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2,300 |
The Mackey-Gleason Problem
|
math.OA
|
Let $A$ be a von Neumann algebra with no direct summand of Type $\roman I_2$,
and let $\scr P(A)$ be its lattice of projections. Let $X$ be a Banach space.
Let $m\:\scr P(A)\to X$ be a bounded function such that $m(p+q)=m(p)+m(q)$
whenever $p$ and $q$ are orthogonal projections. The main theorem states that
$m$ has a unique extension to a bounded linear operator from $A$ to $X$. In
particular, each bounded complex-valued finitely additive quantum measure on
$\scr P(A)$ has a unique extension to a bounded linear functional on $A$.
|
math
|
2,301 |
Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras
|
math.OA
|
We present the analytic foundation of a unified B-D-F extension functor
$\operatorname{Ext}_\tau$ on the category of noncommutative smooth algebras,
for any Fr\'echet operator ideal $\Cal K_\tau$. Combining the techniques
devised by Arveson and Voiculescu, we generalize Voiculescu's theorem to smooth
algebras and Fr\'echet operator ideals. A key notion involved is
$\tau$-smoothness, which is verified for the algebras of smooth functions, via
a noncommutative Sobolev lemma. The groups $\operatorname{Ext}_\tau$ are
computed for many examples.
|
math
|
2,302 |
A splitting property for subalgebras of tensor products
|
math.OA
|
We prove a basic result about tensor products of a $\text{II}_1$ factor with
a finite von Neumann algebra and use it to answer, affirmatively, a question
asked by S. Popa about maximal injective factors.
|
math
|
2,303 |
Bourgain algebras, minimal envelopes, minimal support sets, and some applications
|
math.OA
|
We explicitly compute certain Douglas algebras that are invariant under both
the Bourgain map and the minimal envelope map. We also compute the Bourgain
algebra and the minimal envelope of the maximal subalgebras of a certain singly
generated Douglas algebra.
|
math
|
2,304 |
Relative cohomology of Banach algebras
|
math.OA
|
Let $A$ be a Banach algebra, not necessarily unital, and let $B$ be a closed
subalgebra of $A$. We establish a connection between the Banach cyclic
cohomology group $ {\cal{HC}}^n(A)$ of $A$ and the Banach $B$-relative cyclic
cohomology group $ {\cal{HC}}^n_B(A) $ of $A$. We prove that, for a Banach
algebra $A$ with a bounded approximate identity and an amenable closed
subalgebra $B$ of $A$, up to topological isomorphism, ${\cal{HC}}^n(A) =
{\cal{HC}}^n_B(A) $ for all $n \ge 0$. We also establish a connection between
the Banach simplicial or cyclic cohomology groups of $A$ and those of the
quotient algebra $A/I$ by an amenable closed bi-ideal $I$. The results are
applied to the calculation of these groups for certain operator algebras,
including von Neumann algebras.
|
math
|
2,305 |
Some conditions on Douglas algebras that imply the invariance of the minimal envelope map
|
math.OA
|
We give several conditions on certain families of Douglas algebras that imply
that the minimal envelope of the given algebra is the algebra itself. We also
prove that the minimal envelope of the intersection of two Douglas algebras is
the intersection of their minimal envelope.
|
math
|
2,306 |
Algebras associated with Blaschke products of type {\it G}
|
math.OA
|
Let $\Omega$ and $\Omega_{\fin}$ be the sets of all interpolating Blaschke
products of type $G$ and of finite type $G$, respectively. Let $E$ and
$E_{\fin}$ be the Douglas algebras generated by $H^\infty$ together with the
complex conjugates of elements of $\Omega$ and $\Omega_{\fin}$, respectively.
We show that the set of all invertible inner functions in $E$ is the set of all
finite products of elements of $\Omega$ , which is also the closure of $\Omega$
among the Blaschke products. Consequently, finite convex combinations of finite
products of elements of $\Omega$ are dense in the closed unit ball of the
subalgebra of $H^\infty$ generated by $\Omega$. The same results hold when we
replace $\Omega$ by $\Omega_{\fin}$ and $E$ by $E_{\fin}$.
|
math
|
2,307 |
Fourier-Stieltjes algebras of locally compact groupoids
|
math.OA
|
This paper gives a first step toward extending the theory of
Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact
(second countable) groupoid, we show that B(G), the linear span of the Borel
positive definite functions on G, is a Banach algebra when represented as an
algebra of completely bounded maps on a C^*-algebra associated with G. This
necessarily involves identifying equivalent elements of B(G). An example shows
that the linear span of the continuous positive definite functions need not be
complete. For groups, B(G) is isometric to the Banach space dual of C^*(G). For
groupoids, the best analog of that fact is to be found in a representation of
B(G) as a Banach space of completely bounded maps from a C^*-algebra associated
with G to a C^*-algebra associated with the equivalence relation induced by G.
This paper adds weight to the clues in the earlier study of Fourier-Stieltjes
algebras that there is a much more general kind of duality for Banach algebras
waiting to be explored.
|
math
|
2,308 |
Conjugate operators for finite maximal subdiagonal algebras
|
math.OA
|
Let $\M$ be a von Neumann algebra with a faithful normal trace $\T$, and let
$H^\infty$ be a finite, maximal, subdiagonal algebra of $\M$. Fundamental
theorems on conjugate functions for weak$^*$\!-Dirichlet algebras are shown to
be valid for non-commutative $H^\infty$. In particular the conjugation operator
is shown to be a bounded linear map from $L^p(\M, \T)$ into $L^p(\M, \T)$ for
$1 < p < \infty$, and to be a continuous map from $L^1(\M,\T)$ into $L^{1,
\infty}(\M,\T)$. We also obtain that if an operator $a$ is such that
$|a|\log^+|a| \in L^1(\M,\T)$ then its conjugate belongs to $L^1(\M,\T)$.
Finally, we present some partial extensions of the classical Szeg\"o's theorem
to the non-commutative setting.
|
math
|
2,309 |
Excision in Banach simplicial and cyclic cohomology
|
math.OA
|
We prove that, for every extension of Banach algebras $ 0 \rightarrow B
\rightarrow A \rightarrow D \rightarrow 0 $ such that $B$ has a left or right
bounded approximate identity, the existence of an associated long exact
sequence of Banach simplicial or cyclic cohomology groups is equivalent to the
existence of one for homology groups. It follows from the continuous version of
a result of Wodzicki that associated long exact sequences exist. In particular,
they exist for every extension of $C^*$-algebras.
|
math
|
2,310 |
Fell bundles over groupoids
|
math.OA
|
The author provides some definitions and structural results about Fell
bundles, defined as C^*-algebra bundles over topological groupoids. Such
bundles are a mutual generalization of semi-direct products of groups with
C^*-algebras and C^*-algebra bundles over topological spaces. In particular a
Morita equivalence theorem with semi-direct products is established.
|
math
|
2,311 |
Nonstable K-theory for Z-stable C*-algebras
|
math.OA
|
Let Z denote the simple limit of prime dimension drop algebras that has a
unique tracial state. Let A != 0 be a unital C^*-algebra with A = A tensor Z.
Then the homotopy groups of the group U(A) of unitaries in A are stable
invariants, namely, \pi_i(U(A)) = K_{i-1}(A) for all integers i >= 0.
Furthermore, A has cancellation for full projections, and satisfies the
comparability question for full projections. Analogous results hold for
non-unital Z-stable C^*-algebras.
|
math
|
2,312 |
C*-actions of r-discrete groupoids and inverse semigroups
|
math.OA
|
Groupoid actions on C*-bundles and inverse semigroup actions on C*-algebras
are closely related when the groupoid is r-discrete.
|
math
|
2,313 |
Sub-Riemannian metrics for quantum Heisenberg manifolds
|
math.OA
|
Every Heisenberg manifold has a natural "sub-Riemannian" metric with
interesting properties. We describe the corresponding noncommutative metric
structure for Rieffel's quantum Heisenberg manifolds.
|
math
|
2,314 |
Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules
|
math.OA
|
Pimsner introduced the C*-algebra O_X generated by a Hilbert bimodule X over
a C*-algebra A. We look for additional conditions that X should satisfy in
order to study simplicity and, more generally, the ideal structure of O_X when
X is finite projective. We introduce two conditions: `(I)-freeness' and
`(II)-freeness', stronger than the former, in analogy with [J. Cuntz, W.
