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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 introduce and characterize Bloch functions on bounded symmetric domains, which may be infinite dimensional, by extending several well-known equivalent conditions for Bloch functions on the open unit disc U in C . We also generalize a number of results concerning Bloch functions on U to bounded symmetric domains. Given a holomorphic mapping φ between bounded symmetric domains B X and B Y , we derive criteria for boundedness and compactness of the composition operator C φ between the Bloch spaces B ( B Y ) and B ( B X ) , extending several known results for finite dimensional domains.

We generalize Bonk's distortion theorem on the unit disc in the complex plane to locally biholomorphic mappings on finite dimensional bounded symmetric domains. As an application, we obtain a lower bound for the Bloch constant for various classes of locally biholomorphic Bloch mappings.

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.

Let D = D 1 × ⋯ × D p be a product of Hilbert balls, with coordinate maps π j : D ¯ → D ¯ j on the closure D ¯ , for j = 1 , … , p . Let f be a fixed-point free self-map on D, which is nonexpansive in the Kobayashi distance, and compact for p ⩾ 2 . We describe the horospheres invariant under f and show that there exist a boundary point ( ξ 1 , … , ξ p ) of D and a nonempty set J ⊂ { 1 , … , p } such that each limit function h of the iterates ( f n ) satisfies ξ j ∈ π j ∘ h ( D ) ¯ for all j ∈ J and π j ∘ h ( ⋅ ) = ξ j whenever π j ∘ h ( D ) meets the boundary of D j . For a single Hilbert ball D 1 , either l i m i n f n → ∞ ‖ f 2 n ( 0 ) ‖ 1 or ( f n ) converges locally uniformly to a constant map taking value at the boundary of D 1 .

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 give starlike criteria for a class of rational mappings on the open unit ball of a complex Banach space. We also give a sufficient condition for these mappings to be convex when they are defined in Hilbert spaces. These criteria facilitate the construction of concrete examples of starlike and convex mappings on infinite dimensional domains (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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 describe the spectrum of the Laplacian for a homogeneous graph acted on by a discrete group. This follows from a more general result which describes the spectrum of a convolution operator on a homogeneous space of a locally compact group. We also prove a version of Harnack inequality for a Schrodinger operator on an invariant homogeneous graph.

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 $$n\, \in \,\user2{\mathbb{N}}\, \cup \, \left\{ 0 \right\}$$ .

We study the Jordan structures and geometry of bounded matrix-valued harmonic functions on a homogeneous space and their analogue, the harmonic functionals, in the setting of Fourier algebras of homogeneous spaces.

Let Z and W be JB*-triples and let T be a linear isometry from Z into W. For any z ∈ Z with ||z

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

Let σ be a probability measure on a locally compact groupG. A real Borel functionf onG is called σ-harmonic if it satisfies the convolution equation σ*f=f. Given that σ isnonsingular with its translates, we show that the bounded σ-harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If σ is nondegenerate and absolutely continuous, we solve the more general equation σ*μ=μ for positive measure μ on those groups which are metrizable and separable.

Let D be the open unit ball of a \(JB^*\)-triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and \(A^{**}\) is a type I \(JBW^*\)-triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every \(JB^*\)-triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear.

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 ]).

Given a compact holomorphic fixed-point-free self-map, f f , of the open unit ball of a Hilbert space, we show that the sequence of iterates, ( f n ) (f^n) , converges locally uniformly to a constant map ξ \xi with ‖ ξ ‖ = 1 \Vert \xi \Vert = 1 . This extends results of Denjoy (1926), Wolff (1926), Hervé (1963) and MacCluer (1983). The result is false without the compactness assumption, nor is it true for all open balls of J ∗ J^{*} -algebras.

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.

We show that a C *-algebra A has the weak Banach-Saks property if and only if it is type I and the k th-derivative of its spectrum  is empty for some k . It is equivalent to the existence of a finite chain J 1 ⊂ J 2 ⊂ … ⊂ J n ⊂ A of closed ideals such that J 1 , J 2 / J 1 , …, A / J n are all dual C *-algebras.