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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] * )

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1. CJM Online first

Daws, Matthew
Categorical aspects of quantum groups: multipliers and intrinsic groups
We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group $\mathbb G$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal $C^*$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal $C^*$-algebra level, and that then the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal $C^*$-algebra picture, and then, again, show how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the ``maximal classical'' quantum subgroup of a locally compact quantum group, show that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliser
Categories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25

2. CJM Online first

Klep, Igor; Špenko, Špela
Free function theory through matrix invariants
This paper concerns free function theory. Free maps are free analogs of analytic functions in several complex variables, and are defined in terms of freely noncommuting variables. A function of $g$ noncommuting variables is a function on $g$-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps \emph{with involution} -- free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involution-free counterparts.

Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials
Categories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20

3. CJM 2015 (vol 67 pp. 1290)

Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul
On Two-faced Families of Non-commutative Random Variables
We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

Keywords:free probability, operator algebras, bi-free

4. CJM Online first

Phillips, John; Raeburn, Iain
Centre-valued Index for Toeplitz Operators with Noncommuting Symbols
We formulate and prove a ``winding number'' index theorem for certain ``Toeplitz'' operators in the same spirit as Gohberg-Krein, Lesch and others. The ``number'' is replaced by a self-adjoint operator in a subalgebra $Z\subseteq Z(A)$ of a unital $C^*$-algebra, $A$. We assume a faithful $Z$-valued trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the infinitesimal generator of $\alpha$ and $u$ is invertible in $\operatorname{dom}(\delta)$ then the ``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{-1})\in Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$ to a von Neumann setting $(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$ and $\mathfrak{Z}=Z^{\prime\prime}.$ Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf R}$ corresponding to the non-negative spectrum of the generator of $\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and $\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$ is the embedding then we define for $u\in A^{-1}$, $T_u=P\tilde\pi(u) P$ and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued index of $T_u$ is the negative of the winding operator. In outline the proof follows the proof of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by $\mathfrak{Z}$ in the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development of a $\mathfrak{Z}$-Hilbert Algebra theory. We show that certain of these Fredholm operators fiber as a ``section'' of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

Keywords:index ,Toeplitz operator
Categories:46L55, 19K56, 46L80

5. CJM Online first

Skalski, Adam; Sołtan, Piotr
Quantum families of invertible maps and related problems
The notion of families of quantum invertible maps (C$^*$-algebra homomorphisms satisfying Podleś' condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular Wang's quantum automorphism groups are shown to be universal with respect to quantum families of invertible maps. Further the construction of the Hopf image of Banica and Bichon is phrased in the purely analytic language and employed to define the quantum subgroup generated by a family of quantum subgroups or more generally a family of quantum invertible maps.

Keywords:quantum families of invertible maps, Hopf image, universal quantum group
Categories:46L89, 46L65

6. CJM 2015 (vol 67 pp. 990)

Amini, Massoud; Elliott, George A.; Golestani, Nasser
The Category of Bratteli Diagrams
A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli's notion of equivalence, we obtain in particular a functorial formulation of Bratteli's classification of AF algebras (and at the same time, of Glimm's classification of UHF~algebras). It is shown that the three approaches to classification of AF~algebras, namely, through Bratteli diagrams, K-theory, and abstract classifying categories, are essentially the same from a categorical point of view.

Keywords:C$^{*}$-algebra, category, functor, AF algebra, dimension group, Bratteli diagram
Categories:46L05, 46L35, 46M15

7. CJM Online first

Runde, Volker; Viselter, Ami
On positive definiteness over locally compact quantum groups
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation property
Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89

8. CJM Online first

Izumi, Masaki; Morrison, Scott; Penneys, David
Quotients of $A_2 * T_2$
We study unitary quotients of the free product unitary pivotal category $A_2*T_2$. We show that such quotients are parametrized by an integer $n\geq 1$ and an $2n$-th root of unity $\omega$. We show that for $n=1,2,3$, there is exactly one quotient and $\omega=1$. For $4\leq n\leq 10$, we show that there are no such quotients. Our methods also apply to quotients of $T_2*T_2$, where we have a similar result. The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of $A_2 * T_2$ and $T_2 * T_2$, we anticipate that our technique can be extended to a general method for proving nonexistence of planar algebras with a specified principal graph. During the preparation of this manuscript, we learnt of Liu's independent result on composites of $A_3$ and $A_4$ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch-Haagerup showed that the principal graph of a composite of $A_3$ and $A_4$ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for $n\geq 4$. This is an abridged version of arxiv:1308.5723.

Keywords:pivotal category, free product, quotient, subfactor, intermediate subfactor

9. CJM 2015 (vol 67 pp. 481)

an Huef, Astrid; Archbold, Robert John
The C*-algebras of Compact Transformation Groups
We investigate the representation theory of the crossed-product $C^*$-algebra associated to a compact group $G$ acting on a locally compact space $X$ when the stability subgroups vary discontinuously. Our main result applies when $G$ has a principal stability subgroup or $X$ is locally of finite $G$-orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation $V$ of a stability subgroup is obtained by restricting $V$ to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of $V$. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup, the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the $C^*$-algebra of the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses the classical branching theorem for the special orthogonal group $\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n-1)$. Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Keywords:compact transformation group, proper action, spectrum of a C*-algebra, multiplicity of a representation, crossed-product C*-algebra, continuous-trace C*-algebra, Fell algebra
Categories:46L05, 46L55

10. CJM 2015 (vol 67 pp. 870)

Lin, Huaxin
Minimal Dynamical Systems on Connected Odd Dimensional Spaces
Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one. Let $\Omega$ be a connected compact metric space with finite covering dimension and with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$ Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that $A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on $X\times \Omega,$ where $X$ is the Cantor set.

