Isometric Representations of Totally Ordered Semigroups

Let S be a subsemigroup of an abelian torsion-free group G. If S is a positive cone of G, then all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic. Proved by Murphy, this statement generalized the well-known theorems of Coburn and Douglas. In this note we prove the reverse. If all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic, then S is a positive cone of G. Also we consider G = Z\times Z and prove that if S induces total order on G, then there exist at least two unitarily not equivalent irreducible isometrical representation of S. And if the order is lexicographical-product order, then all such representations are unitarily equivalent.Comment: February 21, 2012. Kazan, Russi

THE EFFECT OF NEBIVOLOL ON BONE MINERAL DENSITY IN POSTMENOPAUSAL WOMEN WITH MILD HYPERTENSION

Aim. To study the effects of nebivolol on bone mineral density (BMD) in postmenopausal women with mild hypertension (HT) and osteopenia.Material and methods. Postmenopausal women (n=56) aged 50-65 years with mild HT фтв osteopenia were included into the randomized controlled study and divided in two groups (28 patients in each). During 12 months patients of the main group received treatment with nebivolol (5-7.5 mg/day) and patients of the control group received treatment with atenolol (12.5-25 mg/day). Clinical and anthropometric examinations, blood pressure measurements, ECG registrations were performed in all patients initially and after 12 months of treatment. Quantitative estimation of BMD was performed by dual energy X-ray absorptiometry with osteodensitometry DELPHI W manufactured by HOLOGIC company (USA) in the lumbar spine (L1-L4), femoral neck and proximal femur in the anterior-posterior projection. In addition, calcium and bone metabolism indices were determined: ionized calcium, total alkaline phosphatase, C-telopeptide of type I collagen (CTX).Results. Therapy of mild HT with nebivolol during 12 months showed increase in BMD in the spine according to the T-test from -1.7±0.4 SD to -1.4±0.53 SD (p&lt;0.001), while in atenolol group this index decreased from -1.5±0.7 SD to -1.6±0.64 SD (p&lt;0.001). When evaluating T-test of the femoral neck the index changed in the main group from - 1.4±0.44 SD to -1.27±0.5 SD (p=0.015), in the control group - from -1.3±0.64 SD to -1.5±0.65 SD (p=0.0005). In the study group T-test of proximal femur changed from -0.58±0.4 SD to -0.49±0.4 SD (p=0.003), and in the control group - from -0.8±0.84 SD to -0.83±0.93 SD (p=0.3). The dynamics of the BMD due to 12 month therapy in all investigated bone segments distinguished significantly between study and control groups. Nebivolol therapy group showed reduction in CTX level from 0.367±0.16 to 0.294±0.12 ng/ml (p&lt;0.001), whereas the control group showed increase in this parameter from 0.369±0.15 to 0.499±0.18 ng/ml (p&lt;0.001). The dynamics of the CTX levels distinguished significantly between groups (p&lt;0.001).Conclusion. The study demonstrated the positive effect of nebivolol on BMD, whereas atenolol had no effect on bone mass.</p