Expand all Collapse all | Results 1 - 25 of 72 |
1. CMB Online first
On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras
as measured by their densely defined lower semicontinuous traces.
We first prove that a unital simple $\mathrm{C}^*$-algebra whose
extreme tracial boundary is nonempty and finite contains positive
operators of every possible rank, independent of the property
of strict comparison. We then turn to nonunital simple algebras
and establish criteria that imply that the Cuntz semigroup is
recovered functorially from the Murray-von Neumann semigroup
and the space of densely defined lower semicontinuous traces.
Finally, we prove that these criteria are satisfied by not-necessarily-unital
approximately subhomogeneous algebras of slow dimension growth.
Combined with results of the first-named author, this shows that
slow dimension growth coincides with $\mathcal Z$-stability,
for approximately subhomogeneous algebras.
Keywords:nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growth Categories:46L35, 46L05, 46L80, 47L40, 46L85 |
2. CMB Online first
A short note on the continuous Rokhlin property and the universal coefficient theorem in E-theory Let $G$ be a metrizable compact group, $A$ a separable $\mathrm{C}^*$-algebra
and $\alpha\colon G\to\operatorname{Aut}(A)$ a strongly continuous action.
Provided that $\alpha$ satisfies the continuous Rokhlin property,
we show that the property of satisfying the UCT in $E$-theory
passes from $A$ to the crossed product $\mathrm{C}^*$-algebra $A\rtimes_\alpha
G$ and the fixed point algebra $A^\alpha$. This extends a similar
result by Gardella for $KK$-theory in the case of unital
$\mathrm{C}^*$-algebras,
but with a shorter and less technical proof. For circle actions
on separable, unital $\mathrm{C}^*$-algebras with the continuous Rokhlin
property, we establish a connection between the $E$-theory equivalence
class of $A$ and that of its fixed point algebra $A^\alpha$.
Keywords:Rokhlin property, UCT, KK-theory, E-theory, circle actions Categories:46L55, 19K35 |
3. CMB 2014 (vol 58 pp. 51)
Spectral Flows of Dilations of Fredholm Operators Given an essentially unitary contraction and an arbitrary unitary
dilation of it, there is a naturally associated spectral flow which is
shown to be equal to the index of the operator. This result is
interpreted in terms of the $K$-theory of an associated mapping
cone. It is then extended to connect $\mathbb{Z}_2$ indices of odd symmetric
Fredholm operators to a $\mathbb{Z}_2$-valued spectral flow.
Keywords:spectral flow, Fredholm operators, Z2 indices Categories:19K56, 46L80 |
4. CMB 2014 (vol 57 pp. 853)
On the Bound of the $\mathrm{C}^*$ Exponential Length Let $X$ be a compact Hausdorff space. In this paper, we give an
example to show that there is $u\in \mathrm{C}(X)\otimes \mathrm{M}_n$
with $\det (u(x))=1$ for all $x\in X$ and $u\sim_h 1$ such that the
$\mathrm{C}^*$ exponential length of $u$
(denoted by $cel(u)$) can not be controlled by
$\pi$. Moreover, in simple inductive limit $\mathrm{C}^*$-algebras,
similar examples also exist.
Keywords:exponential length Category:46L05 |
5. CMB 2014 (vol 58 pp. 207)
Exact and Approximate Operator Parallelism Extending the notion of parallelism we introduce the concept of
approximate parallelism in normed spaces and then substantially
restrict ourselves to the setting of Hilbert space operators endowed
with the operator norm. We present several characterizations of the
exact and approximate operator parallelism in the algebra
$\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a
Hilbert space $\mathscr{H}$. Among other things, we investigate the
relationship between approximate parallelism and norm of inner
derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the
parallel elements of a $C^*$-algebra by using states. Finally we
utilize the linking algebra to give some equivalence assertions
regarding parallel elements in a Hilbert $C^*$-module.
Keywords:$C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-module Categories:47A30, 46L05, 46L08, 47B47, 15A60 |
6. CMB 2014 (vol 57 pp. 708)
Strong Asymptotic Freeness for Free Orthogonal Quantum Groups It is known that the normalized standard generators of the free
orthogonal quantum group $O_N^+$ converge in distribution to a free
semicircular system as $N \to \infty$. In this note, we
substantially improve this convergence result by proving that, in
addition to distributional convergence, the operator norm of any
non-commutative polynomial in the normalized standard generators of
$O_N^+$ converges as $N \to \infty$ to the operator norm of the
corresponding non-commutative polynomial in a standard free
semicircular system. Analogous strong convergence results are obtained
for the generators of free unitary quantum groups. As applications of
these results, we obtain a matrix-coefficient version of our strong
convergence theorem, and we recover a well known $L^2$-$L^\infty$ norm
equivalence for non-commutative polynomials in free semicircular
systems.
Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay Categories:46L54, 20G42, 46L65 |
7. CMB 2014 (vol 58 pp. 110)
Property T and Amenable Transformation Group $C^*$-algebras It is well known that a discrete group which is both amenable and
has Kazhdan's Property T must be finite. In this note we generalize
the above statement to the case of transformation groups. We show
that if $G$ is a discrete amenable group acting on a compact
Hausdorff space $X$, then the transformation group $C^*$-algebra
$C^*(X, G)$ has Property T if and only if both $X$ and $G$ are finite. Our
approach does not rely on the use of tracial states on $C^*(X, G)$.
Keywords:Property T, $C^*$-algebras, transformation group, amenable Categories:46L55, 46L05 |
8. CMB 2013 (vol 57 pp. 546)
Compact Operators in Regular LCQ Groups We show that a regular locally compact quantum group $\mathbb{G}$ is discrete
if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains non-zero compact operators on
$\mathcal{L}^{2}(\mathbb{G})$.
As a corollary we classify all discrete quantum groups among
regular locally compact quantum groups $\mathbb{G}$ where
$\mathcal{L}^{1}(\mathbb{G})$ has the Radon--Nikodym property.
Keywords:locally compact quantum groups, regularity, compact operators Category:46L89 |
9. CMB 2012 (vol 57 pp. 424)
A Note on Amenability of Locally Compact Quantum Groups In this short note we introduce a notion called ``quantum injectivity''
of locally compact quantum groups, and prove that it is equivalent
to amenability of the dual. Particularly, this provides a new characterization
of amenability of locally compact groups.
Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivity Categories:20G42, 22D25, 46L89 |
10. CMB 2012 (vol 57 pp. 90)
Compact Subsets of the Glimm Space of a $C^*$-algebra If $A$ is a $\sigma$-unital $C^*$-algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete
regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists
$\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$. This extends
a result of J. Dauns
to all $\sigma$-unital $C^*$-algebras. However, there are a $C^*$-algebra $A$
and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$, $a\in A$ and $\alpha \gt 0$.
Keywords:primitive ideal space, complete regularization Category:46L05 |
11. CMB 2012 (vol 57 pp. 166)
On Minimal and Maximal $p$-operator Space Structures We show that for $p$-operator spaces, there are natural notions of minimal and maximal
structures. These are useful for dealing with tensor products.
Keywords:$p$-operator space, min space, max space Categories:46L07, 47L25, 46G10 |
12. CMB 2012 (vol 56 pp. 870)
Note on Kasparov Product of $C^*$-algebra Extensions Using the Dadarlat isomorphism, we give a characterization for the
Kasparov product of $C^*$-algebra extensions. A certain relation
between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when
$B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and
$KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general.
Keywords:extension, Kasparov product, $KK$-group Category:46L80 |
13. CMB 2012 (vol 56 pp. 630)
Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres In this paper, we give a different proof of the fact that the odd dimensional
quantum spheres are groupoid $C^{*}$-algebras. We show that the $C^{*}$-algebra
$C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial
isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the
inverse semigroup $T$ by Exel is exactly the same as the groupoid
considered by Sheu.
Keywords:inverse semigroups, groupoids, odd dimensional quantum spheres Categories:46L99, 20M18 |
14. CMB 2011 (vol 56 pp. 337)
Certain Properties of $K_0$-monoids Preserved by Tracial Approximation We show that the following $K_0$-monoid properties of $C^*$-algebras
in the class $\Omega$ are inherited by simple unital $C^*$-algebras in
the class $TA\Omega$: (1) weak comparability, (2) strictly
unperforated, (3) strictly cancellative.
Keywords:$C^*$-algebra, tracial approximation, $K_0$-monoid Categories:46L05, 46L80, 46L35 |
15. CMB 2011 (vol 56 pp. 136)
On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type Product type equivalence relations are hyperfinite measured
equivalence relations, which, up to orbit equivalence, are generated
by product type odometer actions. We give a concrete example of a
hyperfinite equivalence relation of non-product type, which is the
tail equivalence on a Bratteli diagram.
In order to show that the equivalence relation constructed is not of
product type we will use a criterion called property A. This
property, introduced by Krieger for non-singular transformations, is
defined directly for hyperfinite equivalence relations in this paper.
Keywords:property A, hyperfinite equivalence relation, non-product type Categories:37A20, 37A35, 46L10 |
16. CMB 2011 (vol 54 pp. 654)
Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm |
Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)
Keywords:amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotent Categories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25 |
17. CMB 2011 (vol 54 pp. 385)
Irreducible Representations of Inner Quasidiagonal $C^*$-Algebras It is shown that a separable $C^*$-algebra is inner quasidiagonal if and
only if it has a separating family of quasidiagonal irreducible
representations. As a consequence, a separable $C^*$-algebra is a strong
NF algebra if and only if it is nuclear and has a separating family of
quasidiagonal irreducible representations.
We also obtain some permanence properties of the class of inner
quasidiagonal $C^*$-algebras.
Category:46L05 |
18. CMB 2011 (vol 55 pp. 260)
A Note on the Antipode for Algebraic Quantum Groups Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.
In this note, we show that this formula can be proved for any regular multiplier Hopf
algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a
finite-dimensional Hopf algebra, but also that of any
Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that
the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the
theory of regular multiplier Hopf algebras with integrals.
