The norm problem for elementary operators

To Professor Manuel Valdivia on the occasion of his seventieth birthday.
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Abstract

Among the outstanding problems in the theory of elementary operators on Banach algebras is the task to find a formula which describes the norm of an elementary operator in terms of the norms of its coefficients. Here we report on the state-of-the-art of the knowledge on this problem along the lines of our talk at the Functional Analysis Valencia 2000 Conference in July 2000.

Section snippets

Setting the scene

Throughout we denote by A a complex unital Banach algebra and by El(A) the algebra of its elementary operators. By definition, SEl(A) if S is a linear mapping on A of the form Sx=j=1najxbj, where a = (α1, … , αn) and b = (b1, … , bn) are n-tuples of elements of A. Clearly, every elementary operator S is bounded, and it is easy to give upper bounds for the norm of S, e.g. the projective tensor norm of j=1najbj. Many important classes of bounded linear operators on Banach algebras, for

Special cases on special algebras

Suppose A = C(X), the algebra of complex-valued continuous functions on a compact Hausdorff space X with more than one point. Then, there are a, bA, both non-zero, such that LaRb = 0. Clearly, this rules out the possibility of a lower bound for the norm ∥LaRb∥ in terms of ∥a∥ and ∥b∥. The other extreme is the case A = B(E), the bounded operators on a Banach space E, or slightly more general a closed subalgebra A of B(E)which contains all finite rank operators. In this case, the norm of LaRb

General case on very special algebras

The understanding of the behaviour of the norm of elementary operators led to other insights into their structural properties. The Fong-Sourour conjecture [8] stated that there are no non-zero compact elementary operators on C(H) for a separable Hilbert space H, where, for every Banach space E, K(E) denotes the ideal of compact operators and C(E) = B(E)/K(E) stands for the Calkin algebra on E. This was confirmed by different methods in [1], [9], and [12, Part II]. The solution provided in [1]

General case on general C*-algebras

Within the extended Grothendieck programme, elementary operators arise as follows. There is a canonical mapping from the algebraic tensor product AA into B(A) defined as follows:AAB(A),j=1najbjj=1nLajRbj,which extends to a contraction from the projective tensor product A^A onto the closure of El(A) in B(A). In general, ⊖ is not injective. Let us again confine ourselves with C*-algebras. Then, ⊖ is injective if and only if A is prime (see e.g. [3], Section 5.1). However, even in this

General case on good C*-algebras

In view of the results in the previous section the question when the norm and the cb-norm of elementary operators coincide is close at hand. That is, we intend to find a class of ‘good’ C*-algebras A distinguished by the property that ∥S∥ = ∥Scb for every SEl(A). From the general theory we know that commutative C*-algebras are in this class. Magajna showed that very non-commutative C*-algebras can share this property. In [10] he proved that the Calkin algebra on a separable Hilbert space has

The challenge

The above discussion on the norm problem for elementary operators sheds considerable light on the present situation. In addition, it emerges that the solution to the following problem would yield a complete answer, at least in the case of general C*-algebras.

Determine the norm for every elementary operator on B(H), H a Hilbert space!

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