Rough and Ready Summary
My research interests are in operator algebras and operator theory.
This subject is a mix of algebra and analysis.
It has important applications in physics, such as quantum mechanics
and quantum field theory.
There is a long standing theme in mathematics of understanding spaces
in terms of the functions on those spaces.
The commutative self-adjoint operator algebras are the
natural function algebras on a topological space or a measure space.
Topological or measure-theoretic questions about the space can be
translated into algebraic questions about the commutative operator
algebra.
Motivated partly by this and partly by other topics, the study of
non-commutative (selfadjoint) operator algebras can be viewed as
`non-commutative topology' and `non-commutative measure theory'.
My own work is focuses on non-selfadjoint operator algebras.
These correspond, in the sense of the previous paragraph, to the
study of analytic functions.
One interest of mine is limit algebras, which are infinite dimensional
algebras obtained as limits of sequences of finite dimensional algebras,
each embedded inside the next.
Conferences
Great Plains Operator Theory Symposium, 2006
Great Plains Operator Theory Symposium
Canadian Operator Symposium (COSy) 2006
Canadian Operator Symposium (COSy) History
Iowa-Nebraska Functional Analysis Seminar
Papers & Preprints (most recent first)
Because of copyright restrictions, I can only make available some
of published papers.
To read the preprints and papers, you may need the
Adobe Acrobat Reader.
- Norms of Schur Multipliers (with
Ken Davidson),
- Abstract: A subset P of N x N is called Schur bounded if every
infinite matrix with
bounded entries which is zero off of P yields a bounded Schur multiplier on
B(H). Such sets are characterized as being the union of a subset with at most k
entries in each row with another that has at most k entries in each column, for
some finite k. If k is optimal, there is a Schur multiplier supported on the
pattern with norm O(k^(1/2)), which is sharp up to a constant.
The same techniques give a new, more elementary proof of results of
Varopoulos and Pisier on Schur multipliers with given matrix entries of random
sign.
We consider the Schur multipliers for certain matrices which have a large
symmetry group. In these examples, we are able to compute the Schur multiplier
norm exactly. This is carried out in detail for a few examples including the
Kneser graphs.
- This paper is available on the arXiv preprint
server as Math.OA 0506073.
- Analytic Partial Crossed Products (with
Alan Hopenwasser),
Houston Journal of Mathematics 31 (2005) 495-527.
- Abstract: Partial actions of discrete abelian groups can be
used to construct both groupoid C*-algebras and
partial crossed product algebras.
In each case there is a natural notion of an analytic subalgebra.
We show that for countable subgroups of R and free partial actions,
these constructions yield the same C*-algebras and the same analytic
subalgebras.
We also show that under suitable hypotheses an analytic
partial crossed product preserves all the information
in the dynamical system in the sense that two analytic
partial crossed products are isomorphic as Banach algebras
if, and only if, the partial actions are conjugate.
- This paper is available on the arXiv preprint
server as Math.OA 0305337.
- The Jacobson Radical of Analytic Crossed Products (with
Aristides Katavolos and Antonios Manoussos),
Journal of Functional Analysis 187 (2001) 129-145, Math Review
2002k:46170.
- Abstract: We characterise the (Jacobson) radical of the analytic
crossed product of C_0(X) by the non-negative integers (Z_+), answering
a question first raised by Arveson and Josephson in 1969. In fact, we
characterise the radical of analytic crossed products of C_0(X) by (Z_+)^d.
The radical consists of all elements whose `Fourier coefficients' vanish
on the recurrent points of the dynamical system (and the first one is zero).
The multi-dimensional version requires a variation of the notion of
recurrence, taking into account the various degrees of freedom.
- This paper is available on the arXiv preprint
server as Math.OA 0010142.
- Algebraic Isomorphisms and the Spectra of Limit Algebras
(with Steve Power and
David Pitts)
Indiana University Mathematics Journal 50 (2001) 1131-1147,
Math Review
2002k:47148.
- Abstract:We show that the spectrum of a triangular regular limit algebra
(TAF algebra) is an invariant for algebraic isomorphism. Combining this with
previous results provides a striking rigidity property: two triangular
regular limit algebras are algebraically isomorphic if and only if they
are isometrically isomorphic. A consequence of spectral invariance is
a structure theorem for isomorphisms between limit algebras.
