In this article we give an expository account of the holomorphic motion theorem based on work of Mãne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have |ǫ log ǫ| moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz’s lemma and integration over the holomorphic vari...

The Royden ideal boundary of a domain is the set of all points in the maximal ideal space of the domain's Royden algebra that do not lie in the domain. Elements of the Royden ideal boundary can be characterized as nets convergent in both the weak and Euclidean topologies that have no subnet which is a sequence. As with other function algebras, boundary bers can be deened as subsets of all point...

let $fm(x)$ be the  space of  all finite regular borel measures on $x$. a general measure algebra is a subspace  of$fm(x)$,which is an $l$-space and has a multiplication preserving the probability measures. let $clsubseteqfm(x)$ be a general measure algebra on a locallycompact space $x$. in this paper, we investigate the relation between arensregularity of $cl$ and the topology of $x$. we  find...

We prove an effective version of a theorem of Dufresnoy: For any set of 2n + 1 hyperplanes in general position in Pn, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, we have f#(0) ≤ K, where f# denotes the norm of the derivative measured with respect to the Fubini-Study metric. This result gives an explicit lower bo...

We continue the research on structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form than existing ones are provided. As one consequences our study an effective formula for Kobayashi—Royden metric domain $${T_{{\mathbb{B}_n}}}$$ at origin is given. The results natural way provide versions Schwarz Lemma harmonic mappings. also present...

The Arens products are the standard way of extending the product from a Banach algebra A to its bidual A′′. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if A is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known ...

In this paper we define the notion of weak Arens regular Banach algebras and extend the concept of quasi-multipliers to this certain class of Banach algebras. Among other the relationship between Arens regularity of the algebra A∗∗ of a weak Arens regular Banach algebra A and the space QMr(A∗) of all bilinear and separately continuous right quasimultipliers of A∗ is investigated. Further, we st...

In this paper we first introduce a new multiplication on triangular Banach algebras. Then we study their Arens regularity as well as their topological centers . In this paper we first introduce a new multiplication on triangular Banach algebras. Then we study their Arens regularity as well as their topological centers . In this paper we first introduce a new multiplication on triangular Banach ...

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p ∈ C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p then it also extends to the simple Cole extension with respect to p. We...

Let A be a Banach algebra. Then A the second dual of A is a Banach algebra with first (second) Arens product. We study the Arens products of A(= (A∗∗)∗∗). We fined some conditions on A to be a left ideal in A. We fined the biggest two sided ideal I of A, in which I is a left (right) ideal of A∗∗.