Keywords:minimal dynamical systems
Categories:46L35, 46L05

11. CJM 2014 (vol 67 pp. 404)

Hua, Jiajie; Lin, Huaxin
Rotation Algebras and the Exel Trace Formula
We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $\|uv-vu\|\lt 2$ and $uvu^*v^*$ commutes with $u$ and $v,$ then the $C^*$-subalgebra $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of some rotation algebra $A_\theta$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^*$-algebra. Let $\theta\in (-1/2, 1/2)$ be a real number. We prove the following: For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following: if $u$ and $v$ are two unitaries in any unital simple $C^*$-algebra $A$ with tracial rank zero such that \[ \|uv-e^{2\pi i\theta}vu\|\lt \delta \text{ and } {1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta, \] for all tracial state $\tau$ of $A,$ then there exists a pair of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$ such that \[ \tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\, \|u-\tilde{u}\|\lt \epsilon \text{ and } \|v-\tilde{v}\|\lt \epsilon. \]

Keywords:rotation algebras, Exel trace formula

12. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
$K$-theory of Furstenberg Transformation Group $C^*$-algebras
The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism
Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

13. CJM 2013 (vol 66 pp. 596)

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren
The Ordered $K$-theory of a Full Extension
Let $\mathfrak{A}$ be a $C^{*}$-algebra with real rank zero which has the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the corona factorization property. We prove that $ 0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0 $ is a full extension if and only if the extension is stenotic and $K$-lexicographic. {As an immediate application, we extend the classification result for graph $C^*$-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West and the first author, our result may also be used to give a purely $K$-theoretical description of when an essential extension of two simple and stable graph $C^*$-algebras is again a graph $C^*$-algebra.}

Keywords:classification, extensions, graph algebras
Categories:46L80, 46L35, 46L05

14. CJM 2013 (vol 65 pp. 783)

Garcés, Jorge J.; Peralta, Antonio M.
Generalised Triple Homomorphisms and Derivations
We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB$^*$-triples is automatically continuous. When particularised to C$^*$-algebras, we rediscover one of the main theorems established by B.E. Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module, proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$ is automatically continuous.

Keywords:generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$-algebra, JB$^*$-triple
Categories:46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49

15. CJM 2013 (vol 65 pp. 1073)

Kalantar, Mehrdad; Neufang, Matthias
From Quantum Groups to Groups
In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign, to each locally compact quantum group $\mathbb{G}$, a locally compact group $\tilde {\mathbb{G}}$ which is the quantum version of point-masses, and is an invariant for the latter. We show that ``quantum point-masses" can be identified with several other locally compact groups that can be naturally assigned to the quantum group $\mathbb{G}$. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of $\mathbb{G}$ are encoded by $\tilde {\mathbb{G}}$: the latter, despite being a simpler object, can carry very important information about $\mathbb{G}$.

Keywords:locally compact quantum group, locally compact group, von Neumann algebra

16. CJM 2012 (vol 65 pp. 863)

Josuat-Vergès, Matthieu
Cumulants of the $q$-semicircular Law, Tutte Polynomials, and Heaps
The $q$-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings where $q$ follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case $q=0$ of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier-Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

Keywords:moments, cumulants, matchings, Tutte polynomials, heaps
Categories:05A18, 05C31, 46L54

17. CJM 2012 (vol 65 pp. 481)

Ara, Pere; Dykema, Kenneth J.; Rørdam, Mikael
Correction of Proofs in "Purely Infinite Simple $C^*$-algebras Arising from Free Product Constructions'' and a Subsequent Paper
The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple $C^*$-algebras arising from free product constructions}, Canad. J. Math. 50 (1998), 323--341 and of Theorem 3.1 of K. J. Dykema, Purely infinite simple $C^*$-algebras arising from free product constructions, II, Math. Scand. 90 (2002), 73--86 are corrected.

Keywords:C*-algebras, purely infinite

18. CJM 2012 (vol 65 pp. 52)

Christensen, Erik; Sinclair, Allan M.; Smith, Roger R.; White, Stuart
C$^*$-algebras Nearly Contained in Type $\mathrm{I}$ Algebras
In this paper we consider near inclusions $A\subseteq_\gamma B$ of C$^*$-algebras. We show that if $B$ is a separable type $\mathrm{I}$ C$^*$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\mathrm{I}$ and use this to obtain an embedding of $A$ into $B$.

Keywords:C$^*$-algebras, near inclusions, perturbations, type I C$^*$-algebras, similarity problem

19. CJM 2012 (vol 65 pp. 485)

Bice, Tristan Matthew
Filters in C*-Algebras
In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison-Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Lastly we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally we show that consistently all towers on the natural numbers remain towers under this embedding.

Keywords:C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towers
Categories:46L03, 03E35

20. 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 algebra
Categories:46L05, 46L80

21. 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

22. 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 process
Categories:46L53, 60B20, 14L24

23. 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 function
Categories:46L05, 46L08, 46L35

24. 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 density

25. 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 interpolation
Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52
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