B（H）格、密度与算术平均理想
B(H) lattices, density and arithmetic mean ideals
摘要：
这一部分的多论文项目研究的算术平均理想B（H）的Dykema，Figiel，Weiss，Wodzicki介绍的晶格性质。我们证明：所有主理想的格、算术平均的格或无穷稳定主理想的算术平均的格或具有满足Delta_1/2条件的生成元的主理想的格在一般理想的格中都是上稠密的和下稠密的。也就是说，在这些子格之一中的任何理想和一个理想（分别嵌套在上面或下面）之间，存在另一个理想在那个子格中。在这些应用中：一个主理想I是am-稳定的当且仅当它的任何一个一阶算术平均理想是am-稳定的当且仅当该理想满足一阶等式消去性质：J_a = I_a蕴涵J = I.我们表明，这种取消属性可以失败，即使是am稳定可数生成的理想。对于无穷远理想的算术平均也有类似的结果。包含消去一般甚至对主理想也不成立，但对每个理想I都有一个最大理想I^，其中J_a包含I_a，这意味着J包含I^。当我是校长，我也是校长。我们证明了I=I^是比am-稳定性严格更强的性质。例如，对于由diag {1/n}生成的主理想J的p < 1次幂，I^是J的q次幂，其中1/p-1/q = 1。
英文摘要：
This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and lower dense in the lattice of general ideals. That is, between any ideal and an ideal (nested above or below respectively) in one of these sublattices, lies another ideal in that sublattice. Among the applications: a principal ideal I is am-stable (I = I_a) if and only if any of its first order arithmetic mean ideals are am-stable if and only if the ideal satisfies the first order equality cancellation property: J_a = I_a implies J = I. We show that this cancellation property can fail even for am-stable countably generated ideals. Similar results hold for arithmetic mean at infinity ideals. Inclusion cancellations do not hold in general even for principal ideals, but for every ideal I there is a largest ideal I^ for which J_a contains I_a implies that J contains I^. When I is principal, I^ too is principal. We show that I=I^ is a strictly stronger property than am-stability. For example, for I the p < 1 power of the principal ideal J generated by diag {1/n}, I^ is the q power of J where 1/p-1/q = 1.
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