Krieger, Invent. Math. 56, 251-268] and [J. Cuntz, Invent. Math. 63, 25-40]
respectively. (I)-freeness comprehend the case of the bimodules associated with
an inclusion of simple C*-algebras with finite index, real or pseudoreal
bimodules with finite dimension and the case of `Cuntz-Krieger bimodules'. If X
satisfies this condition the C*-algebra O_X does not depend on the choice of
the generators when A is faithfully represented. As a consequence, if X is
(I)-free and A is X-simple, then O_X is simple. In the case of Cuntz-Krieger
algebras, X-simplicity corresponds to irreducibility of the defining matrix. If
A is simple and p.i. then O_X is p.i., if A is nonnuclear then O_X is
nonnuclear. We therefore provide examples of (purely) infinite nonnuclear
simple C*-algebras. Furthermore if X is (II)-free, we determine the ideal
structure of O_X.
|
math
|
2,315 |
Central sequence subfactors and double commutant properties
|
math.OA
|
First, we construct the Jones tower and tunnel of the central sequence
subfactor arising from a hyperfinite type II_1 subfactor with finite index and
finite depth, and prove each algebra has the double commutant property in the
ultraproduct of the enveloping II_1 factor. Next, we show the equivalence
between Popa's strong amenability and the double commutant property of the
central sequence factor for subfactors as above without assuming the finite
depth condition.
|
math
|
2,316 |
Applications of Topological *-Algebras of Unbounded Operators
|
math.OA
|
In this paper we discuss some physical applications of topological *-algebras
of unbounded operators. Our first example is a simple system of free bosons.
Then we analyze different models which are related to this one. We also discuss
the time evolution of two interacting models of matter and bosons. We show that
for all these systems it is possible to build up a common framework where the
thermodynamical limit of the algebraic dynamics can be conveniently studied and
obtained.
|
math
|
2,317 |
Projections in Rotation Algebras and Theta Functions
|
math.OA
|
For each $\alpha \in (0,1)$, $A_\alpha$ denotes the universal $C^*$-algebra
generated by two unitaries $u$ and $v$, which satisfy the commutation relation
$uv=\exp (2\pi i\alpha)vu$. We consider the order four automorphism $\sigma$ of
$A_\alpha$ defined by $\sigma (u)=v$, $\sigma (v)=u^{-1}$ and describe a method
for constructing projections in the fixed point algebra $A_\alpha^\sigma$,
using Rieffel's imprimitivity bimodules and Jacobi's theta functions. In the
case $\alpha =q^{-1}$, $q\in {\mathbf Z}$, $q\geq 2$, we give explicit formulae
for such projections and find a lower bound for the norm of the Harper operator
$u+u^* +v+v^*$.
|
math
|
2,318 |
Almost Representations and Asymptotic Representations of Discrete Groups
|
math.OA
|
We define for discrete finitely presented groups a new property related to
their asymptotic representations. Namely we say that a groups has the property
AGA if every almost representation generates an asymptotic representation. We
give examples of groups with and without this property. For our example of a
group $G$ without AGA the group $K^0(BG)$ cannot be covered by asymptotic
representations of $G$.
|
math
|
2,319 |
Factorization of completely bounded bilinear operators and injectivity
|
math.OA
|
We characterize injectivity of von Neumann algebras in terms of factoring
bilinear maps as products of linear maps.
|
math
|
2,320 |
Multiplicity-free representations of commutative C*-algebras and spectral properties
|
math.OA
|
Let A be a commutative unital C*-algebra and let S denote its Gelfand
spectrum. We give some necessary and sufficient conditions for a nondegenerate
representation of A to be unitarily equivalent to a multiplicative
representation on a space L^2(S, m), where m is a positive measure on the Baire
sets of S. We also compare these conditions with the multiplicity-free property
of a representation.
|
math
|
2,321 |
Viewing AF-algebras as graph algebras
|
math.OA
|
Every AF-algebra arises as a graph algebra in the sense of Kumjian, Pask,
Raeburn, and Renault. For AF-algebras, the diagonal subalgebra defined by
Stratila and Voiculescu is consistent with Kumjian's notion of diagonal, and
the groupoid arising from a well-chosen Bratteli diagram for A coincides with
Kumjian's twist groupoid constructed from a diagonal of A.
|
math
|
2,322 |
The Ext class of an approximately inner automorphism, II
|
math.OA
|
Let A be a simple unital AT algebra of real rank zero and Inn(A) the group of
inner automorphisms of A. In the previous paper we have shown that the natural
map of the group of approximately inner automorphisms into Ext(K_1(A),K_0(A))
oplus Ext(K_0(A),K_1(A)) is surjective; the kernel of this map includes the
subgroup of automorphisms which are homotopic to Inn(A). In this paper we
consider the quotient of the group of approximately inner automorphisms by the
smaller normal subgroup AInn(A) which consists of asymptotically inner
automorphisms and describe it as OrderExt(K_1(A),K_0(A)) oplus
Ext(K_0(A),K_1(A)), where OrderExt(K_1(A),K_0(A)) is a kind of extension group
which takes into account the fact that K_0(A) is an ordered group and has the
usual Ext as a quotient.
|
math
|
2,323 |
Invariant Linear Manifolds for CSL-Algebras and Nest Algebras
|
math.OA
|
Every invariant linear manifold for a CSL-algebra is a closed subspace if,
and only if, each non-zero projection in the projection lattice is generated by
finitely many atoms. In the case of a nest, this condition is equivalent to the
condition that every non-zero projection in the nest has an immediate
predecessor (the nest of orthogonal complements is well ordered).
The invariant linear manifolds of a nest algebra are totally ordered by
inclusion if, and only if, every non-zero projection in the nest has an
immediate predecessor.
|
math
|
2,324 |
Boundary Functions for Ideals in Analytic Limit Algebras
|
math.OA
|
We develop a theory of boundary functions for ideals in trivially analytic
subalgebras of simple AF C*-algebras with an injective 0-cocycle, a class which
includes all full nest algebras. Boundary functions are maps from the spectrum
of the diagonal of the analytic subalgebra to itself. The relation between
boundary functions and ideal sets is explored and a description is given of
meet and join irreducible boundary functions.
|
math
|
2,325 |
The Toeplitz algebra of a Hilbert bimodule
|
math.OA
|
Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X.
Pimsner constructed a C*-algebra O_X which includes, for particular choices of
X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger
algebras O_B. Here we analyse the representations of the corresponding Toeplitz
algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger
algebras of directed graphs, which includes Cuntz's uniqueness theorem for
O_\infty.
|
math
|
2,326 |
Representable bimodules over C*-algebras
|
math.OA
|
Given two C*-algebras A and B, abstract A-B bimodules that can be
isometrically represented as operator bimodules are characterised in terms of
their norm. Various properties of such bimodules are given. Their theory is
very similar to those of classical normed spaces.
|
math
|
2,327 |
Metrics on states from actions of compact groups
|
math.OA
|
Let a compact Lie group act ergodically on a unital $C^*$-algebra $A$. We
consider several ways of using this structure to define metrics on the state
space of $A$. These ways involve length functions, norms on the Lie algebra,
and Dirac operators. The main thrust is to verify that the corresponding metric
topologies on the state space agree with the weak-$*$ topology.
|
math
|
2,328 |
The curvature invariant of a Hilbert module over C[z_1,...,z_d]
|
math.OA
|
A notion of curvature is introduced in multivariable operator theory and an
analogue of the Gauss-Bonnet-Chern theorem is established for graded
(contractive) Hilbert modules over the complex polynomial algebra in d
variables, d=1,2,3,....
The curvature invariant, Euler characteristic, and degree are computed for
some explicit examples based on varieties in (multidimensional) complex
projective space, and applications are given to the structure of graded ideals
in C[z_1,...,z_d] and to the existence of "inner sequences" for closed
submodules of the free Hilbert module H^2(C^d).
|
math
|
2,329 |
Relative positions of matroid algebras
|
math.OA
|
A classification is given for (regular) positions of direct sums of two
matroid algebras (unital algebraic limits of matrix algebras) in a matroid
superalgebra, where the individual summands have index 2 in their associated
corner algebra. A similar classification is obtained for positions of direct
sums of 2-symmetric algebras and, in the odd case, for the positions of sums of
2-symmetric C*-algebras in matroid C*-algebras. The approach relies on an
analysis of intermediate non-self-adjoint operator algebras and the
classifications are given in terms of K0 invariants, partial isometry homology
and scales in the associated composite K0-homology group.
|
math
|
2,330 |
On the classification of nuclear C*-algebras
|
math.OA
|
The mid-seventies' works on C*-algebras of Brown-Douglas-Fillmore and Elliott
both contained uniqueness and existence results in a now standard sense. These
papers served as keystones for two separate theories -- KK-theory and the
classification program -- which for many years parted ways with only moderate
interaction. But recent years have seen a fruitful interaction which has been
one of the main engines behind rapid progress in the classification program.
In the present paper we take this interaction even further. We prove general
existence and uniqueness results using KK-theory and a concept of
quasidiagonality for representations. These results are employed to obtain new
classification results for certain classes of quasidiagonal C*-algebras
introduced by H. Lin. An important novel feature of these classes is that they
are defined by a certain local approximation property, rather than by an
inductive limit construction.
Our existence and uniqueness results are in the spirit of classical
Ext-theory. The main complication overcome in the paper is to control the
stabilization which is necessary when one works with finite C*-algebras. In the
infinite case, where programs of this type have already been successfully
carried out, stabilization is unnecessary. Yet, our methods are sufficiently
versatile to allow us to reprove, from a handful of basic results, the
classification of purely infinite nuclear C*-algebras of Kirchberg and
Phillips.
Indeed, it is our hope that this can be the starting point of a unified
approach to classification of nuclear C*-algebras.
|
math
|
2,331 |
Structure of the group of automorphisms of C$^{*}$-algebras
|
math.OA
|
We obtain a kind of structure theorem for the automorphism group ${\rm
Aut}{\cal A}$ of a unital C$^{*}$-algebra ${\cal A}$. According to it, ${\rm
Aut}{\cal A}$ can be regarded as a subgroup of the semi-direct product of
direct product group consisting of some family of projective unitary groups and
some permutation group on the spectrum of ${\cal A}$.
|
math
|
2,332 |
Normal conditional expectations of finite index and sets of modular generators
|
math.OA
|
Normal conditional expectations E: M --> N in M of finite index on von
Neumann algebras M with discrete center are investigated to find an estimate
for the minimal number of generators of M as a Hilbert N-module. Analyzing the
case of M being finite type I with discrete center we obtain that these von
Neumann algebras M are always finitely generated Hilbert N-modules with a
minimal generator set consisting of at most [K(E)]^2 generators, where [.]
denotes the integer part of a real number and K(E) = {K: K.E-id_M >= 0}. This
result contrasts remarkable examples by P. Jolissaint and S. Popa showing the
existence of normal conditional expectations of finite index on certain type
II_1 von Neumann algebras with center l_\infty which are not algebraically of
finite index in the sense of Y. Watatani. We show that estimates of the minimal
number of module generators by a function of [K(E)] cannot exist for certain
type II_1 von Neumann algebras with non-trivial center.
|
math
|
2,333 |
Compactly-aligned discrete product systems, and generalizations of O_\infty
|
math.OA
|
The universal C*-algebras of discrete product systems generalize the
Toeplitz- Cuntz algebras and the Toeplitz algebras of discrete semigroups. We
consider a semigroup P which is quasi-lattice ordered in the sense of Nica,
and, for a product system p:E\to P, we study those representations of E, called
covariant, which respect the lattice structure of P. We identify a class of
product systems, which we call compactly aligned, for which there is a purely
C*-algebraic characterization of covariance, and study the algebra
C*_{cov}(P,E) which is universal for covariant representations of E. Our main
theorem is a characterization of the faithful representations of C*_{cov}(P,E)
when P is the positive cone of a free product of totally-ordered amenable
groups.
|
math
|
2,334 |
Grothendieck group invariants for partly self-adjoint operator algebras
|
math.OA
|
Various partially ordered Grothendieck group invariants are introduced for
general operator algebras and these are used in the classification of direct
systems and direct limits of finite-dimensional complex incidence algebras with
common reduced digraph H (systems of H-algebras). In particular the dimension
distribution group G(A; C), defined for an operator algebra A and a
self-adjoint subalgebra C, generalises both the K0 group of a sigma unital
C*-algebra B and the spectrum (fundamental relation) R(A) of a regular limit A
of triangular digraph algebras. This invariant is more economical and
computable than the so called regular Grothendieck group which nevertheless
forms the basis for a complete classification of regular systems of H-algebras.
|
math
|
2,335 |
C*-equivalences of graphs
|
math.OA
|
Several relations on graphs, including primitive equivalence, explosion
equivalence and strong shift equivalence, are examined and shown to preserve
either the graph groupoid, a construction of Kumjian, Pask, Raeburn, and
Renault, or the groupoid of a pointed version of the graph. Thus these
relations preserve either the isomorphism class or the Morita equivalence class
of the graph C*-algebra, as defined by Kumjian, Pask, and Raeburn.
|
math
|
2,336 |
The Classification of Limits of 2n-cycle Algebras
|
math.OA
|
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 direct 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.
|
math
|
2,337 |
The K-theory of Cuntz-Krieger algebras for infinite matrices
|
math.OA
|
We compute the K-theory of the Cuntz-Krieger C^*-algebras associated to
infinite matrices.
|
math
|
2,338 |
A Microstates Approach to Relative Free Entropy
|
math.OA
|
We define and study a relative free entropy quantity, analogous in its
properties to Voiculescu's relative free entropy Chi^*(...:B). Our definition
uses matricial microstates, unlike his definition, which involves
non-commutative Hilbert transform. We prove a change of variable formula and
certain maximization results for our quantity. We also exhibit a connection
between the free entropy of a matrix with operator entries, relative to the
algebra of scalar matrices, with the free entropy of the entries of the matrix.
|
math
|
2,339 |
Weight theory for C*-algebraic quantum groups
|
math.OA
|
In this paper, we collect some technical results about weights on C*-algebras
which are useful in de theory of locally compact quantum groups in the
C*-algebra framework. We discuss the extension of a lower semi-continuous
weight to a normal weight following S. Baaj, look into slice weights and their
KSGNS-constructions and investigate the tensor product of weights together with
a partial GNS-construction for such a tensor product. This paper accompanies
our paper 'Locally compact quantum groups' in which we propose a relatively
simple definition of a locally compact quantum group in the C*-algebra
framework.
|
math
|
2,340 |
Cohomology of topological graphs and Cuntz-Pimsner algebras
|
math.OA
|
We compute the sheaf cohomology of a groupoid built from a local
homeomorphism of a locally compact space $X$. In particular, we identify the
twists over this groupoid, and its Brauer group. Our calculations refine those
made by Kumjian, Muhly, Renault and Williams in the case $X$ is the path space
of a graph, and the local homeomorphism is the shift. We also show how the
C*-algebra of a twist may be identified with the Cuntz-Pimsner algebra
constructed from a certain C*-correspondence.
