Search: MSC category 46L ( Selfadjoint operator algebras ($C^$-algebras, von Neumann ($W^*$-) algebras, etc.) [See also 22D25, 47Lxx] *$-algebras, von Neumann ($W^*$-) algebras, etc.) [See also 22D25, 47Lxx] * )  Expand all Collapse all Results 26 - 50 of 84 26. CJM 2011 (vol 64 pp. 755) Brown, Lawrence G.; Lee, Hyun Ho  Homotopy Classification of Projections in the Corona Algebra of a Non-simple$C^*$-algebra We study projections in the corona algebra of$C(X)\otimes K$, where K is the$C^*$-algebra of compact operators on a separable infinite dimensional Hilbert space and$X=[0,1],[0,\infty),(-\infty,\infty)$, or$[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in$K_0$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct. Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebraCategories:46L05, 46L80 27. CJM 2011 (vol 64 pp. 544) Li, Zhiqiang  On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops A K-theoretic classification is given of the simple inductive limits of finite direct sums of the type I$C^*$-algebras known as splitting interval algebras with dimension drops. (These are the subhomogeneous$C^*$-algebras, each having spectrum a finite union of points and an open interval, and torsion$\textrm{K}_1$-group.) Categories:46L05, 46L35 28. CJM 2011 (vol 64 pp. 805) Chapon, François; Defosseux, Manon  Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion. Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor processCategories:46L53, 60B20, 14L24 29. CJM 2011 (vol 64 pp. 573) Nawata, Norio  Fundamental Group of Simple$C^*$-algebras with Unique Trace III We introduce the fundamental group${\mathcal F}(A)$of a simple$\sigma$-unital$C^*$-algebra$A$with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple$C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless$C^*$-algebras. We show that there exist separable stably projectionless$C^*$-algebras such that their fundamental groups are equal to$\mathbb{R}_+^\times$by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian. Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension functionCategories:46L05, 46L08, 46L35 30. CJM 2011 (vol 64 pp. 455) Sherman, David  On Cardinal Invariants and Generators for von Neumann Algebras We demonstrate how most common cardinal invariants associated with a von Neumann algebra$\mathcal M$can be computed from the decomposability number,$\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set,$\operatorname{gen}(\mathcal M)$. Applications include the equivalence of the well-known generator problem, Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, Is every countably-generated von Neumann algebra singly-generated?" and Is the$\operatorname{gen}$invariant monotone?" Modulo the generator problem, we determine the range of the invariant$\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality$\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$. Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation densityCategory:46L10 31. CJM 2011 (vol 63 pp. 798) Daws, Matthew  Representing Multipliers of the Fourier Algebra on Non-Commutative$L^p$Spaces We show that the multiplier algebra of the Fourier algebra on a locally compact group$G$can be isometrically represented on a direct sum on non-commutative$L^p$spaces associated with the right von Neumann algebra of$G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative$L^p$spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative$L^p$spaces, say$A_p(\widehat G)$. It is shown that$A_2(\widehat G)$is isometric to$L^1(G)$, generalising the abelian situation. Keywords:multiplier, Fourier algebra, non-commutative$L^p$space, complex interpolationCategories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 32. CJM 2011 (vol 63 pp. 551) Hadwin, Don; Li, Qihui; Shen, Junhao  Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras In the paper, we introduce a new concept, topological orbit dimension of an$n$-tuple of elements in a unital C$^{\ast}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear C$^{\ast}$-algebra is less than or equal to$1$. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital C$^{\ast}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital MF algebras is also an MF algebra. As an application, we obtain that$\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$is not a group. Keywords: topological free entropy dimension, unital C$^{*}$-algebraCategories:46L10, 46L54 33. CJM 2011 (vol 63 pp. 381) Ji, Kui ; Jiang, Chunlan  A Complete Classification of AI Algebras with the Ideal Property Let$A$be an AI algebra; that is,$A$is the$\mbox{C}^{*}$-algebra inductive limit of a sequence $$A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3} \longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,$$ where$A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$,$X^{i}_n$are$[0,1]$,$k_n$, and$[n,i]$are positive integers. Suppose that$A$has the ideal property: each closed two-sided ideal of$A$is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property. Keywords:AI algebras, K-group, tracial state, ideal property, classificationCategories:46L35, 19K14, 46L05, 46L08 34. CJM 2011 (vol 63 pp. 500) Dadarlat, Marius; Elliott, George A.; Niu, Zhuang  One-Parameter Continuous Fields of Kirchberg Algebras. II Parallel to the first two authors' earlier classification of