Maps Preserving Numerical Radius Or Cross Norms On 2×2 Matrix Algebras

The study of operator algebra theory began in 30s of the 20th century. With the fast development of the theory, now it has become a hot branch playing the role of an initiator in modern mathematics. It has unexpected relations and inter infiltrations with quantum mechanics. noncommutative geometry. linear system and control the-ory, number theory as well as some other important branches of mathematics.Preserver problems on operator algebras are to study the maps leaving some properties of ele-ments in algebras invariant. Nonlinear maps preserving numerical radius or cross norms on matrix algebras and operator spaces has been characterized completely and drawn abundant conclusion. However. most of relevant treatises is supposed that the dimen-sion of Hilbert space is greater than 2. there are absence of paper concerning that the dimension of Hilbert space is equal to 2.In respect that 2×2 Matrix Algebras plays an important and essential role in theory and application, for example,2 x 2 Matrix Algebras is mathematically theoretical frame of single quantum system, it is necessary to research these issues on 2×2 Matrix Algebras.Based on the accomplishments of corresponding issues when the space of dimension is greater than two, the paper mainly discusses nonlinear surjective maps preserving numerical radius or cross norms of matrix products and skew products and Jordan triple skew products on 2 x 2 matrix algebras. It is proved by using fundamental theorem of quantum machanics. namely, Wigner's Theorem. All the maps are characterized completely and classified in order to improve the issues. We obtain the following results: (1) The unital surjective maps on 2 x 2 matix algebra which preserving the numer-ical radius or a cross norm of product of matrices are characterized. These supplement the corresponding results for B(H) with dim(H)≥3.(2) The unital surjective maps on 2 x 2 matix algebra which preserving the nu-merical radius or a cross norm of skew product of matrices are characterized. These supplement the corresponding results for B(H) with dim(H)≥3.(3) The unital surjective maps on 2 x 2 matix algebra which preserving the cross norm of Jordan triple skew product of matrices are characterized.