|
math
|
2,341 |
Skew products and crossed products by coactions
|
math.OA
|
Given a labeling c of the edges of a directed graph E by elements of a
discrete group G, one can form a skew-product graph E cross_c G. We show, using
the universal properties of the various constructions involved, that there is a
coaction delta of G on C*(E) such that C*(E cross_c G) is isomorphic to the
crossed product C*(E) cross_delta G. This isomorphism is equivariant for the
dual action deltahat and a natural action gamma of G on C*(E cross_c G);
following results of Kumjian and Pask, we show that C*(E cross_c G) cross_gamma
G is isomorphic to C*(E cross_c G) cross_{gamma,r} G, which in turn is
isomorphic to C*(E) tensor K(l^2(G)), and it turns out that the action gamma is
always amenable. We also obtain corresponding results for r-discrete groupoids
Q and continuous homomorphisms c: Q -> G, provided Q is amenable. Some of these
hold under a more general technical condition which obtains whenever Q is
amenable or second-countable.
|
math
|
2,342 |
Locally compact quantum groups in the universal setting
|
math.OA
|
In this paper we associate to every reduced C*-algebraic quantum group A a
universal C*-algebraic quantum group. We fine tune a proof of Kirchberg to show
that every *-representation of a modified L1-space is generated by a unitary
corepresentation. By taking the universal enveloping C*-algebra of a dense sub
*-algebra of A we arrive at the uinversal C*-algebra. We show that this
universal C*-algebra carries a quantum group structure which is as rich as its
reduced companion.
|
math
|
2,343 |
On representations of partial ^*-algebras based on B-weights
|
math.OA
|
A generalization of the GNS-representation is investigated that represents
partial ^*-algebras as systems of operators acting on a partial inner product
space (PIP-space). It is based on possibly indefinite B-weights which are
closely related to the positive B-weights introduced by J.-P. Antoine, Y.
Soulet and C. Trapani. Some additional assumptions had to be made in order to
guarantee the GNS-construction. Different partial products of operators on a
PIP-space are considered which allow the GNS-construction under suitable
conditions. Several examples illustrate the argumentation and indicate inherent
problems.
|
math
|
2,344 |
A Note on the Representation Theory of Fell Bundles
|
math.OA
|
We show that every Fell bundle B over a locally compact group G is "proper"
in a sense recently introduced by Ng. Combining our results with those of Ng we
show that if B satisfies the "approximation property" then it is amenable in
the sense that the full and reduced cross-sectional C*-algebras coincide.
|
math
|
2,345 |
Morita-Rieffel Equivalence and Spectral Theory for Integrable Automorphism Groups of C*-Algebras
|
math.OA
|
Given a C*-dynamical system (A,G,\alpha), we discuss conditions under which
subalgebras of the multiplier algebra M(A) consisting of fixed points for
\alpha are Morita-Rieffel equivalent to ideals in the crossed product of A by
G. In case G is abelian we also develop a spectral theory, giving a necessary
and sufficient condition for \alpha to be equivalent to the dual action on the
cross-sectional C*-algebra of a Fell bundle. In our main application we show
that a proper action of an abelian group on a locally compact space is
equivalent to a dual action.
|
math
|
2,346 |
Discrete product systems of Hilbert bimodules
|
math.OA
|
A Hilbert bimodule is a right Hilbert module X over a C*-algebra A together
with a left action of A as adjointable operators on X. We consider families X =
{X_s :s\in P} of Hilbert bimodules, indexed by a semigroup P, which are endowed
with a multiplication which implements isomorphisms X_s\otimes_A X_t \to
X_{st}; such a family is a called a product system. We define a generalized
Cuntz- Pimsner algebra O_X, and we show that every twisted crossed product of A
by P can be realized as O_X for a suitable product system X. Assuming P is
quasi- lattice ordered in the sense of Nica, we analyze a certain Toeplitz
extension T_{cov}(X) of O_X by embedding it in a crossed product B_P
\times_{\tau,X} P which has been ``twisted'' by X; our main Theorem is a
characterization of the faithful representations of B_P \times_{\tau,X} P.
|
math
|
2,347 |
Discrete product systems of finite-dimensional Hilbert spaces, and generalized Cuntz algebras
|
math.OA
|
To each discrete product system E of finite-dimensional Hilbert spaces we
associate a C*-algebra O_E. When E is the n-dimensional product system over N,
O_E is the Cuntz algebra O_n, and the irrational rotation algebras appear as
O_E for certain one-dimensional product systems over N^2. We give conditions
which ensure that O_E is simple, purely infinite, and nuclear. Our main
examples are the lexicographic product systems, for which we obtain slightly
stronger results.
|
math
|
2,348 |
Trace acaling automorphisms of certain stable AF algebras II
|
math.OA
|
Two automorphisms of a simple stable AF algebra with a finite dimensional
lattice of lower semicontinuous traces are shown to be outer conjugate if they
act in the same way on the K-group and the extremal traces are scaled by
numbers which are not equal to 1 and satisfy a certain condition (which always
holds if all the scaling factors are less than 1). The proof goes via the
Rohlin property. As an application we consider the problem of classifying
conjugacy or outer conjugacy classes of certain actions of the circle group on
a separable purely infinite C*-algebra.
|
math
|
2,349 |
On certain extension properties for the space of compact operators
|
math.OA
|
Let $Z$ be a fixed separable operator space, $X\subset Y$ general separable
operator spaces, and $T:X\to Z$ a completely bounded map. $Z$ is said to have
the Complete Separable Extension Property (CSEP) if every such map admits a
completely bounded extension to $Y$; the Mixed Separable Extension Property
(MSEP) if every such $T$ admits a bounded extension to $Y$. Finally, $Z$ is
said to have the Complete Separable Complementation Property (CSCP) if $Z$ is
locally reflexive and $T$ admits a completely bounded extension to $Y$ provided
$Y$ is locally reflexive and $T$ is a complete surjective isomorphism. Let
${\bf K}$ denote the space of compact operators on separable Hilbert space and
${\bf K}_0$ the $c_0$ sum of ${\Cal M}_n$'s (the space of ``small compact
operators''). It is proved that ${\bf K}$ has the CSCP, using the second
author's previous result that ${\bf K}_0$ has this property. A new proof is
given for the result (due to E. Kirchberg) that ${\bf K}_0$ (and hence ${\bf
K}$) fails the CSEP. It remains an open question if ${\bf K}$ has the MSEP; it
is proved this is equivalent to whether ${\bf K}_0$ has this property. A new
Banach space concept, Extendable Local Reflexivity (ELR), is introduced to
study this problem. Further complements and open problems are discussed.
|
math
|
2,350 |
Nest Representations of TAF Algebras
|
math.OA
|
A nest representation of a strongly maximal TAF algebra $A$ is a
representation $\pi$ for which $\operatorname{Lat} \pi(A) is totally ordered.
We prove that if the spectrum of $A$ is totally ordered, or if
$\operatorname{Lat} \pi(A)$ contains an atom, then $\operatorname{ker} \pi$ is
a meet irreducible ideal.
|
math
|
2,351 |
Cuntz-like algebras
|
math.OA
|
The usual crossed product construction which associates to the homeomorphism
$T$ of the locally compact space $X$ the C$^*$-algebra $C^*(X,T)$ is extended
to the case of a partial local homeomorphism $T$. For example, the
Cuntz-Krieger algebras are the C$^*$-algebras of the one-sided Markov shifts.