The proof of the main theorem makes use of a characterization of the
completely meet irreducible ideals of a TAF algebra and a dual space
formulation of the spectrum.
- Automatic closure of invariant linear manifolds for operator
algebras (with
Alan Hopenwasser and
David Pitts),
Illinois Journal of Mathematics 45 (2001) 787-802.
- Abstract: Kadison's transitivity theorem implies that, for irreducible
representations of C*-algebras, every invariant linear manifold is closed.
It is known that CSL algebras have this propery if, and only if, the lattice
is hyperatomic (every projection is generated by a finite
number of atoms). We show several other conditions are equivalent, including
the conditon that every invariant linear manifold is singly generated.
We show that two families of norm closed operator algebras have this property.
First, let L be a CSL and suppose A is a norm closed algebra which is weakly
dense in Alg L and is a bimodule over the (not necessarily closed) algebra
generated by the atoms of L. If L is hyperatomic and the compression of A to
each atom of L is a C*-algebra, then every linear manifold invariant under A
is closed. Secondly, if A is the image of a strongly maximal triangular AF
algebra under a multiplicity free nest representation, where the nest has
order type -N, then every linear manifold invariant under A is closed and
is singly generated.
- This paper is available on the arXiv preprint
server as Math.OA 0005159.
- The Classification of Limits of 2n-cycle
Algebras (with
Steve Power),
Indiana University Mathematics Journal 48 (1999)
411-427, Math Review
2000j:47127.
- Abstract: We obtain a complete classification of the locally finite
algebras and the operator algebras, given as algebraic inductive
limits and Banach algebraic inductive limits, respectively, of
systems:
A_1 contained in A_2 contained in A_3 and so on.
Here the A_k are 2n-cycle algebras, where
n is at least 3 and the inclusions are of rigid type.
The complete isomorphism invariant is essentially
the triple (K_0(A), H_1(A), Sigma(A)) where K_0(A) is
viewed as a scaled ordered group, H_1(A) is a partial isometry
homology group and Sigma(A), contained in the direct sum of
K_0(A) and H_1(A) is the 2n-cycle joint scale.
- This
paper is available in PDF format.
- Algebraic Isomorphisms of Limit Algebras (with
Tim Hudson and
Elias Katsoulis),
Transactions of the American Math. Society 353 (2000)
1169-1182.
- Abstract: We prove that algebraic isomorphisms between limit
algebras are automatically continuous, and consider the consequences of
this result.
In particular, we give partial solutions to a conjecture of Power
(see the Notes to Chapter 8 in Power's book) and to an open
problem (see Problem 7.8 in Power's book).
As a further consequence, we describe epimorphisms between various
classes of limit algebras.
- Meet Irreducible Ideals in Direct Limit Algebras
(with
Alan Hopenwasser,
Tim Hudson,
Michael Lamoureux,
and Baruch
Solel),
Mathematica Scandinavia 87 (2000) 27-62, Math Reviews
2001f:47112.
- Abstract: This paper studies meet-irreducible ideals, that is,
ideals I so that if I is the intersection of two ideals,
then one of the two ideals must equal I, in certain direct
limit algebras. The limit algebras are usually strongly maximal
triangular subalgebras of AF C*-algebras.
We give descriptions of these ideals in terms of coordinates.
Meet-irreducible ideals are interesting because they are closely
related to nest representations; in particular, we show that each
meet-irreducible ideal is the kernel of a nest representation.
We obtain a distance formula for ideals analogous to Arveson's
distance formula for nest algebras.
Finally, the subset of completely meet-irreducible ideals (they satisfy
the same property for an arbitrary intersection) is shown to be
isomorphic to the spectrum of the limit algebra, for certain limit
algebras.
- Algebraic Orders and Chordal Limit Algebras,
Proc. Edinburgh Math. Soc. 41 (1998) 465-485, Math Reviews
2000d:47103.
- Abstract: We develop an isomorphism invariant for limit algebras:
an extension of Power's strong algebraic order on the scale of the
K_0-group (Power, J. Operator Theory 27 1992, 87--106).
This invariant is complete for a certain family of limit algebras:
inductive limits of digraph algebras (a.k.a.\ finite dimensional CSL
algebras) satisfying two conditions:
(1) the inclusions of the digraph algebras respect the order-preserving
normalisers, and
(2) the digraph algebras have chordal digraphs.
