Expand all Collapse all | Results 1 - 25 of 67 |
1. CMB Online first
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 |
2. CMB Online first
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 |
3. CMB Online first
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 |
4. 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 |
5. 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 |
6. 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 |
7. 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 |
8. 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 |
9. 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 |
10. 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 |
11. 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 |
12. 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 |
13. 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 |
14. 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 |
15. 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 |
16. 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 |
17. 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 |
18. 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 |
19. 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 |
20. 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 |
21. CMB 2010 (vol 53 pp. 550)
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations |
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations
In this paper we propose a new technical tool for analyzing
representations of Hilbert $C^*$-product systems. Using this tool,
we give a new proof that every doubly commuting representation
over $\mathbb{N}^k$ has a regular isometric dilation, and we also
prove sufficient conditions for the existence of a regular
isometric dilation of representations over more general
subsemigroups of $\mathbb R_{+}^k$.
Categories:47A20, 46L08 |
22. CMB 2010 (vol 53 pp. 447)
Injective Convolution Operators on l^{∞}(Γ) are Surjective Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results.
Categories:43A20, 46L05, 43A22 |
23. CMB 2010 (vol 53 pp. 256)
Equivalent Definitions of Infinite Positive Elements in Simple C^{*}-algebras We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple $C^*$-algebras.
Keywords:Infinite positive element, Comparison relation Category:46L99 |
24. CMB 2009 (vol 53 pp. 239)
A Note on the Exactness of Operator Spaces In this paper, we give two characterizations of the exactness of operator spaces.
Keywords:operator space, exactness Category:46L07 |
25. CMB 2009 (vol 53 pp. 37)
$C^*$-Crossed-Products by an Order-Two Automorphism We describe the representation theory of $C^*$-crossed-products of a unital $C^*$-algebra A by the cyclic group of order~2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of~A. We characterize each class in term of the restriction of the representations to the fixed point $C^*$-subalgebra of~A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.
Categories:46L55, 46L80 |