A Denjoy-Wolff theorem for bounded symmetric domains
We generalise the Denjoy-Wolff theorem for a fixed-point free holomorphic self-map on the complex unit disc to bounded symmetric domains of finite rank in complex Banach spaces.
math.CV↗
arXiv subjects
Cho-Ho Chu
Publications and source records attributed to Cho-Ho Chu.
Horofunctions and metric compactification of noncompact Hermitian symmetric spaces
Given a Hermitian symmetric space $M$ of noncompact type, we give a complete description of the horofunctions in the metric compactification of $M$ with respect to the Carath\'eodory distance, via the realisation of $M$ as the open unit ball $D$ of a Banach space $(V,\|\cdot\|)$ equipped with a Jordan structure, called a $\mathrm{JB}^*$-triple. The Carath\'eodory distance $\rho$ on $D$ has a Finsler structure. It is the integrated distance of the Carath\'eodory differential metric, and the norm $\|\cdot\|$ in the realisation is the Carath\'eodory norm with respect to the origin $0\in D$. We also identify the horofunctions of the metric compactification of $(V,\|\cdot\|)$ and relate its geometry and global topology to the closed dual unit ball (i.e., the polar of $D$). Moreover, we show that the exponential map $\exp_0 \colon V \longrightarrow D$ at $0\in D$ extends to a homeomorphism between the metric compactifications of $(V,\|\cdot\|)$ and $(D,\rho)$, preserving the geometric structure. Consequently, the metric compactification of $M$ admits a concrete realisation as the closed dual unit ball of $(V,\|\cdot\|)$.
math.DG↗
Infinite dimensional holomorphic homogeneous regular domains
We extend the concept of a finite dimensional {\it holomorphic homogeneous regular} (HHR) domain and some of its properties to the infinite dimensional setting. In particular, we show that infinite dimensional HHR domains are domains of holomorphy and determine completely the class of infinite dimensional bounded symmetric domains which are HHR. We compute the greatest lower bound of the squeezing function of all HHR bounded symmetric domains, including the two exceptional domains. We also show that uniformly elliptic domains in Hilbert spaces are HHR.
math.CV↗
Siegel domains over Finsler symmetric cones
Let $Ω$ be a proper open cone in a real Banach space $V$. We show that the tube domain $V \oplus iΩ$ over $Ω$ is biholomorphic to a bounded symmetric domain if and only if $Ω$ is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that $V$ is a unital JB-algebra in an equivalent norm and $Ω$ is the interior of $\{v^2: v\in V\}$.
math.DG↗
Peeling Property of Bondi-Sachs metrics for nonzero Cosmological Constant
In this paper, we show that the peeling property still holds for Bondi-Sachs metrics with nonzero cosmological constant under the boundary condition given by Sommerfeld's radiation condition together with three nontrivial $Λ$-independent functions $B$, $a$, $b$. This should indicate the new boundary condition is natural. Moreover, we construct some nonstationary vacuum Bondi-Sachs metrics without Bondi news, which Newmann-Penrose quantities fall faster than usual. This provides a new feature of gravitational waves for nonzero cosmological constant.
gr-qc↗
Amenability, Reiter's condition and Liouville property
We show that the Liouville property and Reiter's condition are equivalent for semigroupoids. This result applies to semigroups as well as semigroup actions. In the special case of measured groupoids and locally compact groupoids, our result proves Kaimanovich's conjecture of the equivalence of amenability and the Liouville property.
math.FA↗
Infinite dimensional Jordan algebras and symmetric cones
A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting.
math.RA↗
Horoballs and iteration of holomorphic maps on bounded symmetric domains
Given a fixed-point free compact holomorphic self-map $f$ on a bounded symmetric domain $D$, which may be infinite dimensional, we establish the existence of a family $\{H(ξ, λ)\}_{λ>0}$ of convex $f$-invariant domains at a point $ξ$ in the boundary $\partial D$ of $D$, which generalises completely Wolff's theorem for the open unit disc in $\mathbb{C}$. Further, we construct horoballs at $ξ$ and show that they are exactly the $f$-invariant domains when $D$ is of finite rank. Consequently, we show in the latter case that the limit functions of the iterates $(f^n)$ with weakly closed range all accumulate in one single boundary component of $\partial D$.
math.CV↗
Separably injective $C_σ$-spaces
We show that a (complex) $C_σ$-space is separably injective if and only if it is linearly isometric to the Banach space $C_0(Ω)$ of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space $Ω$.
math.FA↗
Separably injective C*-algebras
We show that a C*-algebra is a $1$-separably injective Banach space if, and only if, it is linearly isometric to the Banach space $C_0(Ω)$ of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space $Ω$.
math.FA↗
Cohomology of Jordan triples via Lie algebras
We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained.
math.OA↗
Eigenvalue decay of operators on harmonic function spaces
Let $Ω$ be an open set in $\R^d$ $(d > 1)$ and $h(Ω)$ the Fréchet space of harmonic functions on $Ω$. Given a bounded linear operator $L :h(Ω)\to h(Ω)$, we show that its eigenvalues $λ_n$, arranged in decreasing order and counting multiplicities, satisfy $|λ_n|\leq K\exp(-cn^{1/(d-1)})$, where $K$ and $c$ are two explicitly computable positive constants.
math.FA↗
Normal contractive projections preserve type
Given a JBW*-triple Z and a normal contractive projection P on Z, we show that the (Murray-von Neumann) type of each summand of P(Z) is dominated by the type of Z.
math.OA↗