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Search: MSC category 46L05 ( General theory of $C^$-algebras *$-algebras * )

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1. CMB Online first

Moslehian, Mohammad Sal; Zamani, Ali
Characterizations of operator Birkhoff--James orthogonality
In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\mathbb{B}(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{{\mathscr{H}}_0}^\perp} \lt \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if and only if there exists a unit vector $\xi\in {\mathscr{H}}_0$ such that $\|T\|\xi = |T|\xi$ and $S^*T\xi = 0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product $C^*$-modules.

Keywords:Hilbert $C^*$-module, Birkhoff--James orthogonality, strong Birkhoff--James orthogonality, approximate orthogonality
Categories:46L05, 46L08, 46B20

2. CMB 2015 (vol 59 pp. 435)

Yao, Hongliang
On Extensions of Stably Finite C*-algebras (II)
For any $C^*$-algebra $A$ with an approximate unit of projections, there is a smallest ideal $I$ of $A$ such that the quotient $A/I$ is stably finite. In this paper, a sufficient and necessary condition is obtained for an ideal of a $C^*$-algebra with real rank zero is this smallest ideal by $K$-theory.

Keywords:extension, stably finite C*-algebra, index map
Categories:46L05, 46L80

3. CMB 2015 (vol 59 pp. 95)

Gonçalves, Daniel; Li, Hui; Royer, Danilo
Faithful Representations of Graph Algebras via Branching Systems
We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz-Krieger uniqueness theorem for graph algebras by means of branching systems.

Keywords:C*-algebra, graph algebra, Leavitt path algebra, branching system, representation
Categories:46L05, 37A55

4. CMB 2015 (vol 58 pp. 402)

Tikuisis, Aaron Peter; Toms, Andrew
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

5. CMB 2014 (vol 57 pp. 853)

Pan, Qingfei; Wang, Kun
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

6. CMB 2014 (vol 58 pp. 207)

Moslehian, Mohammad Sal; Zamani, Ali
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

7. CMB 2014 (vol 58 pp. 110)

Kamalov, F.
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 2012 (vol 57 pp. 90)

Lazar, Aldo J.
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

9. CMB 2011 (vol 56 pp. 337)

Fan, Qingzhai
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 54 pp. 385)

Blackadar, Bruce; Kirchberg, Eberhard
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.


11. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
From Matrix to Operator Inequalities
We generalize Löwner'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

12. CMB 2011 (vol 54 pp. 593)

Boersema, Jeffrey L.; Ruiz, Efren
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 Rørdam. 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

13. CMB 2010 (vol 53 pp. 587)

Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq
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

14. CMB 2010 (vol 53 pp. 447)

Choi, Yemon
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

15. CMB 2009 (vol 52 pp. 39)

Cimpri\v{c}, Jakob
A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings
We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry
Categories:16W80, 46L05, 46L89, 14P99

16. CMB 2007 (vol 50 pp. 460)

Spielberg, Jack
Weak Semiprojectivity for Purely Infinite $C^*$-Algebras
We prove that a separable, nuclear, purely infinite, simple $C^*$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

Keywords:Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebra
Categories:46L05, 46L80, 22A22

17. CMB 2005 (vol 48 pp. 50)

Elliott, George A.; Gong, Guihua; Li, Liangqing
Injectivity of the Connecting Maps in AH Inductive Limit Systems
Let $A$ be the inductive limit of a system $$A_{1}\xrightarrow{\phi_{1,2}}A_{2} \xrightarrow{\phi_{2,3}} A_{3}\longrightarrow \cd $$ with $A_n = \bigoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is a projection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we will prove that $A$ can be written as another inductive limit $$B_1\xrightarrow{\psi_{1,2}} B_2 \xrightarrow{\psi_{2,3}} B_3\longrightarrow \cd $$ with $B_n = \bigoplus_{i=1}^{s_n} Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is a projection in $M_{\{n,i\}}(C(Y_{n,i}))$, with the extra condition that all the maps $\psi_{n,n+1}$ are \emph{injective}. (The result is trivial if one allows the spaces $Y_{n,i}$ to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras (see \cite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ are graphs is due to the third named author \cite{Li1}.

Categories:46L05, 46L35, 19K14

18. CMB 2003 (vol 46 pp. 575)

Marshall, M.
Optimization of Polynomial Functions
This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case.

Categories:14P10, 46L05, 90C22

19. CMB 2003 (vol 46 pp. 588)

Monteiro, Martha Salerno
Weakly Stable Relations and Inductive Limits of $C^\ast$-algebras
We show that if $\mathcal{A}$ is a class of $C^\ast$-algebras for which the set of formal relations $\mathcal{R}$ is weakly stable, then $\mathcal{R}$ is weakly stable for the class $\mathcal{B}$ that contains $\mathcal{A}$ and all the inductive limits that can be constructed with the $C^\ast$-algebras in $\mathcal{A}$. A set of formal relations $\mathcal{R}$ is said to be {\it weakly stable\/} for a class $\mathcal{C}$ of $C^\ast$-algebras if, in any $C^\ast$-algebra $A\in \mathcal{C}$, close to an approximate representation of the set $\mathcal{R}$ in $A$ there is an exact representation of $\mathcal{R}$ in $A$.


20. CMB 2003 (vol 46 pp. 388)

Lin, Huaxin
Tracially Quasidiagonal Extensions
It is known that a unital simple $C^*$-algebra $A$ with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital $C^*$-algebras with tracial topological rank zero that have real rank other than zero. Let $0\to J\to E\to A\to 0$ be a short exact sequence of $C^*$-algebras. Suppose that $J$ and $A$ have tracial topological rank zero. It is known that $E$ has tracial topological rank zero as a $C^*$-algebra if and only if $E$ is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.

Keywords:tracially quasidiagonal extensions, tracial rank
Categories:46L05, 46L80

21. CMB 2000 (vol 43 pp. 418)

Gong, Guihua; Jiang, Xinhui; Su, Hongbing
Obstructions to $\mathcal{Z}$-Stability for Unital Simple $C^*$-Algebras
Let $\cZ$ be the unital simple nuclear infinite dimensional $C^*$-algebra which has the same Elliott invariant as $\bbC$, introduced in \cite{JS}. A $C^*$-algebra is called $\cZ$-stable if $A \cong A \otimes \cZ$. In this note we give some necessary conditions for a unital simple $C^*$-algebra to be $\cZ$-stable.

Keywords:simple $C^*$-algebra, $\mathcal{Z}$-stability, weak (un)perforation in $K_0$ group, property $\Gamma$, finiteness

22. CMB 2000 (vol 43 pp. 320)

Elliott, George; Fulman, Igor
On Classification of Certain $C^\ast$-Algebras
We consider \cst-algebras which are inductive limits of finite direct sums of copies of $ C([0,1]) \otimes \Otwo$. For such algebras, the lattice of closed two-sided ideals is proved to be a complete invariant.

Categories:46L05, 46L35

23. CMB 2000 (vol 43 pp. 193)

Magajna, Bojan
C$^*$-Convexity and the Numerical Range
If $A$ is a prime C$^*$-algebra, $a \in A$ and $\lambda$ is in the numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^* (a-\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of $W(a)$, the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general C$^*$-algebra) $A$, all normal elements in the weak* closure (the norm closure, respectively) of the C$^*$-convex hull of $a$ are characterized.

Categories:47A12, 46L05, 46L10

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