Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Executive functioning in Alzheimer's disease and vascular dementia

Objective: To compare performance of patients with mild-moderate Alzheimer's disease (AD) and vascular dementia (VaD) on tests of executive functioning and working memory.

Methods: Patients with AD (n = 76) and VaD (n = 46) were recruited from a memory clinic along with dementia free participants (n = 28). They underwent specific tests of working memory from the Cognitive Drug Research (CDR) battery and pen and paper tests of executive function including CLOX 1 & 2, EXIT25 and a test of verbal fluency (COWAT). All patients had a CT brain scan which was independently scored for white matter change/ischaemia.

Results: The AD and VaD groups were significantly impaired on all measures of working memory and executive functioning compared to the disease free group. There were no significant differences between the AD and VaD groups on any measure. Z-scores confirmed the pattern of impairment in executive functioning and working memory was largely equivalent in both patient groups. Small to moderate correlations were seen between the MMSE and the neurocognitive scores in both patient groups and the pattern of correlations was also very similar in both patient groups.

Conclusions: This study demonstrates sizeable executive functioning and working memory impairments in patients with mild-moderate AD and VaD but no significant differences between the disease groups. Copyright (C) 2009 John Wiley & Sons, Ltd.

Operator systems from discrete groups

We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group Fn on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group Z2. We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.

SK1 of graded division algebras

The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.

Unitary SK1 of unitary and graded division algebras

The reduced unitary Whitehead group $\SK$ of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary $\SK$ of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary $\SK$ of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary $\SK$ for much wider classes of division algebras over henselian fields.

An isomorphism between group C*-algebras of ax + b-like groups

We give a necessary and sufficient condition for two ax+b-like groups to have isomorphic C*-algebras. In particular, we show that there are many non-isomorphic ax+b -like Lie groups having isomorphic group C*-algebras.

K-Theory of Azumaya Algebras over Schemes

Let X be a connected, noetherian scheme and A{script} be a sheaf of Azumaya algebras on X, which is a locally free O{script}-module of rank a. We show that the kernel and cokernel of K(X) ? K(A{script}) are torsion groups with exponent a for some m and any i = 0, when X is regular or X is of dimension d with an ample sheaf (in this case m = d + 1). As a consequence, K(X, Z/m) ? K(A{script}, Z/m), for any m relatively prime to a. © 2013 Copyright Taylor and Francis Group, LLC.

Association Between Results of a Gene Expression Signature Assay and Recurrence-Free Interval in Patients With Stage II Colon Cancer in Cancer and Leukemia Group B 9581 (Alliance)

PURPOSE: Conventional staging methods are inadequate to identify patients with stage II colon cancer (CC) who are at high risk of recurrence after surgery with curative intent. ColDx is a gene expression, microarray-based assay shown to be independently prognostic for recurrence-free interval (RFI) and overall survival in CC. The objective of this study was to further validate ColDx using formalin-fixed, paraffin-embedded specimens collected as part of the Alliance phase III trial, C9581.

PATIENTS AND METHODS: C9581 evaluated edrecolomab versus observation in patients with stage II CC and reported no survival benefit. Under an initial case-cohort sampling design, a randomly selected subcohort (RS) comprised 514 patients from 901 eligible patients with available tissue. Forty-nine additional patients with recurrence events were included in the analysis. Final analysis comprised 393 patients: 360 RS (58 events) and 33 non-RS events. Risk status was determined for each patient by ColDx. The Self-Prentice method was used to test the association between the resulting ColDx risk score and RFI adjusting for standard prognostic variables.

RESULTS: Fifty-five percent of patients (216 of 393) were classified as high risk. After adjustment for prognostic variables that included mismatch repair (MMR) deficiency, ColDx high-risk patients exhibited significantly worse RFI (multivariable hazard ratio, 2.13; 95% CI, 1.3 to 3.5; P < .01). Age and MMR status were marginally significant. RFI at 5 years for patients classified as high risk was 82% (95% CI, 79% to 85%), compared with 91% (95% CI, 89% to 93%) for patients classified as low risk.

CONCLUSION: ColDx is associated with RFI in the C9581 subsample in the presence of other prognostic factors, including MMR deficiency. ColDx could be incorporated with the traditional clinical markers of risk to refine patient prognosis.

