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We show that every separable complex L1-predual space X is contractively complemented in the CAR-algebra. As an application we deduce that the open unit ball of X is a bounded homogeneous symmetric domain. 1. The main results. This paper is concerned with L1-predual spaces (over C) and their connection to C∗-algebras. Let X be a separable Banach space (over C) such that X∗ is isometrically isomorphic to an L1-space. For example, if K is a compact Hausdorff space then the Banach space C(K) of all complex-valued continuous functions on K has a dual which is isometrically isomorphic to an L1-space. However there are many examples whereX is not isomorphic to any complemented subspace of any C(K)-space ([2]). On the other hand, a separable L1-predual is always isomorphic to a quotient of a C(K)-space ([4]). C(K) is a commutative C∗-algebra. H. P. Rosenthal conjectured that the non-commutative situation might be different, i.e. that X might always be complemented in a (non-commutative) C∗-algebra. Furthermore there might even be a universal C∗-algebra containing all separable L1-preduals as complemented subspaces. The CAR-algebra A might be a candidate for this. The aim of the paper is to confirm Rosenthal’s conjecture. Fix a sequence of integers 0 ≤ m1 < m2 < . . . Then we define the C∗-algebra A(mn) as follows: For a Hilbert space H let L(H) be the space of all linear and bounded operators on H. Moreover letMn = L(ln 2 ) be the space of all n× n-matrices (over C). Identify B ∈ M2mn with 2000 Mathematics Subject Classification: 46B20, 46L05, 46B04, 46G20.

Every separable L1-predual is complemented in a C*-algebra

Every separable L₁-predual is complemented in a C^*-algebra