Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on H_b(V), V absolutely convex open set.     

A Characterization of Homomorphisms Between Banach Algebras

We show that every unital invertibility preserving linear map from a yon Neumann algebra onto a semi-simple Banach algebra is a Jordan homomorphism; this gives an affirmative answer to a problem of Kaplansky for all yon Neumann algebras. For a unital linear map Ф from a semi-simple complex Banach algebra onto another, we also show that the following statements are equivalent: (1)Ф is an homomorphism; (2) Ф is completely invertibility preserving; (3) Ф is 2-invertibility preserving.     

A Characterization of Homomorphisms Between Banach Algebras

We show that every unital invertibility preserving linear map from a yon Neumann algebraonto a semi-simple Banach algebra is a Jordan homomorphism;this gives an affirmative answer to aproblem of Kaplansky for all von Neumann algebras.For a unital linear map Φ from a semi-simplecomplex Banach algebra onto another,we also show that the following statements are equivalent:(1)Φ is an homomorphism;(2)Φ is completely invertibihty preserving;(3)Φ is 2-invertibility preserving.     

Pervasive Algebras of Analytic Functions

We characterize those open U in the sphere such that A(U) is complex-pervasive, and those such that Re A(U) is real-pervasive. Pervasive means, roughly, that the uniform closure on each proper closed subset of bdy U is the space of all continuous functions (to or , as the case may be).     

Gleason Parts and Weakly Compact Homomorphismsbetween Uniform Banach Algebras

 If points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum. We derive from this several consequences about weakly compact homomorphisms and discuss the case of other uniform Banach algebras arising in complex infinite dimensional analysis.     

Gleason Parts and Weakly Compact Homomorphismsbetween Uniform Banach Algebras

 If points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum. We derive from this several consequences about weakly compact homomorphisms and discuss the case of other uniform Banach algebras arising in complex infinite dimensional analysis. (Received 9 March 1998; in revised form 29 June 1998)     

Homomorphisms of Finitary Incidence Algebras

For any field K and poset P, the incidence space I(P) and the finitary incidence algebra FI(P) were introduced in [5 Khripchenko , N. , Novikov , B. ( 2009 ). Finitary incidence algebras . Communications in Algebra 37 : 1670 – 1676 . [Google Scholar]]. The K-vector space I(P) is an FI(P)-bimodule. We investigate K-linear maps from FI(P) to I(P) that preserve submodules. We also consider the idealization FI(P)(+)I(P) of I(P).     

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

Regularity and Algebras of Analytic Functions in Infinite Dimensions

A Banach space is known to be Arens regular if every continuous linear mapping from to is weakly compact. Let be an open subset of , and let denote the algebra of analytic functions on which are bounded on bounded subsets of lying at a positive distance from the boundary of We endow with the usual Fréchet topology. denotes the set of continuous homomorphisms . We study the relation between the Arens regularity of the space and the structure of .

     

解析函数的加权代数中的闭理想问题

该文讨论了单位圆盘上解析函数的加权代数犃犘中的闭理想问题,并且利用加权系给出了一个闭理想成为犃犘中的补子空间的判断准则     

On Homomorphisms from Ringel-Hall Algebras to Quantum Cluster Algebras

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \(\mathcal {A}\) to an appropriate q-polynomial algebra. In the case that \(\mathcal {A}\) is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \(\mathcal {A}\) is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.     

Triple Homomorphisms of Perfect Lie Algebras

Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.     

Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphismsfrom certain commutative Banach algebras was resolved. Let Abe the commutative C*-algebra C(), where is an infinite compactspace. Then, if the continuum hypothesis (CH) be assumed, thereis a discontinuous homomorphism from C() into a Banach algebra[2, 7]. In fact, let A be a commutative Banach algebra. Then(with (CH)) there is a discontinuous homomorphism from A intoa Banach algebra whenever the character space _A of A is infinite[3, Theorem 3] and also whenever there is a non-maximal, primeideal P in A such that |A/P|=2^₀ [4, 8]. (It is an open questionwhether or not every infinite-dimensional, commutative Banachalgebra A satisfies this latter condition.) 1991 MathematicsSubject Classification 46H40.     

Weakly Confluent Quadratic Algebras

Any quadratic algebra endowed with an ordered set of generators can bedescribed by some linear map called a reduction operator. The general lineargroup naturally acts on reduction operators, which allows us to introduceweak and strong confluence. With respect to these notions, a completeclassification for two generators and complex coefficients is obtainedshowing that weak confluence is equivalent to Koszulity in this case. Bycontrast, some Sklyanin algebras with three generators fail to be weaklyconfluent. For an arbitrary number of generators and under some assumptionson the first terms of the Hilbert series, a weak confluence hypothesis isequivalent to some rather drastic conditions which determine the whole ofthe Hilbert series.     

Maps Between Uniform Algebras Preserving Norms of Rational Functions

Let A, B be uniform algebras. Suppose that A ₀, B ₀ are subgroups of A ⁻¹, B ⁻¹ that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A ₀ → B ₀ is a surjection with ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A₀{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A₀{f \in A_0}. This result leads to the following assertion: Suppose that S _A , S _B are subsets of A, B that contain A ⁻¹, B ⁻¹ respectively. If m, n > 0 and a surjection T : S _A → S _B satisfies ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? S_A{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? S_A{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.     

Compact and Weakly Compact Composition Operators on BMOA

Any analytic map φ of the unit disc ${\mathbb{D}}$ into itself induces a composition operator C _φ on BMOA, mapping ${f \mapsto f \circ \varphi}$ , where BMOA is the Banach space of analytic functions ${f\colon \mathbb{D} \to \mathbb{C}}$ whose boundary values have bounded mean oscillation on the unit circle. We show that C _φ is weakly compact on BMOA precisely when it is compact on BMOA, thus solving a question initially posed by Tjani and by Bourdon, Cima and Matheson in the special case of VMOA. As a crucial step of our argument we simplify the compactness criterion due to Smith for C _φ on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. Additional equivalent compactness criteria are established. Furthermore, we prove the unexpected result that compactness of C _φ on VMOA implies compactness even from the Bloch space into VMOA.     

Some Problems on Homomorphisms and Real Function Algebras

 In this paper we solve a problem about the representation of all homomorphisms on a real function algebra as point evaluations and another two about function algebras in which homomorphisms are point evaluations on sequences in the algebra. (Received 4 December 2000; in revised form 2 April 2001)