Elementary operators and invariant subalgebras

We provide an example of an elementary operator which leaves invariant a nest algebra but which cannot be written as a finite sum of multiplications each of which leaves the nest algebra invariant. We also prove that the given operator lies in the completely bounded norm closure of the linear span of the multiplications which leave the nest algebra invariant. Let A be a Banach algebra and let A, B e A. A multiplication (operator) A B: A -A is defined by the equation (A. B)(X) = AXB, X C A. An elementary operator R is an operator of the form R = E 1 Ai Bi, where n is finite. Clearly, R can have different representations E Ct Dt. The length I of R is defined to be the smallest number of multiplication terms required for any representation of R. Elementary operators have been studied in great detail in recent years. Both their spectral theory and their structural theory have received much attention (cf. [AF, Cu, F]). The range inclusion problem for elementary operators [F] can be stated as follows: If I is an ideal in A, and if Ran(R) C I, must R be expressible as A Bi with Ran(Ai Bi) C I, i = 1, 2,.. ., n? Positive results have been obtained with certain hypotheses [AF]. Let C be a subalgebra of A. David Larson has suggested the study of 8c (A), the algebra of elementary operators on A which leave C invariant. With A = B(12) and C = TO, the nest algebra of upper triangular operators with respect to the standard basis in 12, we provide in Theorem 3 an example of an R C 'c (A) which has no representation Z7=1 Ct * Dt such that Ct C ?C(A), t = 1,2,.. ., m. (This example was discussed, without proof, in [Co].) We also demonstrate in Proposition 4 that R lies in the completely bounded norm closed linear span of the length one elements of ?c (A). T, is a commutative subspace lattice (CSL) algebra. Recall that a CSL algebra is a reflexive algebra whose lattice of invariant subspaces is commutative. For finite dimensional CSL algebras there is no operator like the operator R of the previous paragraph. Proposition 1. Let C be a CSL subalgebra of Mn, and let R e ?c(Mn) (R 7& 0). Then there exists a representation R = Em I Ct . Dt such that Ct . Dt E ?C(Mn), 1 2, and let Tn C Mn be the algebra of upper triangular n x n matrices. There exists an R E ?T (MMn) of length two such that for any representation R = t2L Ct Dt with each Ct Dt C ?Tn (Mn), we must have m > log2 n. Proof. Let {a,, a2,... an-1 } be a sequence of distinct complex numbers and define the following four matrices in M7n: n n A1 = ZEji, B1 = Eaj_iEj(j_j), i=2 j=2 n n A2 = ak-1Ekk, B2 = E-E,(1_l, k=2 1=2 Consider the elementary operator R = A1 * B1 + A2 B2. By Proposition 1, R expands as n n Z Z (aj-i ai-1)(Eii Ej(j-.)); i=2 j=2 Rf ?E n (Mn) since it takes strictly upper triangular matrix units to upper triangular matrix units, and the diagonal matrix units are annihilated. Since the ak's are all distinct, any expansion of R must include nonzero multiples of every one of the terms Eii E (j-,) for i j,2 < i,j < n. On the other hand, if CD C ?Tn(Mn) is to contain a nonzero multiple of Eii E (j-i) then C must contain a nonzero multiple of Eiand D must contain a nonzero multiple of Ej(j-1). Thus, if CD contains nonzero multiples of Eii Ej(j-1) and Ekk E.(1_1) then it must also contain a nonzero multiple of Ekk * Ej(j-1). If C. D is to leave T7 invariant, we must have k : j. This content downloaded from 207.46.13.102 on Sat, 30 Jul 2016 05:48:51 UTC All use subject to http://about.jstor.org/terms ELEMENTARY OPERATORS AND INVARIANT SUBALGEBRAS 3439 Now suppose that R = Z:' 1 Ct . Dt. For each i let Wi be the set of t's for which Ct . Dt contains a nonzero multiple of Eii * Ek(k1)We claim the sets Wi are all distinct. If i :& j, the last paragraph implies that there is a to C Wi such that Cto Dto contains a nonzero multiple of Eii-Ej(j-1). But, also by the last paragraph, this means that Cto * Dto cannot contain nonzero multiples of any operator of the form Ejj * El(1i-). This tells us that to v Wj, so Wj =& Wi. Thus, the total number of sets Wi (that is, n 1) is no greater than the total number of distinct nonempty subsets of {1, 2, ... , m}, which is 2' 1. The result follows. El Theorem 3. There exists an elementary operator R C E8 (B(12)) of length two which cannot be written as a finite sum Z1m I Ct Dt of multiplications with Ct Dt C rT (B(12)) for I < t <m. Proof. Let a = (a1, a2,...) be a sequence in lo with the ai's all distinct. Form the following four operators in B(12):