We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.
Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode Categories:16W30, 46L65 |
19. CMB 2011 (vol 55 pp. 339)
From Matrix to Operator Inequalities We generalize LÃ¶wner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition of $C^{*}$-relations
being residually finite dimensional.
Our main result is a meta-theorem about theorems involving relations
on bounded operators. If we can show there are residually finite dimensional
relations involved and verify a technical condition, then such a
theorem will follow from its restriction to matrices.
Applications are shown regarding norms of exponentials, the norms
of commutators, and "positive" noncommutative $*$-polynomials.
Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional Categories:46L05, 47B99 |
20. CMB 2011 (vol 55 pp. 73)
Classification of Inductive Limits of Outer Actions of ${\mathbb R}$ on Approximate Circle Algebras In this paper we present a classification,
up to equivariant isomorphism, of $C^*$-dynamical systems $(A,{\mathbb R},\alpha )$
arising as inductive limits of directed systems
$\{ (A_n,{\mathbb R},\alpha_n),\varphi_{nm}\}$, where each $A_n$
is a finite direct sum of matrix algebras over the continuous
functions on the unit circle, and the $\alpha_n$s are outer actions
generated by rotation of the spectrum.
Keywords:classification, $C^*$-dynamical system Categories:46L57, 46L35 |
21. CMB 2011 (vol 54 pp. 593)
Stability of Real $C^*$-Algebras We will give a characterization of stable real $C^*$-algebras
analogous to the one given for complex $C^*$-algebras by Hjelmborg
and RÃ¸rdam. Using this result, we will prove
that any real $C^*$-algebra satisfying the corona factorization
property is stable if and only if its complexification is stable.
Real $C^*$-algebras satisfying the corona factorization property
include AF-algebras and purely infinite $C^*$-algebras. We will also
provide an example of a simple unstable $C^*$-algebra, the
complexification of which is stable.
Keywords:stability, real C*-algebras Category:46L05 |
22. CMB 2011 (vol 54 pp. 347)
The Haar System in the Preduals of Hyperfinite Factors We shall present examples of Schauder bases in the preduals to the
hyperfinite factors of types~$\hbox{II}_1$, $\hbox{II}_\infty$,
$\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite
(respectively, purely infinite) setting, these systems form Schauder bases
in any associated separable symmetric space of measurable operators
(respectively, in any non-commutative $L^p$-space).
Category:46L52 |
23. CMB 2010 (vol 54 pp. 82)
Lefschetz Numbers for $C^*$-Algebras
Using Poincar\'e duality, we obtain a formula of Lefschetz type
that computes the Lefschetz number of an endomorphism of a separable
nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth
theorem. (The Lefschetz number of an endomorphism is the graded trace
of the induced map on $\textrm{K}$-theory tensored with $\mathbb{C}$, as in the
classical case.) We then examine endomorphisms of Cuntz--Krieger
algebras $O_A$. An endomorphism has an invariant, which is a
permutation of an infinite set, and the contracting and expanding
behavior of this permutation describes the Lefschetz number of the
endomorphism. Using this description, we derive a closed polynomial
formula for the Lefschetz number depending on the matrix $A$ and the
presentation of the endomorphism.
Categories:19K35, 46L80 |
24. CMB 2010 (vol 54 pp. 68)
Non-splitting in Kirchberg's Ideal-related $KK$-Theory
A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's
ideal-related $KK$-theory in the fundamental case of a
$C^*$-algebra with one
specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain
conditions. Employing certain $K$-theoretical information derivable
from the given operator algebras using a method introduced here, we shall
demonstrate that Bonkat's UCT does not split in general. Related
methods lead to information on the complexity of the $K$-theory which
must be used to
classify $*$-isomorphisms for purely infinite $C^*$-algebras with
one non-trivial ideal.
Keywords:KK-theory, UCT Category:46L35 |
25. CMB 2010 (vol 53 pp. 587)
Hulls of Ring Extensions We investigate the behavior of the quasi-Baer and the
right FI-extending right ring hulls under various ring extensions
including group ring extensions, full and triangular matrix ring
extensions, and infinite matrix ring extensions. As a consequence,
we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are
Morita equivalent, then so are the quasi-Baer right ring hulls
$\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of
$R$ and $S$, respectively. As an application, we prove that if
unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings,
then the bounded central closure of $A$ and that of $B$ are
strongly Morita equivalent as $C^*$-algebras. Our results show
that the quasi-Baer property is always preserved by infinite
matrix rings, unlike the Baer property. Moreover, we give an
affirmative answer to an open question of Goel and Jain for the
commutative group ring $A[G]$ of a torsion-free Abelian group $G$
over a commutative semiprime quasi-continuous ring $A$. Examples
that illustrate and delimit the results of this paper are provided.
Keywords:(FI-)extending, Morita equivalent, ring of quotients, essential overring, (quasi-)Baer ring, ring hull, u.p.-monoid, $C^*$-algebra Categories:16N60, 16D90, 16S99, 16S50, 46L05 |