Abstract. We prove that faithful traces on separable and nuclear C * -algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C * -algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C * -algebras.

We introduce the concept of finitely coloured equivalence for unital * -homomorphisms between C * -algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *homomorphisms from separable, unital, nuclear C * -algebras into ultrapowers of simple, unital, nuclear, Z-stable C * -algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data.As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C * -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C * -algebras with finite nuclear dimension.

BackgroundPax7+ satellite cells are required for skeletal muscle fiber growth during post-natal development in mice. Satellite cell-mediated myonuclear accretion also appears to persist into early adulthood. Given the important role of satellite cells during muscle development, we hypothesized that the necessity of satellite cells for adaptation to an imposed hypertrophic stimulus depends on maturational age.MethodsPax7CreER-R26RDTA mice were treated for 5 days with vehicle (satellite cell-replete, SC+) or tamoxifen (satellite cell-depleted, SC-) at 2 months (young) and 4 months (mature) of age. Following a 2-week washout, mice were subjected to sham surgery or 10 day synergist ablation overload of the plantaris (n = 6–9 per group). The surgical approach minimized regeneration, de novo fiber formation, and fiber splitting while promoting muscle fiber growth. Satellite cell density (Pax7+ cells/fiber), embryonic myosin heavy chain expression (eMyHC), and muscle fiber cross sectional area (CSA) were evaluated via immunohistochemistry. Myonuclei (myonuclei/100 mm) were counted on isolated single muscle fibers.ResultsTamoxifen treatment depleted satellite cells by ≥90% and prevented myonuclear accretion with overload in young and mature mice (p < 0.05). Satellite cells did not recover in SC- mice after overload. Average muscle fiber CSA increased ~20% in young SC+ (p = 0.07), mature SC+ (p < 0.05), and mature SC- mice (p < 0.05). In contrast, muscle fiber hypertrophy was prevented in young SC- mice. Muscle fiber number increased only in mature mice after overload (p < 0.05), and eMyHC expression was variable, specifically in mature SC+ mice.ConclusionsReliance on satellite cells for overload-induced hypertrophy is dependent on maturational age, and global responses to overload differ in young versus mature mice.Electronic supplementary materialThe online version of this article (doi:10.1186/s13395-017-0132-z) contains supplementary material, which is available to authorized users.

Recent loss-of-function studies show that satellite cell depletion does not promote sarcopenia or unloading-induced atrophy, and does not prevent regrowth. Although overload-induced muscle fiber hypertrophy is normally associated with satellite cell-mediated myonuclear accretion, hypertrophic adaptation proceeds in the absence of satellite cells in fully grown adult mice, but not in young growing mice. Emerging evidence also indicates that satellite cells play an important role in remodeling the extracellular matrix during hypertrophy.