A pro-C∗-algebra is a (projective) limit of C∗-algebras in the category of topological ∗algebras. From the perspective of non-commutative geometry, pro-C∗-algebras can be seen as non-commutative k-spaces. An element of a pro-C∗-algebra is bounded if there is a uniform bound for the norm of its images under any continuous ∗-homomorphism into a C∗-algebra. The ∗-subalgebra consisting of the bound...

Phosphoinositide-dependent kinase-1 (PDK1) controls the activation of a subset of AGC kinases. Using a conditional knockout of PDK1 in haematopoietic cells, we demonstrate that PDK1 is essential for B cell development. B-cell progenitors lacking PDK1 arrested at the transition of pro-B to pre-B cells, due to a cell autonomous defect. Loss of PDK1 decreased the expression of the IgH chain in pro...