A noncommutative L1-mean ergodic theorem

✍ Scribed by Charles Radin

• Publisher: Elsevier Science
• Year: 1976
• Tongue: English
• Weight: 84 KB
• Volume: 21
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(76)90118-3

✦ Synopsis

Notes

A Noncommutative L,-Mean Ergodic Theorem

We show that operator algebras, as opposed to Banach lattices, provide the more natural structure for &-mean ergodic theory.

We will note below how a result in operator algebras yields a significant generalization of the following classical L,-mean ergodic theorem of Kakutani.

THEOREM [l]. Let (X, M) be a jinite measure space, let L = L,(S, m) (the complex Banach space), and let T be a positive linear contraction on L such that TQ = II. Theu for every f in L there exists f in L such that in the norm topology N-l 1,/N c T"f,;;tf, ?L=O First we need some notation. If L is the predual of a IV*-algebra, let L,& (resp. L,) be the real (resp. positive) part of L. Then we call an element 24 of L-a unit for L if is norm-dense in L, . (A unit for L is just a faithful element u of L, , faithful in the sense that for f in L*, ~(f *f) = 0 implies f = 0.) Note that Ii is a unit for L,(X, m).

Considering L,(X, m) as the predual of the commutative IV*-algebra L&X, m), the following is a noncommutative generalization of Kakutani's result.