Problems in the Theory and Application of Operator Tensor Algebras

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AbstractMuhly Muhly intends to study problems in operator algebra that may be divided into three groups. In the first are found 7 broad questions concerning the theory of general operator algebras and their interactions with quantum Markov semigroups, completely positive maps and quantum information theory. Muhly intends to use the invariants he has discovered for tensor algebras and their representations to analyze semigroups of completely positive maps and related structures. A particular focus will be "continuous tensor algebras" built from product systems. The second group (5 problems) concerns the theory and application of groupoids. First, Muhly proposes to use his recent collaborative work on the Brauer group of a groupoid to build a general cohomology theory that is based on the category of actions of a groupoid on spaces. The second two focus on the structure of operator algebras built from so-called Fell bundles over groupoids. These arise quite commonly "in nature" and a good portion of the effort will be devoted to specific examples. The fourth deals with aspects of operator algebras associated with topological quivers - graphs in which the vertex and edge spaces are topological spaces. The final problem concerns a generalization of the Brauer group that interacts with a variety of investigations of current interest. The third area is part of Muhly's long-term investigations into the foundations of general operator algebra. The focus is on so-called orthoprojective and orthoinjective Hilbert modules with an eye to understanding boundary representations for operator algebras.The projects proposed herein derive naturally from and have an impact upon dynamical systems, particularly irreversible dynamical systems that appear in a variety of settings. Of special interest to us, are certain mathematical models, based on operator algebra, that contribute to the burgeoning area of quantum computing. Also, our work has interactions with the theory of cellular automata (which are, essentially, described in terms of automorphism of shift dynamical systems that have been so thoroughly studied of late), and models for "genetic transmissions", i.e., models that describe how genetic material is passed from generation to generation. Interdisciplinary activity with colleagues in physics, computer science and biology are likely. Discussions are already under way in Iowa's joint mathematical physics seminar and with colleagues in Biology. In addition, the operator algebras associated with graphs are expected to have an impact in mathematical systems theory of the type that appears in computer aided design. They are especially well adapted to handle systems that have built in uncertainties. This projects provides projects at the frontiers of operator algebra and linear algebra for graduate and undergraduate students. Even some high school students may be able to participate. These projects will also be used in the Department's REU and minority recruiting/training efforts supported in part by AGEP funds.