On the Choi-Effros multiplication
A short proof is given for the well-known Choi-Effros theorem on the structure of ranges of completely positive projections.
math.OA↗
arXiv subjects
Bebe Prunaru
Publications and source records attributed to Bebe Prunaru.
Asymptotic stability for a class of Markov semigroups
Let $U\subset K$ be an open and dense subset of a compact metric space and let $\{Φ_t\}_{t\ge0}$ be a Markov semigroup on the space of bounded Borel measurable functions on $U$ with the strong Feller property. Suppose that for each $x\in\bdu$ there exists a barrier $h\in C(K)$ at $x$ such that $Φ_t(h)\ge h$ for all $t\ge0$. Suppose also that every real-valued $g\in C(K)$ with $Φ_t(g)\ge g$ for all $t\ge0$ and which attains its global maximum at a point inside $U$ is constant. Then for each $f\in C(K)$ there exists the uniform limit $F=\lim_{t\to\infty}Φ_t(f)$. Moreover $F$ is continuous on $K$, agrees with $f$ on $\partial{U}$ and $Φ_t(F)=F$ for all $t\ge0$.
math.PR↗
Tsirelson's problem and purely atomic von Neumann algebras
It is shown that if a bipartite behavior admits a field representation in which Alice (or Bob's) observable algebra generates a purely atomic von Neumann algebra then it is non-relativistic.
math.OA↗
Lifting fixed points of completely positive semigroups
Let $\{ϕ_s\}_{s\in S}$ be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra $N$. Assume there exists a semigroup $\{α_s\}_{s\in S}$ of weak*-continuous *-endomorphisms of some larger von Neumann algebra $M\supset N$ and a projection $p\in M$ with $N=pMp$ such that $α_s(1-p)\le 1-p$ for every $s\in S$ and $ϕ_s(y)=pα_s(y)p$ for all $y\in N$. If $\inf_{s\in S}α_s(1-p)=0$ then we show that the map $E:M\to N$ defined by $E(x)=pxp$ for $x\in M$ induces a complete isometry between the fixed point spaces of $\{α_s\}_{s\in S}$ and $\{ϕ_s\}_{s\in S}$.
math.OA↗
Some Exact Sequences for Toeplitz Algebras of Spherical Isometries
A family $\{T_j\}_{j\in J}$ of commuting Hilbert space operators is said to be a spherical isometry if $\sum_{j\in J}T^*_jT_j=1$ in the weak operator topology. We show that every commuting family $\Cal F$ of spherical isometries has a commuting normal extension $\hat{\Cal F}$. Moreover, if $\hat{\Cal F}$ is minimal, then there exists a natural short exact sequence $0\to\Cal C\to C^*(\Cal F)\to C^*(\hat{\Cal F})\to 0$ with a completely isometric cross-section, where $\Cal C$ is the commutator ideal in $C^*(\Cal F)$. We also show that the space of Toeplitz operators associated to $\Cal F$ is completely isometric to the commutant of the minimal normal extension $\hat{\Cal F}$. Applications of these results are given for Toeplitz operators on strictly pseudoconvex or bounded symmetric domains.
math.OA↗
Amenability, completely bounded projections, dynamical systems and smooth orbits
We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representat ions.
math.OA↗