On Some Fascinating Algebras of Operators Constructed from Some 2×2 Matrices
by
Kenneth Driessel
Iowa State University
Coauthors: Irvin R. Hentzel (Iowa State University)

We work with matrices over the real numbers R. Consider the following 2 ×2 matrices:

I : = æ ç è 1 0 0 1 ö ÷ ø , E : = æ ç è 1 0 0 0 ö ÷ ø , Z : = æ ç è 0 0 1 0 ö ÷ ø .

Note that the set { I, E, Z } is closed under multiplication. Hence its span is an associative algebra (contained in the algebra of lower 2 ×2 matrices). Let n be a natural number. Consider the Kronecker/tensor products with lenth n having the following form:

L(k) : = I \otimes... \otimesI \otimesZ \otimesE \otimes... \otimesE

which are formed by a string of k I's (with k=0 and k=n possible), followed by a Z, followed by a string of E's. These L(k) generate an algebra F_n of operators on the tensor product space R^\otimesn : = R² \otimes... \otimesR². The algebra F_n has dimension 2ⁿ (which is the same as the dimension of the tensor product space on which the operators act). Justin Peters suggested that we study these algebras. He noted that there is an embedding from F_n into F_n+1. We shall describe some of our results concerning these algebras. In particular, we discovered that these algebras have an involution. We shall also describe our work on the following optimization problem for such an algebra: Maximize <x * y, x * y > subject to the constraints <x, x > = <y, y > = 1. Here we use x*y to denote the product and <x, y > to denote the usual scalar product in

Date received: May 21, 1999