The generalizations of the Cuntz-Krieger algebras (graph algebras, algebras
$O_A$ where $A$ is an infinite matrix) which have been introduced recently can
also be described as C$^*$-algebras of Markov chains with countably many
states. This is useful to obtain such properties of these algebras as
nuclearity, simplicity or pure infiniteness. One also gives examples of strong
Morita equivalences arising from dynamical systems equivalences.
|
math
|
2,352 |
Stable laws and domains of attraction in free probability theory
|
math.OA
|
In this paper we determine the distributional behavior of sums of free (in
the sense of Voiculescu) identically distributed, infinitesimal random
variables. The theory is shown to parallel the classical theory of independent
random variables, though the limit laws are usually quite different. Our work
subsumes all previously known instances of weak convergence of sums of free,
identically distributed random variables. In particular, we determine the
domains of attraction of stable distributions in the free theory. These freely
stable distributions are studied in detail in the appendix, where their
unimodality and duality properties are demonstrated.
|
math
|
2,353 |
Amenability of Hopf C^*-algebras
|
math.OA
|
Three natural definitions for amenability of general Hopf C^*-algebras (all
of them being generalizations of the case of locally compact groups) were given
and the relations between them were studied. Moreover, amenability in the
situation of duality of Hopf C^*-algebras was also studied.
|
math
|
2,354 |
Approximation property of $C^*$-algebraic Bundles
|
math.OA
|
In this paper, we will define the reduced cross-sectional $C^*$-algebras of
$C^*$-algebraic bundles over locally compact groups and show that if a
$C^*$-algebraic bundle has the approximation property (defined similarly as in
the discrete case), then the full cross-sectional $C^*$-algebra and the reduced
one coincide. Moreover, if a semi-direct product bundle has the approximation
property and the underlying $C^*$-algebra is nuclear, then the cross-sectional
$C^*$-algebra is also nuclear. We will also compare the approximation property
with the amenability of Anantharaman-Delaroche in the case of discrete groups.
|
math
|
2,355 |
Stable Ranks, K-Groups and Witt Groups of some Banach and C-star Algebras
|
math.OA
|
We show that certain dense and spectral invariant subalgebras of a
$C^*$-algebra have the same bilateral Bass stable rank. This is a partial
answer for (a version of) an open problem raised by R.G. Swan. Then, for
certain Banach algebras, we indicate when the homotopy groups
$\pi_{i}(GL_{n}(A))$ stabilize for large $n$. This is an improvement of a
result due to G. Corach and A. Larotonda. Using some results due to M. Karoubi,
we show the isomorphism of the Witt group of a symmetric Banach algebra with
the $K_0$-group of its enveloping $C^*$-algebra. The question if this is true
for all involutive Banach algebras was raised by A. Connes.
|
math
|
2,356 |
Modules over operator algebras, and the maximal C^*-dilation
|
math.OA
|
We continue our study of the general theory of possibly nonselfadjoint
algebras of operators on a Hilbert space, and modules over such algebras,
developing a little more technology to connect `nonselfadjoint operator
algebra' with the C$^*-$algebraic framework. More particularly, we make use of
the universal, or maximal, C$^*-$algebra generated by an operator algebra, and
C$^*-$dilations. This technology is quite general, however it was developed to
solve some problems arising in the theory of Morita equivalence of operator
algebras, and as a result most of the applications given here (and in a
companion paper) are to that subject. Other applications given here are to
extension problems for module maps, and characterizations of C$^*-$algebras.
|
math
|
2,357 |
Regular Operators on Hilbert C^*-modules
|
math.OA
|
A regular operator T on a Hilbert C^*-module is defined just like a closed
operator on a Hilbert space, with the extra condition that the range of
(I+T^*T) is dense. Semiregular operators are a slightly larger class of
operators that may not have this property. It is shown that, like in the case
of regular operators, one can, without any loss in generality, restrict oneself
to semiregular operators on C^*-algebras. We then prove that for abelian
C^*-algebras as well as for subalgebras of the algebra of compact operators,
any closed semiregular operator is automatically regular. We also determine how
a regular operator and its extensions (and restrictions) are related. Finally,
using these results, we give a criterion for a semiregular operator on a
liminal C^*-algebra to have a regular extension.
|
math
|
2,358 |
Extremal richness of multiplier and corona algebras of simple C*-algebras with real rank zero
|
math.OA
|
In this paper we investigate the extremal richness of the multiplier algebra
$M(A)$ and the corona algebra $M(A)/A$, for a simple C*-algebra $A$ with real
rank zero and stable rank one. We show that the space of extremal quasitraces
and the scale of $A$ contain enough information to determine whether $M(A)/A$
is extremally rich. In detail, if the scale is finite, then $M(A)/A$ is
extremally rich. In important cases, and if the scale is not finite, extremal
richness is characterized by a restrictive condition: the existence of only one
infinite extremal quasitrace which is isolated in a convex sense.
|
math
|
2,359 |
Regularity of operators on essential extensions of the compacts
|
math.OA
|
A semiregular operator on a Hilbert C^*-module, or equivalently, on the
C^*-algebra of `compact' operators on it, is a closable densely defined
operator whose adjoint is also densely defined. It is shown that for operators
on extensions of compacts by unital or abelian C^*-algebras, semiregularity
leads to regularity. Two examples coming from quantum groups are discussed.
|
math
|
2,360 |
Pure infiniteness, stability and C*-algebras of graphs and dynamical systems
|
math.OA
|
Pure infiniteness (in sense of E.Kirchberg and M.R{\o}rdam) is considered for
C*-algebras arising from singly generated dynamical systems. In particular,
Cuntz-Krieger algebras and their generalizations, i.e., graph-algebras and O_A
of an infinite matrix A, admit characterizations of pure infiniteness. As a
consequence, these generalized Cuntz-Krieger algebras are traceless if and only
if they are purely infinite. Also, a characterization of AF-algebras among
these C*-algebras is given. In the case of graph-algebras of locally finite
graphs, characterizations of stability are obtained.
|
math
|
2,361 |
Topological Entropy for the Canonical Endomorphism of Cuntz-Krieger Algebras
|
math.OA
|
It is shown that Voiculescu's toplogical entropy for the canonical
endomorphism of a simple Cuntz-Krieger algebra O_A equals the logarithm of the
spectral radius of A.
|
math
|
2,362 |
On the Toeplitz algebras of right-angled and finite-type Artin groups
|
math.OA
|
The graph product of a family of groups lies somewhere between their direct
and free products, with the graph determining which pairs of groups commute and
which do not. We show that the graph product of quasi-lattice ordered groups is
quasi-lattice ordered, and, when the underlying groups are amenable, that it
satisfies Nica's amenability condition for quasi-lattice orders. As a
consequence the Toeplitz algebras of these groups are universal for covariant
isometric representations on Hilbert space, and their representations are
faithful if the isometries satisfy a properness condition given by Laca and
Raeburn. An application of this to right-angled Artin groups gives a uniqueness
theorem for the C^*-algebra generated by a collection of isometries such that
any two of them either *-commute or else have orthogonal ranges. In contrast,
the nonabelian Artin groups of finite type considered by Brieskorn and Saito
and Deligne have canonical quasi-lattice orders that are not amenable in the
sense of Nica, so their Toeplitz algebras are not universal and the C^*-algebra
generated by a collection of isometries satisfying the Artin relations fails to
be unique.
|
math
|
2,363 |
On unbounded p-summable Fredholm modules
|
math.OA
|
We prove that odd unbounded p-summable Fredholm modules are also bounded
p-summable Fredholm modules (this is the odd counterpart of a result of A.
