非单位C* 代数的Blackadar-Handelman定理
The Blackadar-Handelman theorem for non-unital C*-algebras
摘要：
Blackadar和Handelman的一个著名定理指出，每个单位稳定有限的C* 代数都有一个有界拟竞赛。稳定有限性对非酉情形的相当强的推广可以通过要求乘子代数是稳定有限的，或者要求它至少是稳定非真无限的来获得。本文讨论了Blackadar-Handelman结果是否可以推广到非幺正情形。对于适当行为良好的C* 代数-代数，有一个肯定的结果，但没有一个非单位元的版本完全普遍成立。构造了C* 代数的两个例子。第一个是一个非单位的、稳定交换的C* 代数$A$，它与Blackadar-Handelman定理的最弱可能推广相矛盾：$A$上所有矩阵代数的乘子代数是有限的，而$A$没有有界拟迹。第二个例子是一个稳定非稳定的非单位的简单C*-代数$B$，即在$B$上没有矩阵代数是稳定的C* 代数。事实上，该C* 代数的所有矩阵代数上的乘子代数并不是真正无限的。此外，C* 代数$B$没有有界拟迹，因此给出了Blackadar-Handelman定理的一个可能推广的简单反例。
英文摘要：
A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. This paper deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra $A$ that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over $A$ are finite, while $A$ has no bounded quasitrace. The second example is a non-unital, simple C*-algebra $B$ that is stably non-stable, i.e. no matrix algebra over $B$ is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra $B$ has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.
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