S - Numbers of Elementary Operators on C ∗ - Algebras

We study the s-numbers of elementary operators acting on C∗algebras. The main results are the following: If τ is any tensor norm andA,B ∈ B(H) are such that the sequences s(A), s(B) of their singular numbers belong to a stable Calkin space i then the sequence of approximation numbers ofA⊗τB belongs to i. If A is a C∗-algebra, i is a stable Calkin space, s is an s-number function, and ai, bi ∈ A, i = 1, 2, . . . ,m are such that s(π(ai)), s(π(bi)) ∈ i, i = 1, 2, . . . ,m for some faithful representation π of A then s ( ∑m i=1 Mai,bi) ∈ i. The converse implication holds if and only if the ideal of compact elements of A has finite spectrum. We also prove a quantitative version of a result of Ylinen. introduction Let A be a C∗-algebra. If a, b ∈ A we denote by Ma,b the operator on A given byMa,b(x) = axb. An operator Φ : A → A is called elementary if Φ = ∑m i=1 Mai,bi for some ai, bi ∈ A, i = 1, . . . , m. Let H be a separable Hilbert space and B(H) the C∗-algebra of all bounded linear operators on H. A theorem of Fong and Sourour [10] asserts that an elementary operator Φ on B(H) is compact if and only if there exists a representation ∑m i=1 MAi,Bi of Φ such that the symbols Ai, Bi, i = 1, . . . , m of Φ are compact operators. An element a of a C∗-algebra A is called compact if the operator Ma,a is compact. Ylinen [21] showed that a ∈ A is a compact element if and only if there exists a faithful *-representation π of A such that the operator π(a) is compact. The result of Fong and Sourour was extended by Mathieu [14] who showed that if A is a prime C∗-algebra, then an elementary operator Φ on A is compact if and only if there exist compact elements ai, bi ∈ A, i = 1, . . . , m, such that Φ = ∑m i=1 Mai,bi. Recently Timoney [20] extended this result to general C ∗algebras. In this paper we investigate quantitative aspects of the above results. It is well-known that a bounded operator on a Banach space is compact if and only if its Kolmogorov numbers form a null sequence. In our approach we use the more general notion of the s-function introduced by Pietsch and the theory of ideals Date: 2 October 2008. 2000 Mathematics Subject Classification. Primary 46L05; Secondary 47B47, 47L20. 1 2 M. ANOUSSIS, V. FELOUZIS AND I. G. TODOROV of B(H) developed by von Neumann, Schatten, Calkin and others. A detailed study of these notions is presented in the monographs [16],[5], [11] and [18]. In Section 1 of the paper we recall the definitions of Calkin spaces and the basic properties of s-functions. In Section 2 we study stable Calkin spaces. An analogous property for ideals of B(H), called “tensor product closure property”, was considered by Weiss [22]. We give a necessary and sufficient condition for the stability of a singly generated Calkin space. We also provide a sufficient condition for the stability of a Lorentz sequence space. If a, b ∈ A and C is a C∗-subalgebra of A such that Ma,b(C) ⊆ C we denote by MC a,b the operator C → C defined by MC a,b(x) = axb. In Section 3 we prove inequalities relating s-number functions of the operators Ma,b and M C a,b. In Section 4 we study elementary operators acting on B(H). Some of our results can be presented in a more general setting. Namely, we show that if τ is any tensor norm and A,B ∈ B(H) are such that s(A), s(B) belong to a stable Calkin space i then the sequence of approximation numbers of A ⊗τ B belongs to i. A result of this type for i = lp,q was proved by König in [12] who used it to prove results concerning tensor stability of s-number ideals in Banach spaces. We also show that if Φ is an elementary operator on B(H), i is a stable Calkin space and s is an s-function then s(Φ) ∈ i if and only if there exist Ai, Bi ∈ B(H), i = 1, . . . , m, such that Φ = ∑m i=1 MAi,Bi and s(Ai), s(Bi) ∈ i. It is well known that all s-functions coincide for operators acting on Hilbert spaces. It follows from our result that if Φ is an elementary operator on B(H), i is a stable Calkin space and s, s′ are s-number functions, then the sequence s(Φ) belongs to i if and only if the sequence s′(Φ) belongs to i. In Section 5 we study elementary operators acting on C∗-algebras. We show that if A is a C∗-algebra, i is a stable Calkin space, s is an s-number function, and ai, bi ∈ A, i = 1, . . . , m, are such that s(π(ai)), s(π(bi)) ∈ i, i = 1, . . . , m for some faithful representation π of A then s( ∑m i=1 Mai,bi) ∈ i. The converse implication holds if and only if the ideal of compact elements of A has finite spectrum. Finally, we prove that if a ∈ A and d(Ma,a) ∈ i for some Calkin space i then s(ρ(a)) ∈ i, where ρ is the reduced atomic representation of A. This result may be viewed as a quantitative version of the aformentioned result of Ylinen. 1. calkin spaces and s-functions In this section we recall some notions and results concerning the ideal structure of the algebra of all bounded linear operators acting on a separable Hilbert space. We also recall the definition of an s-function. We will denote by B the class of all bounded linear operators between Banach spaces. If X and Y are Banach spaces, we will denote by B(X ,Y) the set of all bounded linear operators from X into Y . If X = Y we set B(X ) = B(X ,X ). S-NUMBERS OF ELEMENTARY OPERATORS ON C∗-ALGEBRAS 3 Ideals of B(X ) or, more generally, of a normed algebra A, will be proper, twosided and not necessarily norm closed. By K(X ) (resp. F(X )) we denote the ideal of all compact (resp. finite rank) operators on X . By ‖T‖ we denote the operator norm of a bounded linear operator T . We denote by l∞ the space of all bounded complex sequences, by c0 the space of all sequences in l∞ converging to 0 and by c00 the space of all sequences in c0 that are eventually zero. The space of all p-summable complex sequences is denoted by lp. For a subspace  of l∞, we let  be the subset of  consisting of all sequences with non-negative terms. We denote by c0 the subset of c + 0 consisting of all non-negative decreasing sequences. If α = (αn) ∞ n=1 and β = (βn) ∞ n=1 are sequences of real numbers, we write α ≤ β if αn ≤ βn for each n ∈ N. For every α = (αn)n=1 ∈ c0 we define α = (α n) ∞ n=1 ∈ c0 to be the sequence given by α 1 = max{|αn| : n ∈ N}, α 1 + · · ·+ α n = max {∑ i∈I |αi| : I ⊆ N, |I| = n } . The sequence α is the rearrangement of the sequence (|αn|)n=1 in decreasing order including multiplicities. A Calkin space [18] is a subspace i of c0 which has the following property: If α ∈ i and β ∈ c0 then β ≤ α∗ implies that β ∈ i. Let H be a separable Hilbert space. If e, f ∈ H we denote by f ∗ ⊗ e the rank one operator on H given by f ∗ ⊗ e(x) = (x, f)e, x ∈ H. Given an operator T ∈ K(H) there exist orthonormal sequences (fn)n=1 ⊆ H and (en)n=1 ⊆ H and a unique sequence (sn(T )) ∞ n=1 ∈ c0 such that