Nonlinear Expectation In Operator Algebras

In classic probability theory, expectation characterizes some average of random variables'value and the linearity of expectation is very important. In this framework, we have the law of large numbers and central limit theorem. In the theoretical study of nonlinear expectation, Shige Peng's research achieved an international advanced level and obtained the corresponding new law of large numbers and central limit theorem under the nonlinear situation. This paper transfer the linear expectation into opera-tor algebras, and thus introduced the nonlinear expectation in operator algebras. We constructed the nonlinear functional on a self-adjoint von Neumann algebra by taking the supremum of some states on the self-adjoint von Neumann algebra. We will show that the constructed nonlinear functional can be seen as a sublinear expectation on the self-adjoint von Neumann algebra. We present some related notions, new law of large numbers and its generalization under sublinear expectations in operator algebras. We also prove the theorem. This can be seen as a small combination of operator algebra and probability theory. In fact it is the development of free probability theory in operator algebra fields.