Hiroaki YOSHIDA, Department of Mathematics and Information Science, Professor

[ Japanese / English ]

It can be said that noncommutative analysis is the main tool for describing the quantum objects mathematically, which is classified as one of the area of functional analysis. It is, however, related to the many fields of mathematics and physics. The studies of the operators acting on a Hilbert space, and of the algebra of operators are important problems in the studies of quantum system as von Nuemann investigated the operators acting on a Hilbert space to describe the quantum objects mathematically.

My current field of research is noncommutative probability space and related topics. Roughly speaking, noncommutative probability space is the operator algebra together with some linear functional, called the expectation. In this frame, we can regard the operator as random variable, especially the spectrum density of a self-adjoint operator as the probability density on the real. But the usual independence should be replaced by an appropriate notion in noncommutative case. The theory of noncommutative probability space is not abstract object but really applicable to many other mathematical problems. For instance, we can apply it to the spectrum problem of the infinite graphs. The above figures illustrate the spectrum densities of the discrete Laplacians of some infinite graphs.

At present, we are now collaborated with the national institute of statistical mathematics on theory of quantum entropy and quantum information geometry. It can be expected that we will have many developments in these areas.

REFERENCES

  1. O. Hiwatashi, M. Nagisa, H. Yoshida The characterizations of a semicircle law by the certain freeness in a C^* probability space, Probab. Theory Relat. Fields 113 (1999),115-133.
  2. O. Hiwatashi, T.Kuroda, M. Nagisa, H. Yoshida The free analogu of noncentral chi-square distributions and quadratic forms in free random variables, Math. Zeit. 230 (1999), 79-105.

KEY WORDS

noncommutative analysis; quantum probability space; quantum entropy; C*-algebra; random walks on graphs.

ADDRESS

2-1-1, Otsuka, Bunkyo-ku, Tokyo, 112-8610, Japan.
E-mail:
yoshida@is.ocha.ac.jp
Phone:+81-3-5978-5386
Fax:+81-3-5978-5705

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