Abstract
In this paper we consider the problem of determining the maximum dimension of P⊥(A⊕B)P, where A and B are unital, semi-simple subalgebras of the set Mn of n×n complex matrices, and P∈M2n is a projection of rank n. We exhibit a number of equivalent formulations of this problem, including the one which occupies the majority of the paper, namely: determine the minimum dimension of the space A∩S−1BS, where S is allowed to range over the invertible group ▪ of Mn. This problem in turn is seen to be equivalent to the problem of finding two automorphisms α and β of Mn for which the dimension of α(A)+β(B) is maximised. It is this phenomenon which gives rise to the title of the paper.
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