Non-commutative inequalities in operator algebras
by
George Popescu
University of Craiova

Standard positivity and inequalities in operator algebras are centered around the selfadjoint part of commutative C ^* -algebras. Completely positive maps could be viewed as a step towards inequalities that may be called ``non-commutative''. Following this idea we consider Hilbert C <sup>*</sup> -modules with ``completely positive'' inner products, i.e. instead of the standard positivity required <x, y> >= 0 , we ask that

æ ç ç ç ç ç ç ç è <x1, x1> <x1, x2> * ... <x1, xn> * <x2, x1> <x2, x2> ... <x2, xn> * ... ... ... ... <x1, xn> <x2, xn> ... <xn, xn> ö ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø i, j=1, n  >= 0

for all square matrices of all orders n >= 1. The standard inner product on a C ^* -algebra <a, b> = b ^* a verifies the above condition. We may equaly consider the ''skew'' condition using the transpose of the above matrices:

æ ç ç ç ç ç ç ç è <x1, x1> <x1, x2> ... <x1, xn> <x2, x1> * <x2, x2> ... <x2, xn> ... ... ... ... <x1, xn> * <x2, xn> * ... <xn, xn> ö ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø i, j=1, n  >= 0

The ''skew'' inner product on a C ^* -algebra < a, b > =ab ^* verifies this second condition. This leads to the idea that there are two types of completely positive inner products (say ''left'' and ''right'' type) which emphasize the non-commutative nature of the inequalities obtained. We give a characterization for inner products on a C ^* -algebra to decide whether they are left or right completely positive or neither. For n=2 the positivity of the above matrix gives a non-commutative version of the classical Schwarz inequality for inner products. Other classical inequalities have also corresponding non-commutative versions.

References

[1] Lance E.C., Hilbert C ^* -modules, Cambridge Univ. Press, 1995.

[2] Paulsen V.I., Completely bounded maps and dilations, Longman Scientific 1986.

Date received: May 30, 2002