Name	Ken Miyake
	Department	Department of Mathematics, School of Fundamental Science and Technology
	Research Fields	Non-commutative Geometry

There is a research field deals with the generalized notion of the random variables and the expected values in probability theory which is called a non-commutative probability theory. I have been trying the formulation using the techniques of the non-commutative probability theory to the problem of quantization of space. A non-commutative probability space is defined as a pair of a *-algebra and a state on it. At this time, a set of the whole states corresponding to a certain *-algebra is a convex set. For example, when a n-dimensional complex space is made into a *-algebra, the whole states can be identified with the (n−1)-dimensional simplex and the whole states in the case of (2×2)-matrix algebra M(2,C) with a 3-dimensional unit ball. Then, the whole states on a non-commutative n-dimensional complex space can be interpreted as a non-commutative simplex. Similarly, the objects like a non-commutative 3-dimensional unit ball can be considered. At present, I have a trial of the classifications of states corresponding to several θ-deformed function algebras from this viewpoint. On the other hand, the studies of the already constructed non-commutative probability spaces, and the θ-deformed homogeneous spaces are also advanced.