Mathematical Sciences: Infinite Simple C*-Algebras and TheirInvariants

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DMS-9626592 PI: Shuang Zhang Univ of Cincinnati During the next three years the Principal Investigator, Shuang Zhang, proposes to continue his research on the following three problems. (1) Is there a simple C*-algebra that contains both finite and infinite nonzero projections? (2) Determine the K-theory of the simple C*-algebras generated by the reduced group C*-algebras associated with free products of cyclic groups and certain projections, and simple C*-algebras generated by a weighted bilateral shift and an isometry. (3) Classifying these non-nuclear, purely infinite, simple C*-algebras up to isomorphism. The problem (1) is an outstanding problem in the theory of simple C*-algebras and the program of classifying separable simple C*-algebras. The proposer intends to attack this problem by some techniques in the associated multiplier algebras. The C*-algebras in (2) and (3) include non-nuclear, purely infinite, simple C*-algebras which might serve as standard models in certain classes of non-nuclear simple C*-algebras. Investigation on the structure of these C*-algebras will supplement the expansion of classification program to the category of non-nuclear simple C*-algebras. Encouraged by the interests of several leading researchers on the projects above, the proposer expects that this investigation will stimulate fruitful research activities on the subjects. The collection of all linear transformations in the three dimensional space, and in any finite dimensional space, is generated by translations and certain rotations (via addition, scalar multiplication, and composition). This collection is the basic example of simple C*-algebras and can be identified with the algebra of all square matrices. These elementary simple C*-algebras are fully classified by the dimension of the space. Many problems in the quantum physics and in modern technologies (for example, computer programming) are reduced to the investigation of simple C*-algebras of this sort, and more generally, a variety of simple C*-algebras generated by certain linear transformations in the infinite dimensional space. The proposed project is concerned with the structure of simple C*-algebras generated by some square matrices of infinite size. In the process of increasing knowledge about these simple C*-algebras, one important step is to understand their structure and then to classify them. The proposed project is especially concerned with studying invariants that distinguish these algebras. The classical dimension is no longer the distinguishing invariant, and thus, finding and studying new distinguishing invariants become two key problems. The proposed project aims at attacking these outstanding problems in this area.