2007 Fiscal Year Final Research Report Summary
Flows and their symmetries on operator algebras
| Project/Area Number |
16540181
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Hokkaido University |
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Principal Investigator |
KISHIMOTO Akitaka Hokkaido University, Fac. of Sci., Professor (00128597)
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| Co-Investigator(Kenkyū-buntansha) |
ARAI Asao Hokkaido University, Fac. of Sci., Professor (80134807)
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| Project Period (FY) |
2004 – 2007
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| Keywords | C^*-algebra / cocyle / Kirchberg algebra / Rohlin property / AF flow / multiplier / approximately inner / core symmetry |
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| Research Abstract |
We have been trying to find and solve problems concerning flows on operator algebras by regarding them as a model in quantum statistical mechanics. We referred to symmetries in the title because we anticipated a contribution to phase transition might follow from this research. Although what we hoped for was not attained, we shall list and explain three major results obtained in this process.
Rohlin flows: This is the farthest kind of flows from what we encounter in statistical mechanics but probably is the simplest kind mathematically because the 1-cocycles are almost coboundary. Although there are known many examples of flows with this property, we have shown that there are for all Kirchberg algebras and remarked that they may be unique up to cocycle conjugacy with a partial result (with O. Bratteli and D.W. Robinson). The proof originates in the research on another 1-cocycle property of the shift automorphism on the infinite tensor product of two-by-two matrices and its restriction to a so-called gauge-invariant part.
AF flows: This is a flow which can be inductively defined based on flows on matrix algebras and is supposed to correspond to a classical model instead of a quantum model. A basic question is still unsolved of whether a quantum flow is really far from a classical flow (or AF flow), but we showed that if the flow looks close to an AF flow on appearance then it is close on principle. The proof uses physical properties derived from statistical mechanics.
Flows, restrictions to invariant hereditary subalgebras, and perturbations by multipliers: This had not been touched upon in the C^*-case in spite of the well-known results in some the W^*-case. This reduces to problems on cocycle multipliers for a flow on a stable C^*-algebra. A main difficulty lies in the fact such a cocycle is not norm-continuous. By adopting functional analytic methods, we showed that the cocycle can be approximated by a norm-continuous cocycle in a weak sense, thus achieving our goal.
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