Banach and operator space structure of C ∗ - algebras

Introduction A C∗-algebra is often thought of as the non-commutative generalization of a C(K)-space, i.e. the space of continuous functions on some locally compact Hausdorff space, vanishing at infinity. We go one step further, for we seek to compare the Banach space properties of C∗-algebras and their naturally complemented subspaces, with those of C(K)-spaces. (For a recent survey on C(K) spaces or Banach spaces, see [Ro3].) This involves the recent theory of operator spaces, or quantized Banach spaces. We briefly review this concept at the beginning of section 1; the reader is referred to [ER] and [Pi] for in depth coverage. For the definition of complete boundedness of maps, complete isomorphisms, etc., see section 1. Our presentation here is expository; only simple deductions are given, often from rather deep principles. Section 1 shows how C∗-algebras share certain Banach space properties of C(K)-spaces. For example, we state Pfitzner’s theorem that C∗-algebras have Pe lczyński’s property (V ) as Theorem 1.1, and then deduce that nonreflexive completely complemented subspaces of C∗-algebras contain complete isomorphic copies of c0 in Corollary 1.2. We then use an old result