Completeness in the Mackey topology

Bonet and Cascales [Non-complete Mackey topologies on Banach spaces, Bulletin of the Australian Mathematical Society, 81, 3 (2010), 409-413], answering a question of M. Kunze and W. Arendt, gave an example of a norming norm-closed subspace N of the dual of a Banach space X such that mu(X, N) is not complete,where mu(X, N) denotes the Mackey topology associated with the dual pair aEuroX, NaEuro parts per thousand. We prove in this note that we can decide on the completeness or incompleteness of topologies of this form in a quite general context, thus providing large classes of counterexamples to the aforesaid question. Moreover, our examples use subspaces N of X* that contain a predual P of X (if exists), showing that the phenomenon of noncompleteness that Kunze and Arendt were looking for is not only relatively common but illustrated by "well-located" subspaces of the dual. We discuss also the situation for a typical Banach space without a predual-the space c (0)-and for the James space J.The first author is supported in part by MICINN and FEDER (project no. MTM2008-05396), by Fundacion Seneca (project no. 08848/PI/08), by Generalitat Valenciana (GV/2010/036), and by Universitat Politecnica de Valencia (project no. PAID-06-09-2829). The second author is supported in part by MICINN project no. MTM2011-22417, by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project no. PAID-06-09-2829).Guirao Sánchez, AJ.; Montesinos Santalucia, V. (2015). Completeness in the Mackey topology. Functional Analysis and Its Applications. 49(2):97-105. https://doi.org/10.1007/s10688-015-0091-2S97105492J. Bonet and B. Cascales, “Non-complete Mackey topologies on Banach spaces,” Bull. Aust. Math. Soc., 81:3 (2010), 409–413.M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Math., Springer-Verlag, New York, 2011.P. Pérez-Carreras and J. Bonet, Barreled Locally Convex Spaces, North-Holland Mathematical Studies, vol. 131, North-Holland, Amsterdam, 1987.P. Civin and B. Yood, “Quasi-reflexive spaces,” Proc. Amer. Math. Soc., 8:5 (1957), 906–911.J. Diestel, Sequences and Series in Banach Spaces, Graduate Text in Math., vol. 92, Springer-Verlag, New York, 1984.K. Floret, Weakly Compact Sets, Lecture Notes in Math., vol. 801, Springer-Verlag, Berlin, 1980.G. Godefroy, “Boundaries of convex sets and interpolation sets,” Math. Ann., 277:2 (1987), 173–184.R. C. James, “On nonreflexive Banach space isometric with its second conjugate,” Proc. Nat. Acad. Sci. USA, 37 (1951), 174–177.G. Köthe, Topological Vector Spaces I, Springer-Verlag, New York, 1969

Further phenotypic characterization of the primitive lineage− CD34+CD38−CD90+CD45RA− hematopoietic stem cell/progenitor cell sub-population isolated from cord blood, mobilized peripheral blood and patients with chronic myelogenous leukemia

The most primitive hematopoietic stem cell (HSC)/progenitor cell (PC) population reported to date is characterized as being Lin−CD34+CD38−CD90+CD45R. We have a long-standing interest in comparing the characteristics of hematopoietic progenitor cell populations enriched from normal subjects and patients with chronic myelogenous leukemia (CML). In order to investigate further purification of HSCs and for potential targetable differences between the very primitive normal and CML stem/PCs, we have phenotypically compared the normal and CML Lin−CD34+CD38−CD90+CD45RA− HSC/PC populations. The additional antigens analyzed were HLA-DR, the receptor tyrosine kinases c-kit and Tie2, the interleukin-3 cytokine receptor, CD33 and the activation antigen CD69, the latter of which was recently reported to be selectively elevated in cell lines expressing the Bcr-Abl tyrosine kinase. Notably, we found a strikingly low percentage of cells from the HSC/PC sub-population isolated from CML patients that were found to express the c-kit receptor (<1%) compared with the percentages of HSC/PCs expressing the c-kitR isolated from umbilical cord blood (50%) and mobilized peripheral blood (10%). Surprisingly, Tie2 receptor expression within the HSC/PC subset was extremely low from both normal and CML samples. Using in vivo transplantation studies, we provide evidence that HLA-DR, c-kitR, Tie2 and IL-3R may not be suitable markers for further partitioning of HSCs from the Lin−CD34+CD38−CD90+CD45RA− sub-population