Centrality in BL (X)‎, the banach algebra of bounded linear operators

Abstract EN

In this paper we show that for a complex Banach algebra A = BL(X) of all bounded linear operators on the Banach space X, the set of all ρ-quasi central elements of A is a subset of each of: (i) the set of all discrete ρ-quasi central elements; (ii) the set of all continuous ρ-quasi central elements; (iii) the set of all residual ρ-quasi central elements; and (iv) the set of all approximate discrete ρ-quasi central elements of A, and this set inclusion may be proper, but in the case that X is finite dimensional we see that the set of all discrete ρ-quasi central elements, the set of all approximate discrete ρ-quasi central elements and the set of all ρ-quasi central elements of A all are identical, in the other hand the set of all continuous ρ-quasi central elements, the set of all residual ρ-quasi central elements of A and A are equal.

Also we show that any normal operator on a Hilbert space H is a residual ρ-quasi central element of BL(H), and any compact operator T on a Banach space X with zero as an eigenvalue of T, must be a residual and a continuous ρ-quasi central element of A.

Finally we show that for a unital complex Banach algebra B, Z(B) = Qσ(1,B).

Moreover, we give two examples: one is a unital complex Banach algebra B that shows the set of all σ-quasi center of B need not be topologically closed; and the other is a non-commutative Banach algebra B with Q(B) = Qσ(B) = Qρ(B) = B.