Noncommutative Lp spaces, Operator spaces and Applications
نویسندگان
چکیده
Overview of the field. NoncommutativeLp-spaces are at the heart of this conference. These spaces have a long history going back to pioneering works by von Neumann, Dixmier and Segal. They are the analogues of the classical Lebesgue spaces of pintegrable functions, where now functions are replaced by operators. These spaces have been investigated for operator algebras with a trace, and then around 1980 generalizations to type III von Neumann algebras have appeared (Kosaki, Haagerup, Terp, Hilsum). These algebras have no trace and therefore the integration theory has to be entirely redone. These generalizations were motivated and made possible by the great progress in operator algebra theory, in particular the Tomita-Takesaki theory and Connes’s spectacular results on the classification of type III factors. Since the early nineties and the arrival of new theories like those of operator spaces and free probability, noncommutative integration is living another period of stimulating new developments. In particular, noncommutative Khintchine and martingale inequalities have opened new perspectives. It is well-known nowadays that the theory of noncommutativeLp-spaces is intimately related with many other fields such as:
منابع مشابه
Operator space embedding of Lq into Lp
The idea of replacing functions by linear operators, the process of quantization, goes back to the foundations of quantum mechanics and has a great impact in mathematics. This applies for instance to representation theory, operator algebra, noncommutative geometry, quantum and free probability or operator space theory. The quantization of measure theory leads to the theory of Lp spaces defined ...
متن کاملTHE NORM OF SUMS OF INDEPENDENT NONCOMMUTATIVE RANDOM VARIABLES IN Lp(l1)
We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class S q in Sp(lmq ) with optimal order m ∼ n. Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for Lp[0, 1] is min(p, p) (max(p, p)). Simi...
متن کاملA reduction method for noncommutative Lp-spaces and applications
We consider the reduction of problems on general noncommutative Lp-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is an unpublished result of the first-named author which approximates any noncommutative Lpspace by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products...
متن کامل9 NONCOMMUTATIVE L p - SPACE AND OPERATOR SYSTEM
We show that noncommutative Lp-spaces satisfy the axioms of the (nonunital) operator system with a dominating constant 2 1 p . Therefore, noncommutative Lpspaces can be embedded into BpHq 2 1 p -completely isomorphically and complete order isomorphically.
متن کامل