The first condition is also used to show that the invariant depends
only on the limit algebra and not the direct system.
We give an intrinsic characterisation of the limit algebras satisfying
both~(1) and~(2).
- Dilations of Limit Algebras and Interpolating
Spectrum, Pacific
J. of Math. 184 (1998) 75-93, Math Reviews
99f:47053.
- Abstract: We show that limit algebras having interpolating
spectrum are characterized by the property that all locally
contractive representations have *-dilations.
This extends a result for digraph algebras by Davidson.
It is an open question if such a limit algebra is the limit of
a direct system of digraph algebras with interpolating digraphs,
although a positive answer would allow one to obtain one direction
of our result directly from Davidson's.
Instead, we give a `local' construction of digraph algebras with
interpolating digraphs and use this to extend representations.
Tree algebras (in the sense of Davidson, Paulsen, and Power) have been
characterized by a commutant lifting property among digraph algebras
with interpolating digraphs.
We show that the analogous result holds for limit algebras, i.e.,
limit algebras with the analogous spectral condition are characterized
by the same commutant lifting property among the limit algebras
with interpolating spectrum.
- Norm-closed Bimodules of Nest Algebras, (with
Ken Davidson and
Tim Hudson)
J. Operator Theory 39 (1998) 59-87,
Math Reviews
99e:47053.
- Abstract: We study the support and essential support functions
of a norm--closed bimodule of a nest algebra.
An allowable support function pair determines a maximal bimodule.
There is also a natural candidate for the minimal bimodule for a given
support function pair.
We determine precisely when this candidate is the minimal element.
In the other cases, this module is still the intersection of all
bimodules with a given support function pair, but it is not in
this class itself.
- Homology for Operator Algebras IV: On the Regular
Classification of Limits of 4-Cycle Algebras (with
Steve Power),
J. of Functional Analysis 150 (1997) 240-287,
Math Reviews
99g:47109.
- Abstract: A 4-cycle algebra is a finite dimensional digraph
algebra (CSL algebra) whose reduced digraph is a 4-cycle. A rigid
embedding between such algebras is a direct sum of certain
nondegenerate multiplicity one star-extendible embeddings.
A complete classification is obtained for the regular
isomorphism classes of direct systems $A$ of 4-cycle algebras
with respect to rigid embeddings.
The critical invariant is a binary relation in the direct sum of
K_{0} A and H_{1} A,
generalising the scale of the $K_{0}$-group, called the joint
scale.
The joint scale encapsulates other invariants and compatibility
conditions of regular isomorphism.
These include the scale of H_{1} A, the scale of the
direct sum of H_{0} A and H_{1} A
sign compatibility, congruence compatibility and
H_{0} H_{1} coupling classes.
These invariants are also important for lifting (K_{0}
direct sum H_{1})
isomorphisms to algebra isomorphisms; we resolve this lifting problem
for various classes of 4-cycle algebra direct systems.
- On Derivations of Semi-nest Algebras (with
B.E.
Forrest and
L.W.
Marcoux),
Houston J. of Mathematics 22 (1996) 375-398,
Math Reviews
97h:47042.
- Abstract: A seminest algebra can be decomposed a two by two
block matrix, where the (1,1) entry is a nest algebra, the (4,4)
entry is a reflexive algebra and the (1,2) entry is a reflexive
subspace. In this paper, we study the derivations from a semi-nest
algebra into itself and show that such derivations are always
continuous. If the algebra is also a CSL algebra, we scharacterize
when the first homology group of the algebra is contained in the
first homology group of the (4,4) entry; in these cases, the only
obstruction to a derivation being inner arises from the (4,4) entry.
In particular, the H_{1}-group of the algebra vanishes
if the (4,4) entry is a direct sum of nest algebras or is selfadjoint.
If the algebra is not a CSL algebra, then this fails; for example,
if the (4,4) entry is multiples of I_{n}, n>1,
we give outer derivations on the algebra inspite of the (4,4) entry
being selfadjoint.
- The Failure of Approximate Inner Conjugacy for Standard
Diagonals in Regular Limit Algebras (with
Steve Power),
Canadian Math. Bulletin, 39 (1996) 420-428,
Math Reviews
97i:46096.
- Abstract: AF C^{*}-algebras contain natural AF masas
which, here, we call standard diagonals.
