论文标题
标准的晶格同态
Norm-attaining lattice homomorphisms
论文作者
论文摘要
在本文中，我们研究了从Banach Lattice $ x $ to $ \ mathbb {r} $中的所有晶格同构的集合的结构。利用晶格同构和脱节家庭之间的关系，我们证明了免费的Banach lattice $ fbl（a）$ fbl（a）$ a $ a $ a $的拓扑双重偶发，其中包含一个不连接的基数$ 2^{| a |} $，回答了B. de Pagter and A.W.的问题。威克斯特德。我们还处理规范的晶格同态。对于古典Banach晶格，AS $ C_0 $，$ L_P $ - 和$ C（k）$ - 空间，其上的每个格子同构都达到了其标准，特别是表明此类功能没有James定理。我们证明，实际上，每当$ x $都有订单连续规范时，$ x $和$ c（k，x）$上的每一个晶格同态都会达到其规范。另一方面，我们提供了晶格同态文献中的第一个例子，该典型同态无法达到其规范。通常，我们研究了在自由Banach晶格中未达到其规范的晶格同态的存在和表征。结果，结果表明，在Banach晶格设置中，没有主教型定理是正确的，即，并非每个晶格同构可以通过标准的晶格同态近似。
In this paper we study the structure of the set $\mbox{Hom}(X,\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice $FBL(A)$ generated by a set $A$ contains a disjoint family of cardinality $2^{|A|}$, answering a question of B. de Pagter and A.W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as $c_0$, $L_p$-, and $C(K)$-spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on $X$ and $C(K,X)$ attains its norm whenever $X$ has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop-Phelps type theorem holds true in the Banach lattice setting, i.e. not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.
