Representation of Contractively Complemented Hilbertian Operator Spaces on the Fock Space

The operator spaces Hk n 1 ≤ k ≤ n, generalizing the row and column Hilbert spaces, and arising in the authors’ previous study of contractively complemented subspaces of C∗-algebras, are shown to be homogeneous and completely isometric to a space of creation operators on a subspace of the anti-symmetric Fock space. The completely bounded Banach-Mazur distance from Hk n to row or column space is explicitly calculated. Introduction and Preliminaries A well-known result of Friedman and Russo ([4, Theorem 2]) states that if a subspace X of a C-algebra A is the range of a contractive projection on A, then X is isometric to a JC-triple, that is, a norm closed subspace of B(H,K) stable under the triple product abc + cba. If X is atomic (in particular, finite-dimensional), then it is isometric to a direct sum of Cartan factors of types 1 to 4. The authors showed in [7] that this latter result fails, as it stands, in the category of operator spaces. In that paper, we defined a family of n-dimensional Hilbertian operator spaces H n , 1 ≤ k ≤ n, generalizing the row and column Hilbert spaces Rn and Cn and showed that in the above result, if X is atomic, the word “isometric” can be replaced by “completely semi-isometric,” provided the spacesH n are allowed as summands along with the Cartan factors ([7, Theorem 2]). It is pointed out in [7] that the space H n is contractively complemented in some B(K), and for 1 < k < n, is not completely (semi-)isometric to either of the Cartan factors B(C,C) = H n or B(C,C) = H n , and that these spaces appeared in a slightly different form and context in [1]. It is also shown in [7, Theorem 3] that finite dimensional JCtriples which are contractively complemented in a C-algebra can be classified up to complete isometry. In this paper, we study the operator space structure of the spaces H n. Besides being a generalization of the row and column Hilbert spaces, as shown in Lemma 2.1 below, they are completely isometric to the span of creation operators on a subspace of the anti-symmetric Fock space. Thus they are related to the operator space denoted by Φn in [9, section 9.3], which is the span of the creation operators on the full anti-symmetric Fock space. Φn is the unique operator space which is completely isometric to the span of n operators satisfying the canonical anticommutation relations (CAR), [9, Theorem 9.3.1], and ∩k=1H n is completely isometric Date: January 1, 1994 and, in revised form, June 22, 1994. 1991 Mathematics Subject Classification. Primary 46L07.