Approximate Diagonals and Cohomology of Certain Annihilator Banach Algebras
Approximate Diagonals and Cohomology of Certain Annihilator Banach Algebras
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某些歼灭巴纳赫代数的近似对角线和上同调
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发表时间：
1972
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通讯作者：
B. Johnson
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作者：
B. Johnson
为了总结本文的内容，有必要使用[5]中的一些定义。这些定义在下文第1节的开头重复。尽管本文增加了[5]中的材料，但我们试图将引用[o]的次数减少到最低限度。在Banach代数31中的近似对角线是31 a中的有界网{ma}，对于31 a中的所有a，在范数拓扑中31？31是通常的射影张量积[2]和-k是连续线性映射31？31->31由ir（a？B）=ab。网{m31）**，其中aM = Ma和？r**（M）a = a对所有a？31.在代数的代数上同调的存在的元素m的A A与ma = am和ir（m）的单位A是等价的声明，H1（A，X）=0的所有31个模块X几乎从来没有成立？事实上，在交换的情况下，对于所有的A双模X。在Banach代数中，命题jEP^S^X）=0（见[3;定理1]和[5;命题8.1]），我们总能找到一个Banach 31模？其中S^1（3 I，36）=t^O，除非31 zz？n.存在一种近似？配偶对角线是不寻常的，相当于声明！M*（31，36 *）= 0对所有的Banach 31模X，即31是一个可使能Banach代数.元素M在顺从代数理论中的作用与不变平均在顺从群中的作用相同。文[5]中所用的定义和连接顺从性与近似对角线的定理在？1.在哪？2，我们表明，如果31有一个虚拟的对角线与马？31岁？对于3 T中的所有a，则为913（31，3E）？.0对于所有的Banach 31模3？.满足这个条件的代数是紧拓扑群的grdup代数和复数收敛序列的代数c 0。虽然我们没有证明它的结果也适用于任何零化子B* 代数只有有限维极小理想。在哪？3.证明了如果31是半单可换服从Banach代数，则存在一个Banach 31模X，其中3/2（3 I，3E）^0，除非31是有限的
In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite