Abstract. Let F m×n be the set of m × n matrices over a field F. Consider a graph G = (F m×n , ∼)with F m×n as the vertex set such that two vertices A, B ∈ F m×n are adjacent if rank(A − B) = 1. We study graph properties of G when F is a finite field. In particular, G is a regular connected graph with diameter equal to min{m, n}; it is always Hamiltonian. Furthermore, we determine the independence number, chromatic number and clique number of G. These results are used to characterize the graph endomorphisms of G, which extends Hua's fundamental theorem of geometry on F m×n .
Let X, Y be real or complex Banach spaces with dimension greater than 2 and let A, B be standard operator algebras on X and Y , respectively. In this paper, we show that every map completely preserving idempotence from A onto B is either an isomorphism or (in the complex case) a conjugate isomorphism; every map completely preserving square-zero from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism.
Let A and B be standard operator algebras on complex Banach spaces X and Y , respectively. In this paper, we characterize the surjective maps completely preserving the invertibility in both directions and the surjective maps completely preserving the spectrum from A to B. We show that a surjective map from A to B is a ring isomorphism if and only if it is unital and completely preserves the invertibility of operators in both directions; is an isomorphism if and only if it completely preserves the spectrum of operators.
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