Connes for the case of even Fredholm modules).
|
math
|
2,364 |
Hilbert bimodules with involution
|
math.OA
|
We examine Hilbert bimodules which possess a (generally unbounded)
involution. Topics considered include a linking algebra representation,
duality, locality, and the role of these bimodules in noncommutative
differential geometry.
|
math
|
2,365 |
On $C^*$-algebras related to asymptotic homomorphisms
|
math.OA
|
We study the $C^*$-algebras related to Mishchenko's version of asymptotic
homomorphisms. In particular we show that their different versions are weakly
homotopy equivalent but not isomorphic to each other. We give also the
continuous version for these algebras.
|
math
|
2,366 |
Interactions in noncommutative dynamics
|
math.OA
|
A mathematical notion of interaction is introduced for noncommutative
dynamical systems, i.e., for one parameter groups of *-automorphisms of $\Cal
B(H)$ endowed with a certain causal structure. With any interaction there is a
well-defined "state of the past" and a well-defined "state of the future". We
describe the construction of many interactions involving cocycle perturbations
of the CAR/CCR flows and show that they are nontrivial. The proof of
nontriviality is based on a new inequality, relating the eigenvalue lists of
the "past" and "future" states to the norm of a linear functional on a certain
C^*-algebra.
|
math
|
2,367 |
Exactness of reduced amalgamated free product C*-algebras
|
math.OA
|
Some completely positive maps on reduced amalgamated free products of
C*-algebras are constructed; these allow a proof that the class of exact unital
C*-algebras is closed under taking reduced amalgamated free products.
Consequently, the class of exact discrete groups is closed under taking
amalgamated free products.
|
math
|
2,368 |
Exactness of Cuntz-Pimsner C*-algebras
|
math.OA
|
Let H be a full Hilbert bimodule over a C*-algebra A. We show that the
Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact.
Using this result, we give alternative proofs for exactness of reduced
amalgamated free products of exact C*-algebras. In the case that A is a finite
dimensional C*-algebra, we also show that the Brown-Voiculescu topological
entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to
an A,A Hilbert bimodule is zero.
|
math
|
2,369 |
Purely infinite, simple C*-algebras arising from free product constructions, II
|
math.OA
|
Certain reduced free products of C*-algebras,
(A,phi)=(A_1,phi_1)*(A_2,\phi_2), taken with respect to faithful states, at
least one of which is not a trace, are shown to be purely infinite and simple.
It is assumed that one of the A_i contain a partial isometry in the spectral
subspace of phi_i corresponding to a positive number not equal to one. For
example, if A_1 and A_2 are copies of the two-by-two complex matrices and if
phi_1 and phi_2 are not unitarily conjugate, it is shown that A is simple and
purely infinite.
|
math
|
2,370 |
Purely infinite, simple C*-algebras arising from free product constructions, III
|
math.OA
|
In the reduced free product of C*-algebras (A,phi)=(A_1,phi_1)*(A_2,phi_2), A
is shown to be purely infinite and simple under the hypothesis that A_1 is the
crossed product of a C*-algebra by a discrete infinite group, phi_1 is well
behaved with respect to this crossed product and A_2 is not one dimensional.
|
math
|
2,371 |
Projections in free product C*-algebras, II
|
math.OA
|
Let (A,phi) be the reduced free product of infinitely many pairs (A_i,phi_i)
of C*-algebras with faithful states. Assume that the A_i are not too small, in
a specific sense. It is shown that if phi is a trace then K_0(A) is determined
entirely by K_0(phi). If, furthermore, the image of K_0(phi) is dense in the
reals then A has real rank zero. On the other hand, if phi is not a trace then
A is simple and purely infinite.
|
math
|
2,372 |
Topological entropy of some automorphisms of reduced amalgamated free product C*-algebras
|
math.OA
|
Certain classes of automorphisms of recued amalgamated free products of
C*-algebras are shown to have Brown-Voiculescu topological entropy zero. Also,
for automorphisms of exact C*-algebras, the Connes-Narnhofer-Thirring entropy
is shown to be bounded above by the Brown-Voiculescu entropy. These facts are
applied to generalize Stormer's result about entropy of automorphisms of the
II_1-factor of a free group.
|
math
|
2,373 |
Compressions of free products of von Neumann algebras
|
math.OA
|
A reduction formula for compressions of von Neumann algebras arising as free
products is proved. This shows that the fundamental group is all of the
positive reals for some such algebras. Additionally, by taking a sort of free
product with an unbounded semicircular element, continuous one parameter groups
of trace scaling automorphisms on II_infinity factors are constructed; this
produces type III_1 factors with core M tensor B(H), where M can be a full
II_1-factor without the Haagerup approximation property.
|
math
|
2,374 |
Embeddings of reduced free products of operator algebras
|
math.OA
|
Given reduced amalgamated free products of C$^*$-algebras,
$(A,phi)=*_i(A_i,phi_i)$ and $(D,psi)=*_i(D_i,psi_i)$, an embedding $A\to D$ is
shown to exist assuming there are conditional expectation preserving embeddings
$A_i\to D_i$. This result is extended to show the existance of the reduced
amalgamated free product of certain classes of unital completely positive maps.
Analogues of the above mentioned results are proved for von Neumann algebras.
|
math
|
2,375 |
Index of $Γ$-equivariant Toeplitz operators
|
math.OA
|
Let $\Gamma$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and
let M denote the quotient of the unit disc by $\Gamma$. We prove that a
Toeplitz operator with $\Gamma$-invariant symbol f in C(M) is Brauer Fredholm
if its symbol is invertible on the boundary of M and its Brauer index is equal
to the winding number of f at the boundary. We construct the associated
extension of the algebra of functions continuous on the boundary of M by the
Brauer ideal in the C*-algebra generated by such operators.
|
math
|
2,376 |
Homotopy of state orbits
|
math.OA
|
Let M be a von Neumann algebra, f a faithful normal state and denote by M^f
the fixed point algebra of the modular group of f. Let U_M and U_{M^f} be the
unitary groups of M and M^f. In this paper we study the quotient U_M/U_{M^f}
endowed with two natural topologies: the one induced by the usual norm of M
(called here usual topology), and the one induced by the pre-Hilbert C*-module
norm given by the f-invariant conditional expectation E_f:M \to M^f (called the
modular topology). It is shown that U_M/U_{M^f} is simply connected with the
usual topology. Both topologies are compared, and it is shown that they
coincide if and only if the Jones index of E_f is finite. The set U_M/U_{M^f}
can be regarded as a model for the unitary orbit {f \circ Ad(u^*): u\in U_M} of
f, and either with the usual or the modular it can be embedded continuously in
the conjugate space M* (although not as a topological submanifold).
|
math
|
2,377 |
Geometry of oblique projections
|
math.OA
|
Let A be a unital C*-algebra. Denote by P the space of selfadjoint
projections of A. We study the relationship between P and the spaces of
projections P_a determined by the different involutions #_a induced by positive
invertible elements a in A. The maps f_p: P \to P_a sending p to the unique q
in P_a with the same range as p and \Omega_a: P_a \to P sending q to the
unitary part of the polar decomposition of the symmetry 2q-1 are shown to be
diffeomorphisms. We characterize the pairs of idempotents q, r in A with
|q-r|<1 such that there exists a positive element a in A verifying that q, r
are in P_a. In this case q and r can be joined by an unique short geodesic
along the space of idempotents Q of A.
|
math
|
2,378 |
The ideal structure of the Hecke C*-algebra of Bost and Connes
|
math.OA
|
We compute explicitly the primitive ideal space of the Bost-Connes Hecke
C*-algebra by embedding it as a full corner in a transformation group
C*-algebra and applying a general theorem of Williams. This requires the
computation of the quasi-orbit space for the action of the multiplicative
positive rationals on the space of finite adeles. We then carry out a similar
computation for the action of the nonzero rationals on the space of full
adeles.
|
math
|
2,379 |
From endomorphisms to automorphisms and back: dilations and full corners
|
math.OA
|
When S is a discrete subsemigroup of a discrete group G such that G = S^{-1}
S, it is possible to extend circle-valued multipliers from S to G; to dilate
(projective) isometric representations of S to (projective) unitary
representations of G; and to dilate/extend actions of S by injective
endomorphisms of a C*-algebra to actions of G by automorphisms of a larger
C*-algebra. These dilations are unique provided they satisfy a minimality
condition. The (twisted) semigroup crossed product corresponding to an action
of S is isomorphic to a full corner in the (twisted) crossed product by the
dilated action of G. This shows that crossed products by semigroup actions are
Morita equivalent to crossed products by group actions, making powerful tools
available to study their ideal structure and representation theory. The
dilation of the system giving the Bost-Connes Hecke C*-algebra from number
theory is constructed explicitly as an application: it is the crossed product
corresponding to the multiplicative action of the positive rationals on the
additive group of finite adeles.