separable, unita one-parameter, continuous fields of Kirchberg algebras with torsion free$\mathrm{K}$-groups supported in one dimension, one-parameter, separable, uni continuous fields of AF-algebras are classified by their ordered$\mathrm{K}_0$-sheaves. Effros-Handelman-Shen type theorems are pr separable unital one-parameter continuous fields of AF-algebras and Kirchberg algebras. Keywords:continuous fields of C$^*$-algebras,$\mathrm{K}_0$-presheaves, Effros--Handeen type theoremCategory:46L35 35. CJM 2010 (vol 63 pp. 222) Wang, Jiun-Chau  Limit Theorems for Additive Conditionally Free Convolution In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution. Keywords:additive c-free convolution, limit theorems, infinitesimal arraysCategories:46L53, 60F05 36. CJM 2010 (vol 63 pp. 3) Banica, T.; Belinschi, S. T.; Capitaine, M.; Collins, B.  Free Bessel Laws We introduce and study a remarkable family of real probability measures$\pi_{st}$that we call free Bessel laws. These are related to the free Poisson law$\pi$via the formulae$\pi_{s1}=\pi^{\boxtimes s}$and${\pi_{1t}=\pi^{\boxplus t}}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups. Keywords:Poisson law, Bessel function, Wishart matrix, quantum groupCategories:46L54, 15A52, 16W30 37. CJM 2010 (vol 62 pp. 845) Samei, Ebrahim; Spronk, Nico; Stokke, Ross  Biflatness and Pseudo-Amenability of Segal Algebras We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra,$L^1(G)$, and the Fourier algebra,$A(G)$, of a locally compact group~$G$. Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebraCategories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07 38. CJM 2010 (vol 62 pp. 889) Xia, Jingbo  Singular Integral Operators and Essential Commutativity on the Sphere Let${\mathcal T}$be the$C^\ast $-algebra generated by the Toeplitz operators$\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$on the Hardy space$H^2(S)$of the unit sphere in$\mathbf{C}^n$. It is well known that${\mathcal T}$is contained in the essential commutant of$\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of$\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$is strictly larger than${\mathcal T}$. Categories:32A55, 46L05, 47L80 39. CJM 2009 (vol 61 pp. 1239) Davidson, Kenneth R.; Yang, Dilian  Periodicity in Rank 2 Graph Algebras Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of$\mathrm{C}^*(\mathbb{F}^+_{\theta})$. The periodic$\mathrm{C}^*$-algebras are characterized, and it is shown that$\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq \mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$where$\mathfrak{A}$is a simple$\mathrm{C}^*$-algebra. Keywords:higher rank graph, aperiodicity condition, simple$\mathrm{C}^*$-algebra, expectationCategories:47L55, 47L30, 47L75, 46L05 40. CJM 2009 (vol 61 pp. 1262) Dong, Z.  On the Local Lifting Properties of Operator Spaces In this paper, we mainly study operator spaces which have the locally lifting property (LLP). The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the LLP. Several homological characterizations of the LLP and weak expectation property are given. We also prove that for any operator space$V$,$V^{**}$has the LLP if and only if$V$has the LLP and$V^{*}$is exact. Keywords:operator space, locally lifting property, strongly locally reflexiveCategory:46L07 41. CJM 2009 (vol 61 pp. 241) Azamov, N. A.; Carey, A. L.; Dodds, P. G.; Sukochev, F. A.  Operator Integrals, Spectral Shift, and Spectral Flow We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general \vNa s. For semifinite \vNa s we give applications to the Fr\'echet differentiation of operator functions that sharpen existing results, and establish the Birman--Solomyak representation of the spectral shift function of M.\,G.\,Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow. Categories:47A56, 47B49, 47A55, 46L51 42. CJM 2008 (vol 60 pp. 975) Boca, Florin P.  An AF Algebra Associated with the Farey Tessellation We associate with the Farey tessellation of the upper half-plane an AF algebra$\AA$encoding the cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of$\AA$. Using the path algebra model for AF algebras we construct, for each$\tau \in \big(0,\frac{1}{4}\big]$, projections$(E_n)$in$\AA$such that$E_n E_{n\pm 1}E_n \leq \tau E_n$. Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20 43. CJM 2008 (vol 60 pp. 703) Toms, Andrew S.; Winter, Wilhelm $\mathcal{Z}$-Stable ASH Algebras The Jiang--Su algebra$\mathcal{Z}$has come to prominence in the classification program for nuclear$C^*$-algebras of late, due primarily to the fact that Elliott's classification conjecture in its strongest form predicts that all simple, separable, and nuclear$C^*$-algebras with unperforated$\mathrm{K}$-theory will absorb$\mathcal{Z}$tensorially, i.e., will be$\mathcal{Z}$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and$\mathcal{Z}$-stable$C^*$-algebras. We prove that virtually all classes of nuclear$C^*$-algebras for which the Elliott conjecture has been confirmed so far consist of$\mathcal{Z}$-stable$C^*$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible$C^*$-algebras are$\mathcal{Z}$-stable. Keywords:nuclear$C^*$-algebras, K-theory, classificationCategories:46L85, 46L35 44. CJM 2008 (vol 60 pp. 189) Lin, Huaxin  Furstenberg Transformations and Approximate Conjugacy Let$\alpha$and$\beta$be two Furstenberg transformations on$2$-torus associated with irrational numbers$\theta_1,\theta_2,$integers$d_1, d_2$and Lipschitz functions$f_1$and$f_2$. It is shown that$\alpha$and$\beta$are approximately conjugate in a measure theoretical sense if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z.