Leibniz Seminorms and Best Approximation from C*-subalgebras

We show that if B is a C*-subalgebra of a C*-algebra A such that B contains a bounded approximate identity for A, and if L is the pull-back to A of the quotient norm on A/B, then L is strongly Leibniz. In connection with this situation we study certain aspects of best approximation of elements of a unital C*-algebra by elements of a unital C*-subalgebra.Comment: 24 pages. Intended for the proceedings of the conference "Operator Algebras and Related Topics". v2: added a corollary to the main theorem, plus several minor improvements v3: much simplified proof of a key lemma, corollary to main theorem added v4: Many minor improvements. Section numbers increased by

Nutraceutical therapies for atherosclerosis

Atherosclerosis is a chronic inflammatory disease affecting large and medium arteries and is considered to be a major underlying cause of cardiovascular disease (CVD). Although the development of pharmacotherapies to treat CVD has contributed to a decline in cardiac mortality in the past few decades, CVD is estimated to be the cause of one-third of deaths globally. Nutraceuticals are natural nutritional compounds that are beneficial for the prevention or treatment of disease and, therefore, are a possible therapeutic avenue for the treatment of atherosclerosis. The purpose of this Review is to highlight potential nutraceuticals for use as antiatherogenic therapies with evidence from in vitro and in vivo studies. Furthermore, the current evidence from observational and randomized clinical studies into the role of nutraceuticals in preventing atherosclerosis in humans will also be discussed