175 research outputs found
Progress in noncommutative function theory
In this expository paper we describe the study of certain non-self-adjoint
operator algebras, the Hardy algebras, and their representation theory. We view
these algebras as algebras of (operator valued) functions on their spaces of
representations. We will show that these spaces of representations can be
parameterized as unit balls of certain -correspondences and the
functions can be viewed as Schur class operator functions on these balls. We
will provide evidence to show that the elements in these (non commutative)
Hardy algebras behave very much like bounded analytic functions and the study
of these algebras should be viewed as noncommutative function theory
On projective equivalence of invariant subspace lattices
AbstractFor i = 1,2, let Ai be a linear transformation on a complex vector space and let σ be a lattice isomorphism from the invariant subspace lattice of A1 onto the invariant subspace lattice of A2. We determine the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds)
Strong Shift Equivalence of -correspondences
We define a notion of strong shift equivalence for -correspondences and
show that strong shift equivalent -correspondences have strongly Morita
equivalent Cuntz-Pimsner algebras. Our analysis extends the fact that strong
shift equivalent square matrices with non-negative integer entries give stably
isomorphic Cuntz-Krieger algebras.Comment: 26 pages. Final version to appear in Israel Journal of Mathematic
Composition Operators and Endomorphisms
If is an inner function, then composition with induces an
endomorphism, , of that leaves
invariant. We investigate the structure of the
endomorphisms of and that implement
through the representations of and
in terms of multiplication operators on
and . Our analysis, which is based on work
of R. Rochberg and J. McDonald, will wind its way through the theory of
composition operators on spaces of analytic functions to recent work on Cuntz
families of isometries and Hilbert -modules
Wieler solenoids, Cuntz-Pimsner algebras and K-theory
We study irreducible Smale spaces with totally disconnected stable sets and their associated -theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one -theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the groupoids of Deaconu-Renault and Putnam-Spielberg, extending results of Thomsen. The Deaconu-Renault groupoid and the explicit Morita equivalence leads to a Cuntz-Pimsner model for the stable Ruelle algebra. The -theoretic invariants of Cuntz-Pimsner algebras are then studied using the Cuntz-Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions we characterize the KMS weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples
Noncommutative ball maps
In this paper, we analyze problems involving matrix variables for which we
use a noncommutative algebra setting. To be more specific, we use a class of
functions (called NC analytic functions) defined by power series in
noncommuting variables and evaluate these functions on sets of matrices of all
dimensions; we call such situations dimension-free. These types of functions
have recently been used in the study of dimension-free linear system
engineering problems.
In this paper we characterize NC analytic maps that send dimension-free
matrix balls to dimension-free matrix balls and carry the boundary to the
boundary; such maps we call "NC ball maps". We find that up to normalization,
an NC ball map is the direct sum of the identity map with an NC analytic map of
the ball into the ball. That is, "NC ball maps" are very simple, in contrast to
the classical result of D'Angelo on such analytic maps over C. Another
mathematically natural class of maps carries a variant of the noncommutative
distinguished boundary to the boundary, but on these our results are limited.
We shall be interested in several types of noncommutative balls, conventional
ones, but also balls defined by constraints called Linear Matrix Inequalities
(LMI). What we do here is a small piece of the bigger puzzle of understanding
how LMIs behave with respect to noncommutative change of variables.Comment: 46 page
Cartan subalgebras in C*-algebras of Hausdorff etale groupoids
The reduced -algebra of the interior of the isotropy in any Hausdorff
\'etale groupoid embeds as a -subalgebra of the reduced
-algebra of . We prove that the set of pure states of with unique
extension is dense, and deduce that any representation of the reduced
-algebra of that is injective on is faithful. We prove that there
is a conditional expectation from the reduced -algebra of onto if
and only if the interior of the isotropy in is closed. Using this, we prove
that when the interior of the isotropy is abelian and closed, is a Cartan
subalgebra. We prove that for a large class of groupoids with abelian
isotropy---including all Deaconu--Renault groupoids associated to discrete
abelian groups--- is a maximal abelian subalgebra. In the specific case of
-graph groupoids, we deduce that is always maximal abelian, but show by
example that it is not always Cartan.Comment: 14 pages. v2: Theorem 3.1 in v1 incorrect (thanks to A. Kumjain for
pointing out the error); v2 shows there is a conditional expectation onto
iff the interior of the isotropy is closed. v3: Material (including some
theorem statements) rearranged and shortened. Lemma~3.5 of v2 removed. This
version published in Integral Equations and Operator Theor
Groupoids and an index theorem for conical pseudo-manifolds
We define an analytical index map and a topological index map for conical
pseudomanifolds. These constructions generalize the analogous constructions
used by Atiyah and Singer in the proof of their topological index theorem for a
smooth, compact manifold . A main ingredient is a non-commutative algebra
that plays in our setting the role of . We prove a Thom isomorphism
between non-commutative algebras which gives a new example of wrong way
functoriality in -theory. We then give a new proof of the Atiyah-Singer
index theorem using deformation groupoids and show how it generalizes to
conical pseudomanifolds. We thus prove a topological index theorem for conical
pseudomanifolds
Operator theory and function theory in Drury-Arveson space and its quotients
The Drury-Arveson space , also known as symmetric Fock space or the
-shift space, is a Hilbert function space that has a natural -tuple of
operators acting on it, which gives it the structure of a Hilbert module. This
survey aims to introduce the Drury-Arveson space, to give a panoramic view of
the main operator theoretic and function theoretic aspects of this space, and
to describe the universal role that it plays in multivariable operator theory
and in Pick interpolation theory.Comment: Final version (to appear in Handbook of Operator Theory); 42 page
Quantized reduction as a tensor product
Symplectic reduction is reinterpreted as the composition of arrows in the
category of integrable Poisson manifolds, whose arrows are isomorphism classes
of dual pairs, with symplectic groupoids as units. Morita equivalence of
Poisson manifolds amounts to isomorphism of objects in this category.
This description paves the way for the quantization of the classical
reduction procedure, which is based on the formal analogy between dual pairs of
Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with
correspondences between von Neumann algebras. Further analogies are drawn with
categories of groupoids (of algebraic, measured, Lie, and symplectic type). In
all cases, the arrows are isomorphism classes of appropriate bimodules, and
their composition may be seen as a tensor product. Hence in suitable categories
reduction is simply composition of arrows, and Morita equivalence is
isomorphism of objects.Comment: 44 pages, categorical interpretation adde
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