|
math
|
2,380 |
Projective spaces of a C*-algebra
|
math.OA
|
Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we
study the notion of projective space associated to a C*-algebra A with a fixed
projection p. The resulting space P(p) admits a rich geometrical structure as a
holomorphic manifold and a homogeneous reductive space of the invertible group
of A. Moreover, several metrics (chordal, spherical, pseudo-chordal,
non-Euclidean - in Schwarz-Zaks terminology) are considered, allowing a
comparison among P(p), the Grassmann manifold of A and the space of positive
elements which are unitary with respect to the bilinear form induced by the
reflection e = 2p-1. Among several metrical results, we prove that geodesics
are unique and of minimal length when measured with the spherical and
non-Euclidean metrics.
|
math
|
2,381 |
Polar decomposition under perturbations of the scalar product
|
math.OA
|
Let A be a unital C* algebra with involution * represented in a Hilbert space
H, G the group of invertible elements of A, U the unitary group of A, G^s the
set of invertible selfadjoint elements of A, Q={e in G : e^2 = 1} the space of
reflections and P = Q\cap U. For any positive a in G consider the a-unitary
group U_a={g in G : a^{-1} g^* a = g^{-1}}, i.e. the elements which are unitary
with respect to the scalar product <\xi,\eta>_a = <a \xi,\eta> for \xi, \eta in
H. If \pi denotes the map that assigns to each invertible element its unitary
part in the polar decomposition, we show that the restriction \pi|_{U_a}: U_a
\to U is a diffeomorphism, that \pi(U_a \cap Q) = P and that \pi(U_a\cap G^s) =
U_a\cap G^s = {u in G: u=u^*=u^{-1} and au = ua}.
|
math
|
2,382 |
Orbits of conditional expectations
|
math.OA
|
Let N \subseteq M be von Neumann algebras and E:M\to N a faithful normal
conditional expectation. In this work it is shown that the similarity orbit
S(E) of E by the natural action of the invertible group of G_M of M has a
natural complex analytic structure and the map given by this action: G_M\to
S(E) is a smooth principal bundle. It is also shown that if N is finite then
S(E) admits a reductive structure. These results were known previously under
the conditions of finite index and N'\cap M \subseteq N, which are removed in
this work. Conversely, if the orbit S(E) has an homogeneous reductive structure
for every expectation defined on M, then M is finite. For every algebra M and
every expectation E, a covering space of the unitary orbit U(E) is constructed
in terms of the connected component of 1 in the normalizer of E. Moreover, this
covering space is the universal covering in any of the following cases: 1) M is
a finite factor and Ind(E) < \infty; 2) M is properly infinite and E is any
expectation; 3) E is the conditional expectation onto the centralizer of a
state. Therefore, in those cases, the fundamental group of U(E) can be
characterized as the Weyl group of E.
|
math
|
2,383 |
Projective space of a C*-module
|
math.OA
|
Let X be a right Hilbert C*-module over A. We study the geometry and the
topology of the projective space P(X) of X, consisting of the orthocomplemented
submodules of X which are generated by a single element. We also study the
geometry of the p-sphere S_p(X) and the natural fibration S_p(X) \to P(X),
where S_p(X)={x\in X: <x,x>=p}, for p in A a projection. The projective space
and the p-sphere are shown to be homogeneous differentiable spaces of the
unitary group of the algebra L_A(X) of adjointable operators of X. The homotopy
theory of these spaces is examined.
|
math
|
2,384 |
Inclusions of second quantization algebras
|
math.OA
|
In this note we study inclusions of second quantization algebras, namely
inclusions of von Neumann algebras on the Fock space of a separable complex
Hilbert space H, generated by the Weyl unitaries with test functions in closed,
real linear subspaces of H. We show that the class of irreducible inclusions of
standard second quantization algebras is non empty, and that they are depth two
inclusions, namely the third relative commutant of the Jones' tower is a
factor. When the smaller vector space has codimension n into the bigger, we
prove that the corresponding inclusion of second quantization algebras is given
by a cross product with R^n. This shows in particular that the inlcusions
studied in hep-th/9703129, namely the inclusion of the observable algebra
corresponding to a bounded interval for the (n+p)-th derivative of the current
algebra on the real line into the observable algebra for the same interval and
the n-th derivative theory is given by a cross product with R^p. On the
contrary, when the codimension is infinite, we show that the inclusion may be
non regular (cf. M. Enock, R. Nest, J. Funct. Anal. 137 (1996), 466-543), hence
do not correspond to a cross product with a locally compact group.
|
math
|
2,385 |
Asymptotically split extensions and E-theory
|
math.OA
|
We show that the E-theory of Connes and Higson can be formulated in terms of
C*-extensions in a way quite similar to the way in which the KK-theory of
Kasparov can. The essential difference is that the role played by split
extensions should be taken by asymptotically split extensions. We call an
extension of a C*-algebra $A$ by a stable C*-algebra $B$ asymptotically split
if there exists an asymptotic homomorphism consisting of right inverses for the
quotient map. An extension is called semi-invertible if it can be made
asymptotically split by adding another extension to it. Our main result is that
there exists a one-to-one correspondence between asymptotic homomorphisms from
$SA$ to $B$ and homotopy classes of semi-invertible extensions of $S^2A$ by
$B$.
|
math
|
2,386 |
Universal C*-algebra of real rank zero
|
math.OA
|
It is well-known that every commutative separable unital C*-algebra of real
rank zero is a quotient of the C*-algebra of all compex continous functions
defined on the Cantor cube. We prove a non-commutative version of this result
by showing that the class of all separable unital C*-algebras of real rank zero
concides with the class of quotients of a certain separable unital C*-algebra
of real-rank zero.
|
math
|
2,387 |
Microstates free entropy and cost of equivalence relations
|
math.OA
|
We define an analog of Voiculescu's free entropy for n-tuples of unitaries
(u_{1},...,u_{n}) in a tracial von Neumann algebra M, normalizing a unital
diffuse abelian subalgebra B in M. Using this quantity, we define the free
dimension \delta_{0}(u_{1},..,u_{n}\btw B). This number depends on (u_{1},...
,u_{n}) only up ``orbit equivalence'' over B. In particular, if R is an
measirable equivalence relation on [0,1] generated by n automorphisms
\alpha_{1},...,\alpha_{n}, let u_{1},..., u_{n} be the unitaries implementing
\alpha_{1},..,\alpha_{n} in the Feldman-Moore crossed product algebra
M=W^{*}([0,1],R), and let B be the canonical copy of L^\infty functions on
[0,1] inside M. In this way, we obtain an invariant \delta (R)=\delta
_{0}(u_{1},...,u_{n},\btw B) of the equivalence relation (R). If R is treeable,
\delta (R) coincides with the cost C(R) of R in the sense of Gaboriau. For a
general equivalence relation R posessing a finite graphing, \delta(R)\leq C(R).
Using the notion of free dimension, we define an dynamical entropy invariant
for an automorphism of a measurable equivalence relation (or more generally of
an r-discrete measure groupoid), and give examples.
|
math
|
2,388 |
Three Bimodules for Mansfield's Imprimitivity Theorem
|
math.OA
|
There are at least three imprimitivity bimodules naturally associated to a
maximal coaction of a discrete group G on a C*-algebra and a normal subgroup of
G: Mansfield's bimodule; the bimodule assembled by Ng from Green's
imprimitivity bimodule and Katayama duality; and a bimodule assembled from
Green's bimodule and a crossed-product Mansfield bimodule. We show that all
three of these are isomorphic, so that the corresponding inducing maps on
representations are identical. This can be interpreted as saying that Mansfield
and Green induction are inverses of one another ``modulo Katayama duality''.