Standard diagonals are unique, in the sense that two standard diagonals
in an AF C^{*}-algebra are conjugate by an approximately
inner automorphism.
We show that this uniqueness fails for non-selfadjoint AF~operator
algebras.
Precisely, we construct two standard diagonals in a particular
non-selfadjoint AF~operator algebra which are not conjugate by an
approximately inner automorphism of the non-selfadjoint algebra.
- Nest Algebras with Locally Constant Cocycles, (with
J.R. Peters)
J. of the London Math. Society 55 (1997) 569-587,
Math Reviews
98d:47095.
- Abstract: The first examples of triangular AF algebras to
be studied were the refinement algebra and the standar algebra.
Both are analytic algebras with the property that the cocycle can
be taken to be constant on the matrix units of the algebra.
The latter property is quite special and still quite ill-understood.
In the present paper, we weamine the clase of nest algebras
T in AF C^{*}-algebras which share the
distinctive properties of the refinement algebra:
(1) T is a nest algebra in which the nest generated the
diagonal, (2) T admits a locally constant cocycle.
There are many such algebras and we classify them in terms of
source-ordered Bratteli diagrams.
- This
paper is available in PDF format.
- On the Lattice of Ideals of Triangular AF Algebras
(with Tim Hudson),
J. of Functional Analysis 138 (1996) 1-39,
Math Reviews
97e:47068.
- Abstract: We study triangular AF (TAF) algebras in terms of
their lattices of closed two-sided ideals. Not (isometrically)
isomorphic algebras can have isomorphic lattices of ideals; indeed,
there is an uncountable family of pairwise non-isomorphic algebras,
all with isomorphic lattices of ideals. In the positive direction,
if A and B are strongly maximal TAF algebras with
isomorphic lattices of ideals, then there is a bijective isometry
between the subalgebras of A and B generated by
their order-preserving normalizers. This bijective isometry is
the sum of an algebra isomorphism and an anti-isomorphism. Using
this, we show that if the TAF algebras are generated by their
order-preserving normalizers and are triangular subalgebras of
primitive C^{*}-algebras, then the lattices of ideals
are isomorphic if and only if the algebras are either (isometrically)
isomorphic or anti-isomorphic. Finally, we use complete distributivity
to show that there are TAF algebras whose lattices of ideals can not
arise from TAF algebras generated by their order preserving
normalizers. Our techniques rely on constructing a topological
binary relation based on the lattice of ideals; this realtion is
closely connected to the spectrum or fundamental relation (also a
topological binary relation) of the TAF algebra.
- Order Preservation in Limit Algebras (with
Alan Hopenwasser),
J. of Functional Analysis 133 (1995) 342-394,
Math Reviews
96k:46099.
- Abstract: The matrix units of a digraph algebra, A,
induce a relation, known as the diagonal order, on the projections
in a masa in the algebra. Normalizing partial isometries in
A act on these projections by conjugation; they are said
to be order preserving when they repect the diagonal order.
Order preserving embeddings, in turn, are those embeddings which
carry order preserving normalizers to order preserving normalizers.
This paper studies operatre algebras which are direct limits of
finite dimensional algebras with order preserving embeddings.
We give a complete classification of direct limits of full triangular
matrix algebras with order preserving embeddings. We also
investigate the problem of characterizing algebras with order
preserving embeddings.
- Semisimple Triangular AF Algebras,
J. of Functional Analysis
111 (1993) 323-349, Math Reviews
94b:46084.
- Abstract: We give several necessary and sufficient conditions
for a triangular AF algebra to be semisimple. In particular,
a triangular AF algebra which can be written using the standard
embedding infinitely often is semisimple; we also give a semisimple
triangular AF algebra which does not have a presentation of this
form. If two triangular AF algebras have the same Peters-Poon-Wagner
diagonal invariant, then either both are semisimple or both are not.
However, we give two algebras with the same diagonal invariant
where one has Jacobson radical equal to the strong radical and the
other does not. Semisimplicity can be characterized in terms of
Power's fundamental relation. We give generalizations of these
results to triangular sublagebras of groupoid
C^{*}-algebras. It is shown that there is a unique
maximal bimodule over the diagonal which does not intersect the
radical. Using this, we show that the Wedderburn principal theorem
does not hold for triangular AF algebras. A necessary condition for
the theorem to hold is that the above mentioned unique bimodule is a
subalgebra.
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