Math. Phys. seminar at SISSA

[Emanuele Tuillier Illingworth]

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MATHEMATICAL PHYSICS SECTOR ACTIVITIES
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Michael Skeide
Università del Molise

Titolo:
Bimodules over B(H) vs Bimodules over L^\infty --- or essentially 
commutative
bimodules vs the most noncommutative bimodules that exist

Abstract

B(H), the algebra of all bounded operators on a Hilbert space H, is 
often considered the most noncommutative object. However, when we 
consider Hilbert bimodules (more precisely, von Neumann bimodules) over 
B(H), they are all trivial in the sense that they have an orthnormal 
bimodule basis. In other words, they appear as tensor products of B(H) 
with a another Hilbert space. They are also centered in the sense that 
they are generated by their central elements.

On the contrary, Hilbert or von Neumann bimodules over commutative 
algebras are centered, if and only if left and right action coincide. 
Although in commutative algebra it is sometimes a comon habit to 
consider modules over a commutative ring as bimodules (with left action 
equal to right action), in practise, such a behaviour happens almost 
never in practical application. For instance, such natural objects as 
Brownian motion or Ornstein-Uhlenbeck processes give naturally rise to 
bimodules over commutative algebras that do not have a single nonzero 
element that would commute with the algebra. Bimodules over commutative 
algebras are, therefore, among the most noncommutative objects one can 
imagine: Classical probability in terms of Hilbert modules reveals to be 
true quantum probability.


Thu. 02 April 2009 at 14.30 in SISSA lecture room D



-- 
Dr. Emanuele Tuillier Illingworth
Secretary of Mathematical Physics and Elementary Particle sectors
(+39) 040 37 87 598
SISSA
Via Beirut 2-4
Trieste, ITALY
tuillier at sissa.it