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Let Ω be a proper open cone in a real Banach space V. We show that the tube domain V ⊕ i ⁢ Ω {V\oplus i\Omega} 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 ∈ V } {\{v^{2}:v\in V\}} .

We characterise the Tits–Kantor–Koecher Lie algebra of a JB*-triple which can be infinite dimensional. In particular, we show that a complex Lie algebra is the Tits–Kantor–Koecher Lie algebra of a JB*-triple if and only if it has a real form which is isomorphic to the reduced Lie algebra of a bounded symmetric domain. Matrix representations of these Lie algebras are also presented.

We generalize a number of finite dimensional results on Bloch functions to infinite dimensional bounded symmetric domains. In particular, we characterize the Bloch space as well as the little Bloch space of a Hilbert ball, and give one sufficient and several necessary conditions for a composition operator on a Bloch space to be an isometry. We also answer some open questions of Allen and Colonna concerning Bloch functions and composition operators.

We correct an error in Lemma 2.3 in our paper with the above title, published in J. Geom. Anal. 30: 223–247.

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 $\{H(\xi, \lambda)\}_{\lambda >0}$ of convex $f$-invariant domains at a point $\xi$ 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 $\xi$ 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 $\partial D$., Comment: 30 pages

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 \({C_0(\Omega)}\) of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space \({\Omega}\).

We investigate iterations of fixed-point free holomorphic self-maps on a Lie ball of any dimension, where a Lie ball is a bounded symmetric domain and the open unit ball of a spin factor which can be infinite dimensional. We describe the invariant domains of a holomorphic self-map f on a Lie ball D when f is fixed-point free and compact, and show that each limit function of the iterates ( f n ) has values in a one-dimensional disc on the boundary of D. We show that the Mobius transformation g a induced by a nonzero element a in D may fail the Denjoy–Wolff-type theorem, even in finite dimension. We determine those which satisfy the theorem.

We discuss how Jordan algebraic structures arise from the geometry of bounded symmetric domains and their useful role in the study of holomorphic functions on these domains.

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.

Let B be the open unit ball of a complex Banach space X and let B be homogeneous. We prove distortion results for normalized convex mappings f : B → X which generalize various finite dimensional distortion theorems and improve some infinite dimensional ones. In particular, our results are valid for the open unit balls of complex Hilbert spaces and the Cartan domains.

Let G be a metric group, not necessarily locally compact, acting on a metric space X, for instance, a right coset space of G. We introduce and develop a basic structure theory for harmonic functions on X which is applicable to infinite dimensional Riemannian symmetric spaces.

We introduce a class of Riemannian symmetric spaces, called Jordan symmetric spaces, which correspond to real Jordan triple systems and may be infinite dimensional. This class includes the symmetric R-spaces as well as the Hermitian symmetric spaces.

We introduce a class of real Jordan triple systems, called JH-triples, and show, via the Tits–Kantor–Koecher construction of Lie algebras, that they correspond to a class of Riemannian symmetric spaces including the Hermitian symmetric spaces and the symmetric R-spaces.

We show that, in a JB-algebra, the projections form a Banach manifold and also, the rank-n projections in a JBW-factor form a Riemannian symmetric space of compact type, for $$n\, \in \,\user2{\mathbb{N}}\, \cup \, \left\{ 0 \right\}$$ .

Let D be an irreducible bounded symmetric domain in a complex Hilbert space and let f: D->D be a holomorphic map withf(O) = 0. We give conditions under whichf is linear, injective or isometric.

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., Comment: 27 pages, to appear in: Topics in Functional Analysis and Algebra, Proceedings of a special session of the USA-Uzbekistan Conference on Analysis and Mathematical Physics, CSU Fullerton, May 20-23, 2014, Contemporary Mathematics, 2016

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., To appear in Journal of Functional Analysis

We describe the affine connections, geodesics and symmetries of various Banach manifolds of tripotents in JB*-triples which include the C*-algebras and Hilbert spaces where the nonzero tripotents are respectively the partial isometries and the extreme points of the closed unit ball.

Let ai be a positive matrix-valued measure on a locally compact abelian group G such that a(G) is the identity matrix. We give a necessary and sufficient condition on a for the absence of a bounded non-constant matrixvalued function f on G satisfying the convolution equation f * a = f. This extends Choquet and Deny's theorem for real-valued functions on G.

Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.

Complex (Banach) Jordan triples occurred in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on (complex) C∗-algebras. These spaces are equipped with a ternary product, the Jordan triple product, and are essentially geometric objects in that the linear isometries between them are exactly the linear maps preserving the triple product [13].

JB*-triples occur in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on C*-algebras. These spaces are equipped with a ternary product {·,·,·}, the Jordan triple product , and are essentially geometric objects in that the linear isometries between them are exactly the linear bijections preserving the Jordan triple product (cf. [ 23 ]).

G. Choquet and J. Deny have characterized the positive solutions μ of the convolution equation σ*μ=μ of measures on locally compact abelian groups, for a given positive measure σ. By elementary methods, we extend their characterization to locally compact nilpotent groups which complements the various existing results on the equation, and we work out the solutions μ explicitly for the Heisenberg groups and some nilpotent matrix groups, by finding all the exponential functions on these groups.

We obtain some results on the unique extension of (factor) states of C ∗ C^* -algebras which complement various existing results. Our results also lead to a class of C ∗ C^* -algebras whose states are σ \sigma -convex sums of factor states.

Let K and S be compact convex sets and let A(K) and A(S) be the corresponding Banach spaces of continuous affine functions. If the Banach-Mazur distance between A(K) and A(S) is less than 2, then under certain geometric conditions, the extreme boundaries of K and S are homeomorphic. This extends a result of Amir and Cambern and has applications to function algebras

Let $\Omega$ be an open set in $\R^d$ $(d > 1)$ and $h(\Omega)$ the Fr\'echet space of harmonic functions on $\Omega$. Given a bounded linear operator $L :h(\Omega)\to h(\Omega)$, we show that its eigenvalues $\lambda_n$, arranged in decreasing order and counting multiplicities, satisfy $|\lambda_n|\leq K\exp(-cn^{1/(d-1)})$, where $K$ and $c$ are two explicitly computable positive constants., Comment: AMS-LaTeX, 14 pages

Let H∞(B) be the Banach algebra of bounded holomorphic functions on the open unit ball B of a Banach space. We show that the identity operator is an isolated point in the space of composition operators on H∞(B). This answers a conjecture of Aron, Galindo and Lindstrom. Swiss National Science Foundation

2.1 Preliminaries and notation 2.2 Poisson representation of harmonic functions 2.3 Semigroup structures of the Poisson space 2.4 Almost periodic harmonic functions 2.5 Distal harmonic functions 2.6 Transitive group actions on Poisson spaces 2.7 Examples

Let 1 ≤ p < oo. We show that a weighted translation operator on the L P space of a homogeneous space is hypercyclic under some condition on the weight. This condition is also necessary in the discrete case and is equivalent to hereditary hypercyclicity of the operator. The condition can be strengthened to characterise topologically mixing weighted translation operators on discrete spaces.

Complex Banach spaces for which the group of biholomorphic automorphisms of the open unit ball acts transitively, alias JB*-triples, possess a ternary algebraic structure uniquely determined by the holomorphic properties of the open unit ball [21]. A large and important class of these spaces is comprised of the JC*-triples of [17] (known also as J*-algebras) which are up to isometry the norm closed subspaces of B…H;K†, where H and K are complex Hilbert spaces, that are closed under the ternary product

We show that the predual of a JBW *-triple is complemented in the predual of a von Neumann algebra. Hence a quotientof a JB*-triple does not contain 11 if and only if its dual enjoys the Radon-Nikodym property. We also show that a JB*-triple either contains c0 or is reflexive.

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).

Banach spaces whose duals possess the Radon-Nikodym property have been studied extensively in the past (cf. [5]). It has been shown recently in [4] that a C*-algebra is scattered if and only if its Banach dual possesses the Radon-Nikodym property. This result extends the well-known result of Pełczynski and Semandini [8] that a compact Hausdorff space Ωis dispersed if and only if C(Ω)* has the Radon-Nikodym property. The purpose of this note is to give a transparent proof of a more general result for Jordan algebras which unifies the aforementioned results. We prove that the dual of a JB-algebra A possesses the Radon-Nikodym property if and only if the state space of A is the cr-convex hull of its pure states. We also consider the projective tensor products of the duals of JB-algebras in this context.