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We establish domains of invariance for holomorphic self-maps of a bounded symmetric domain of arbitrary dimension, generalising results of Wol¨and Julia in the open unit disc h of C and well known results on the Hilbert ball. The holomorphically invariant domains are analogues of the Poincare  balls and their limiting horocycles in h. The results appear to be new even in the ®nite dimensional (non-Hilbert space) case. completely describe our invariant domains as the limit in a natural sense of a sequence of Kobayashi balls. Our principal result presented in Section 3 is a generalization of Wol¨'s classical theorem, on invariant domains of a ®xed point free holomorphic self map of h, to compact holomorphic ®xed point free maps on a bounded symmetric domain, giving in particular an exact analogue of Wol¨'s theorem in the ®nite dimensional case. This result, surprisingly, appears to be new even in ®nite dimensions (apart from the Hilbert ball) andBrought to you by | University of Iowa Libraries Authenticated Download Date | 5/31/15 6:48 AM 1.1. De®nition. A JB à -triple is a complex Banach space Z with a real trilinear mapping fÁ Y Á Y Ág X Z  Z  Z 3 Z satisfying:(i) fxY yY zg is complex linear and symmetric in the outer variables x and z, and is complex anti-linear in y.(ii) The map z U 3 fxY xY zg, denoted x a x, is Hermitian, sx a x 0 and kx a xk kxk 2 for all x e Z, where s denotes the spectrum.(iii) The product satis®es the following``triple identity'' faY bY fxY yY zgg ffaY bY xgY yY zg À fxY fbY aY ygY zg fxY yY faY bY zggXwhere y à denotes the usual adjoint of y.(ii) CX , the continuous C-valued functions on a compact Hausdor¨space X, is a JB à -triple for the product fxY yY zg xyz. By a deep result of Kaup [18] every bounded symmetric domain is biholomorphically equivalent to the open unit ball of a JB à -triple and vice versa. Mellon, Holomorphic invariance on bounded symmetric domains200 Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/31/15 6:48 AM

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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 o:, :, :q, 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]).A JB*-triple is a complex Banach space and its open unit ball admits many biholomorphic automorphisms, which play a fundamental role in the theory of JB*-triples and bounded symmetric domains. In fact, a Banach space is a JB*-triple if, and only if, the biholomorphic automorphisms of its open unit ball act transitively [23]. Recently, Isidro and Kaup [22] studied the question of when these holomorphic automorphisms are weakly continuous, and a notion of weakly continuous JB*-triples was introduced in [24]. The weak continuity of these automorphisms turns out to be closely related to a well-known Banach property, namely the Dunford-Pettis property (which will be recalled below). Indeed, using [22] one can show that a JB*-triple with a unitary tripotent has the Dunford-Pettis property if, and only if, every biholomorphic automorphism of its open unit ball is sequentially weakly continuous in the sense that it preserves weak convergence of sequences (see Proposition 8 below). It is therefore of interest to know which JB*-triples have the Dunford-Pettis property.In this paper, we characterise JB*-triples having the Dunford-Pettis property. We show that, among other results, a JB*-triple Z has the Dunford-Pettis property if, and only if, for every weakly null sequence (z n ) in Z, the sequence (oz n , z n , zq) is also weakly null for all z ? Z**. It follows that the Dunford-Pettis property is inherited by subtriples and that a JBW*-triple W has the Dunford-Pettis property if, and only if, W is an l _ -sum G α L _ (Ω α , µ α , C α ), where C α is a Cartan factor and sup α dim C α _.We also show that the predual WJ has the Dunford-Pettis property if, and only if, W l l _ -sum G α L _ (Ω α , µ α , C α ) with dim C α _ for all α. These results subsume those in [5, 9, 10]. Dunford-Pettis property and weak continuity of automorphismsLet C(X ) be the Banach space of continuous functions on a compact Hausdorff space X. It is a celebrated result of Grothendieck [15] that every weakly compact linear operator on C(X ) is completely continuous. He called this property of C(X ) the Dunford-Pettis property (DPP for short) referring, of course, to an earlier result of Dunford and Pettis [13] that L " -spaces enjoy the same property. Grothendieck [15] has also shown that a Banach space E has DPP if, and only if, whenever (x n ) and

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n-dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n×n matrices {M1,…,Mn} describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.