SK1-like functors for division algebras

We investigate the group valued functor G(D) = D*/F*D' where D is a division algebra with center F and D' the commutator subgroup of D*. We show that G has the most important functorial properties of the reduced Whitehead group SK1. We then establish a fundamental connection between this group, its residue version, and relative value group when D is a Henselian division algebra. The structure of G(D) turns out to carry significant information about the arithmetic of D. Along these lines, we employ G(D) to compute the group SK1(D). As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method.

Reduced K-theory for Azumaya algebras

Abstract In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps Ki(F)?Ki(D), where Ki are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK1. In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.

Neighboring Group-Controlled Hydrolysis: Towards “Designer” Drug Release Biomaterials

To give the first demonstration of neighboring group-controlled drug delivery rates, a series of novel, polymerizable ester drug conjugates was synthesized and fully characterized. The monomers are suitable for copolymerization in biomaterials where control of drug release rate is critical to prophylaxis or obviation of infection. The incorporation of neighboring group moieties differing in nucleophilicity, geometry, and steric bulk in the conjugates allowed the rate of ester hydrolysis, and hence drug liberation, to be rationally and widely controlled. Solutions (2.5 x 10-5 mol dm-3) of ester conjugates of nalidixic acid incorporating pyridyl, amino, and phenyl neighboring groups hydrolyzed according to first-order kinetics, with rate constants between 3.00 ( 0.12 10-5 s -1 (fastest) and 4.50 ( 0.31 10- 6 s-1 (slowest). The hydrolysis was characterized using UV-visible spectroscopy. When copolymerized with poly(methyl methacrylate), free drug was shown to elute from the resulting materials, with the rate of release being controlled by the nature of the conjugate, as in solution. The controlled molecular architecture demonstrated by this system offers an attractive class of drug conjugate for the delivery of drugs from polymeric biomaterials such as bone cements in terms of both sustained, prolonged drug release and minimization of mechanical compromise as a result of release. We consider these results to be the rationale for the development of 'designer' drug release biomaterials, where the rate of required release can be controlled by predetermined molecular architecture.

An analogue of the Magnus problem for associative algebras

We prove an analogue of Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by $n+k$ generators and $k$ relations and has an $n$-element system of generators, then this algebra is a free algebra of rank $n$.

Electroreduction of oxygen in a series of room temperature ionic liquids composed of group 15-centered cations and anions

The electrochemical reduction of oxygen is reported in four room temperature ionic liquids (RTILs) based on quaternary alkyl -onium cations and heavily fluorinated anions in which the central atom is either nitrogen or phosphorus. Data were collected using cyclic voltammetry and potential step chronoamperometry at gold, platinum, and glassy carbon disk electrodes of micrometer dimension under water-free conditions at a controlled temperature. Analysis via fitting, to appropriate theoretical equations was then carried out to obtain kinetic and thermodynamic information pertaining to the electrochemical processes observed. In the quaternary ammonium electrolytes, reduction of oxygen was found to occur reversibly to give stable superoxide, in an analogous manner to that seen in conventional aprotic solvents such as dimethyl sufoxide and acetonitrile. The most significant difference is in the relative rate of diffusion; the diffusion coefficients of oxygen in the RTILs are an order of magnitude lower than in common organic solvents, and for superoxide these values are reduced by a further factor of 10. In the quaternary phosphonium ionic liquids, however, more complex voltammetry is observed, akin to that expected for the reduction of oxygen in acidified organic media. This is shown to be consistent with the occurrence of a proton abstraction reaction between the electrogenerated superoxide and quaternary alkyl phosphonium cations following the initial electron transfer.

Non-trivial stably free modules over crossed products

We consider the class of crossed products of noetherian domains with universal enveloping algebras of Lie algebras. For algebras from this class we give a sufficient condition for the existence of projective non-free modules. This class includes Weyl algebras and universal envelopings of Lie algebras, for which this question, known as noncommutative Serre's problem, was extensively studied before. It turns out that the method of lifting of non-trivial stably free modules from simple Ore extensions can be applied to crossed products after an appropriate choice of filtration. The motivating examples of crossed products are provided by the class of RIT algebras, originating in non-equilibrium physics.