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Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an exam …

I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange. Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there …

Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve co …

I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does …

For a (unital) C$^*$-algebra $A$ with centre $Z(A)$ the bidual $A^{**}$ is a von Neumann algebra with centre $Z(A^{**})$ for which I believe that $Z(A)^{**} \subset Z(A^{**})$ holds by extending the p …

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real num …

Let $X$ be a Banach space where the closed unit ball equals the convex hull of its extreme points. Is it true that this implies $X$ is reflexive?

I have a question about commutative direct summands of $C$*-algebras. Let $A$ be a $C$*-algebra (with unit) and suppose that its bidual $A^{**}$ has a commutative direct summand, that is, $A^{**}=B\o …

Suppose $f_1\colon K\to [0,\infty)$ and $f_2\colon K\to[0,\infty)$ are two upper semi-continuous affine functions, $$ f_i(\lambda x+(1-\lambda)y)=\lambda f_i(x)+(1-\lambda)f_i(y)\ \mbox{ for all }\ x, …