These results pass to twisted coactions; dual results starting with an action
are also given.
|
math
|
2,389 |
Naturality and Induced Representations
|
math.OA
|
We show that induction of covariant representations for C*-dynamical systems
is natural in the sense that it gives a natural transformation between certain
crossed-product functors. This involves setting up suitable categories of
C*-algebras and dynamical systems, and extending the usual constructions of
crossed products to define the appropriate functors. From this point of view,
Green's Imprimitivity Theorem identifies the functors for which induction is a
natural equivalence. Various spcecial cases of these results have previously
been obtained on an ad hoc basis.
|
math
|
2,390 |
Entropy of Bogoliubov automorphisms of CAR and CCR algebras with respect to quasi-free states
|
math.OA
|
We compute the dynamical entropy of Bogoliubov automorphisms of CAR and CCR
algebras with respect to arbitrary gauge-invariant quasi-free states. This
completes the research started by Stormer and Voiculescu, and continued in
works of Narnhofer-Thirring and Park-Shin.
|
math
|
2,391 |
On the K-property of quantized Arnold cat maps
|
math.OA
|
We prove that some quantized Arnold cat maps are entropic K-systems. This
result was formulated by H. Narnhofer[1], but the fact that the optimal
decomposition for the multi-channel entropy constructed there is not strictly
local was not appropriately taken care of. We propose a strictly local
decomposition based on a construction of Voiculescu.
|
math
|
2,392 |
Entropy in type I algebras
|
math.OA
|
It is shown that if (M,phi,alpha) is a W*-dynamical system with M a type I
von Neumann algebra then the entropy of alpha w.r.t. phi equals the entropy of
the restriction of alpha to the center of M. If furthermore (N,psi,beta) is a
W*-dynamical system with N injective then the entropy of the tensor product
system is the sum of the entropies.
|
math
|
2,393 |
Entropy of automorphisms of II_1-factors arising from the dynamical systems theory
|
math.OA
|
Let a countable amenable group G acts freely and ergodically on a Lebesgue
space (X,mu), preserving the measure mu. If T is an automorphism of the
equivalence relation defined by G then T can be extended to an automorphism
alpha_T of the II_1-factor M=L^\infty(X,\mu)\rtimes G. We prove that if T
commutes with the action of G then H(alpha_T)=h(T), where H(alpha_T) is the
Connes- Stormer entropy of alpha_T, and h(T) is the Kolmogorov-Sinai entropy of
T. We prove also that for given s and t, 0\le s\le t\le\infty, there exists a T
such that h(T)=s and H(alpha_T)=t.
|
math
|
2,394 |
Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem
|
math.OA
|
For each \beta\in(0,+\infty) there exists a canonical measure \mu_\beta on
the ring A_f of finite adeles. We show that the positive rationals act
ergodically on (A_f,\mu_\beta) for \beta\in(0,1], and then deduce from this the
uniqueness of KMS_\beta-states for the Bost-Connes system.
|
math
|
2,395 |
Asymptotic homomorphisms into the Calkin algebra
|
math.OA
|
Let $A$ be a separable $C^*$-algebra and let $B$ be a stable $C^*$-algebra
with a strictly positive element. We consider the (semi)group $\Ext^{as}(A,B)$
(resp. $\Ext(A,B)$) of homotopy classes of asymptotic (resp. of genuine)
homomorphisms from $A$ to the corona algebra $M(B)/B$ and the natural map
$i:\Ext(A,B)\ar\Ext^{as}(A,B)$. We show that if $A$ is a suspension then
$\Ext^{as}(A,B)$ coincides with $E$-theory of Connes and Higson and the map $i$
is surjective. In particular any asymptotic homomorphism from $SA$ to $M(B)/B$
is homotopic to some genuine homomorphism.
|
math
|
2,396 |
Dual group actions on C*-algebras and their description by Hilbert extensions
|
math.OA
|
Given a C*-algebra $A$, a discrete abelian group $X$ and a homomorphism
$\Theta: X\to$ Out$A$ defining the dual action group $\Gamma\subset$ aut$A$,
the paper contains results on existence and characterization of Hilbert
$\{A,\Gamma\}$, where the action is given by $\hat{X}$. They are stated at the
(abstract) C*-level and can therefore be considered as a refinement of the
extension results given for von Neumann algebras for example by Jones
[Mem.Am.Math.Soc. 28 Nr 237 (1980)] or Sutherland [Publ.Res.Inst.Math.Sci. 16
(1980) 135]. A Hilbert extension exists iff there is a generalized 2-cocycle.
These results generalize those in [Commun.Math.Phys. 15 (1969) 173], which are
formulated in the context of superselection theory, where it is assumed that
the algebra $A$ has a trivial center, i.e. $Z=C1$. In particular the well-known
``outer characterization'' of the second cohomology $H^2(X,{\cal
U}(Z),\alpha_X)$ can be reformulated: there is a bijection to the set of all
$A$-module isomorphy classes of Hilbert extensions. Finally, a Hilbert space
representation (due to Sutherland in the von Neumann case) is mentioned. The
C*-norm of the Hilbert extension is expressed in terms of the norm of this
representation and it is linked to the so-called regular representation
appearing in superselection theory.
|
math
|
2,397 |
Serre-Swan theorem for non-commutative C$^{*}$-algebras
|
math.OA
|
We generalize the Serre-Swan theorem to non-commutative C$^{*}$-algebras. For
a Hilbert C$^{*}$-module $X$ over a C$^{*}$-algebra ${\cal A}$, we introduce a
hermitian vector bundle $\exx$ associated to $X$. We show that there is a
linear subspace $\Gamma_{X}$ of the space of all holomorphic sections of ${\cal
E}_{X}$ and a flat connection $D$ on ${\cal E}_{X}$ with the following
properties: (i) $\Gamma_{X}$ is a Hilbert ${\cal A}$-module with the action of
${\cal A}$ defined by $D$, (ii) the C$^{*}$-inner product of $\Gamma_{X}$ is
induced by the hermitian metric of ${\cal E}_{X}$, (iii) ${\cal E}_{X}$ is
isomorphic to an associated bundle of an infinite dimensional Hopf bundle, (iv)
$\Gamma_{X}$ is isomorphic to $X$.
|
math
|
2,398 |
Boundary actions for affine buildings and higher rank Cuntz-Krieger algebras
|
math.OA
|
Let $\G$ be a group of type rotating automorphisms of an affine building
$\cB$ of type $\wt A_2$. If $\G$ acts freely on the vertices of $\cB$ with
finitely many orbits, and if $\Omega$ is the (maximal) boundary of $\cB$, then
$C(\Om)\rtimes \G$ is a p.i.s.u.n. $C^*$-algebra. This algebra has a structure
theory analogous to that of a simple Cuntz-Krieger algebra and is the
motivation for a theory of higher rank Cuntz-Krieger algebras, which has been
developed by T. Steger and G. Robertson. The K-theory of these algebras can be
computed explicitly in the rank two case. For the rank two examples of the form
$C(\Om)\rtimes \G$ which arise from boundary actions on $\wt A_2$ buildings,
the two K-groups coincide.
|
math
|
2,399 |
The generalized Chern character and Lefschetz numbers in W*-modules
|
math.OA
|
We define N-theory being some analogue of K-theory on the category of von
Neumann algebras such that $K_0(A)\subset N_0(A)$ for any von Neumann algebra
A. Moreover, it turns out to be possible to construct the extension of the
Chern character to some homomorphism from $N_0(A)$ to even Banach cyclic
homology of A. Also, we define generalized Lefschetz numbers for an arbitrary
unitary endomorphism U of an A-elliptic complex. We study them in the situation
when U is an element of a representation of some compact Lie group.
|
math
|
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