$Closely related to the classification of simple amenable \CAs, it is shown that$\af$and$\bt$are approximately$K$-conjugate if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z$and$|d_1|=|d_2|.$This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic. Keywords:Furstenberg transformations, approximate conjugacyCategories:37A55, 46L35 45. CJM 2007 (vol 59 pp. 966) Forrest, Brian E.; Runde, Volker; Spronk, Nico  Operator Amenability of the Fourier Algebra in the$\cb$-Multiplier Norm Let$G$be a locally compact group, and let$A_{\cb}(G)$denote the closure of$A(G)$, the Fourier algebra of$G$, in the space of completely bounded multipliers of$A(G)$. If$G$is a weakly amenable, discrete group such that$\cstar(G)$is residually finite-dimensional, we show that$A_{\cb}(G)$is operator amenable. In particular,$A_{\cb}(\free_2)$is operator amenable even though$\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if$G$is a discrete group such that$A_{\cb}(G)$is operator amenable, a closed ideal of$A(G)$is weakly completely complemented in$A(G)$if and only if it has an approximate identity bounded in the$\cb$-multiplier norm. Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenabilityCategories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 46. CJM 2007 (vol 59 pp. 343) Lin, Huaxin  Weak Semiprojectivity in Purely Infinite Simple$C^*$-Algebras Let$A$be a separable amenable purely infinite simple \CA which satisfies the Universal Coefficient Theorem. We prove that$A$is weakly semiprojective if and only if$K_i(A)$is a countable direct sum of finitely generated groups ($i=0,1$). Therefore, if$A$is such a \CA, for any$\ep>0$and any finite subset${\mathcal F}\subset A$there exist$\dt>0$and a finite subset${\mathcal G}\subset A$satisfying the following: for any contractive positive linear map$L: A\to B$(for any \CA$B$) with$ \|L(ab)-L(a)L(b)\|<\dt$for$a, b\in {\mathcal G}$there exists a homomorphism$h\from A\to B$such that$ \|h(a)-L(a)\|<\ep$for$a\in {\mathcal F}$. Keywords:weakly semiprojective, purely infinite simple$C^*$-algebrasCategories:46L05, 46L80 47. CJM 2006 (vol 58 pp. 1144) Hamana, Masamichi  Partial$*$-Automorphisms, Normalizers, and Submodules in Monotone Complete$C^*$-Algebras For monotone complete$C^*$-algebras$A\subset B$with$A$contained in$B$as a monotone closed$C^*$-subalgebra, the relation$X = AsA$gives a bijection between the set of all monotone closed linear subspaces$X$of$B$such that$AX + XA \subset X$and$XX^* + X^*X \subset A$and a set of certain partial isometries$s$in the normalizer" of$A$in$B$, and similarly for the map$s \mapsto \Ad s$between the latter set and a set of certain partial$*$-automorphisms" of$A$. We introduce natural inverse semigroup structures in the set of such$X$'s and the set of partial$*$-automorphisms of$A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough$B$the homomorphism becomes surjective and all the partial$*$-automorphisms of$A$are realized via partial isometries in$B$. In particular, the inverse semigroup associated with a type${\rm II}_1$von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the$C^*$-algebra version of these results. Categories:46L05, 46L08, 46L40, 20M18 48. CJM 2006 (vol 58 pp. 1268) Sims, Aidan  Gauge-Invariant Ideals in the$C^*$-Algebras of Finitely Aligned Higher-Rank Graphs We produce a complete description of the lattice of gauge-invariant ideals in$C^*(\Lambda)for a finitely alignedk$-graph$\Lambda$. We provide a condition on$\Lambda$under which every ideal is gauge-invariant. We give conditions on$\Lambda$under which$C^*(\Lambda)$satisfies the hypotheses of the Kirchberg--Phillips classification theorem. Keywords:Graphs as categories, graph algebra,$C^*$-algebraCategory:46L05 49. CJM 2006 (vol 58 pp. 768) Hu, Zhiguo; Neufang, Matthias  Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability number of a von Neumann algebra$\m$(denoted by$\dec(\m)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in$\m$. In this paper, we explore the close connection between$\dec(\m)$and the cardinal level of the Mazur property for the predual$\m_*$of$\m$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group$G$as the group algebra$\lone$, the Fourier algebra$A(G)$, the measure algebra$M(G)$, the algebra$\luc^*$, etc. We show that for any of these von Neumann algebras, say$\m$, the cardinal number$\dec(\m)$and a certain cardinal level of the Mazur property of$\m_*$are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of$G$: the compact covering number$\kg$of$G$and the least cardinality$\bg$of an open basis at the identity of$G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra$\ag^{**}$. Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centreCategories:22D05, 43A20, 43A30, 03E55, 46L10 50. CJM 2006 (vol 58 pp. 39) Exel, R.; Vershik, A. $C^*$-Algebras of Irreversible Dynamical Systems We show that certain$C^*\$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed-product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity. Categories:46L55, 37A55