a unitary from
L
(
H
)
. We are also able to identify the finite trace
L
(
H
)
-bimodules in
ℓ
2
(
G
)
as double cosets which are also finite unions of left cosets.
This paper addresses a conjecture in the work by Kadison and Kastler Kadison RV, Kastler D (1972)
Am J Math
94:38–54 that a von Neumann algebra
M
on a Hilbert space
should be unitarily equivalent to ...each sufficiently close von Neumann algebra
N
, and, moreover, the implementing unitary can be chosen to be close to the identity operator. This conjecture is known to be true for amenable von Neumann algebras, and in this paper, we describe classes of nonamenable factors for which the conjecture is valid. These classes are based on tensor products of the hyperfinite II
1
factor with crossed products of abelian algebras by suitably chosen discrete groups.
This paper addresses a conjecture in the work by Kadison and Kastler Kadison RV, Kastler D (1972) Am J Math 94:38-54 that a von Neumann algebra M on a Hilbert space ... should be unitarily equivalent ...to each sufficiently close von Neumann algebra N, and, moreover, the implementing unitary can be chosen to be close to the identity operator. This conjecture is known to be true for amenable von Neumann algebras, and in this paper, we describe classes of nonamenable factors for which the conjecture is valid. These classes are based on tensor products of the hyperfinite II1 factor with crossed products of abelian algebras by suitably chosen discrete groups. (ProQuest: ... denotes formulae/symbols omitted.)
PDF
