Let M n be the algebra of all n x n matrices over a field IF, where n > 2. Let : S -*• M n such that F((A)(j>(B)) = F(AB)for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A i-> iJ.(A)S(a(ajj))S~{ for all A = (a,-y) e S for some invertible S e M n , field monomorphism a of F, and an P-valued mapping \x defined on S. For real matrices, a is often the identity map; for complex matrices, a is often the identity map or the conjugation map: z i-> z. A key idea in our study is reducing the problem to the special case when F : M n ->• {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize : S -> M n such that 4>(A)4>(B) = 0 if and only if AS = 0. We show that such a map has the standard form described above on rank one matrices in S.2000 Mathematics subject classification: primary 15A04, 15A60, 15A18.

Abstract. Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H.where s i (A) denotes the ith s-numbers of A. In this paper we study some basic properties of this norm and give a characterization of the extreme points of its closed unit ball. Using these results, we obtain a description of the corresponding isometric isomorphisms on B(H).

Let A be a unital C Ã -algebra. An element u of A is unitary and belongs to the centre of A if and only if jfuj 1 for every pure state f. Using this fact we show that a numerical radius preserving linear isomorphism on A is a C Ã -isomorphism multiplied by a fixed unitary element in the centre of A.

Abstract. Let H be a complex Hilbert space and B(H) the Banach space of all bounded linear operators on H . For any A ∈ B(H) , let w(A) denote the numerical radius of A . Then A is normaloid if w(A) = A . In this note, we show that A is normaloid if there is a sequence of unit vectors (x n ) such that lim n→∞ Ax n = A and lim n→∞ | Ax n ,x n | = w(A) simultaneously. The result is then used to study the Davis-Wielandt radius.Mathematics subject classification (2010): 47A12, 47B20.

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