Operator Spaces

Operator space theory provides a synthesis of Banach space theory with the non-commuting 'quantum' variables of operator algebra theory, and it has led to exciting new approaches in both disciplines. The authors begin by giving completely elementary proofs of the basic representation theorems for abstract operator spaces and their mappings. This is followed by a discussion of tensor products and the analogue of Grothendieck's approximation property. In the next section, the operator space analogues of the nuclear, integral and absolutely summing mappings are discussed. In what is perhaps the deepest part of the book, the authors present the remarkable 'non-classical' phenomena that occur when one considers local reflexivity and exactness for operator spaces. They have included the recent proof that, in contrast to C*-algebras themselves, C*-algebraic duals are always locally reflexive. In the final section of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory.

Operator space theory provides a synthesis of Banach space theory with the non-commuting 'quantum' variables of operator algebra theory, and it has led to exciting new approaches in both disciplines. The authors begin by giving completely elementary proofs of the basic representation theorems for abstract operator spaces and their mappings. This is followed by a discussion of tensor products and the analogue of Grothendieck's approximation property. In the next section, the operator space analogues of the nuclear, integral and absolutely summing mappings are discussed. In what is perhaps the deepest part of the book, the authors present the remarkable 'non-classical' phenomena that occur when one considers local reflexivity and exactness for operator spaces. They have included the recent proof that, in contrast to C*-algebras themselves, C*-algebraic duals are always locally reflexive. In the final section of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory.

Booknews

An overview of quantization of Banach space theory whose goal is to explain the deep analogy between linear spaces of bounded functions and linear spaces of bounded operators. Effros (UCLA) and Ruan (U. of Illinois at Urbana-Champaign) are confident that the extraordinary array of techniques developed by Banach space theorists will have many applications in non-commutative analysis, and that operator space theory in turn will provide Banach space theorists with new vistas for research. "More generally," they write, "it is our hope that this new formalism will help to unravel some of the difficulties associated with quantization." Annotation c. Book News, Inc., Portland, OR (booknews.com)

University of California, Los Angeles

University of Illinois at Urbana-Champaign