Difference between revisions of "Tomita-Takesaki theory"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (moved Tomita–Takesaki theory to Tomita-Takesaki theory: ascii title) |
(No difference)
|
Revision as of 18:54, 24 March 2012
M. Tomita [a4] defined the notion of a left Hilbert algebra as follows: An involutive algebra over the field
of complex numbers, with involution
, that admits an inner product
satisfying the following conditions:
i) the mapping is continuous for every
;
ii) for all
;
iii) is total in the Hilbert space
obtained by completion of
.
iv) is a closeable conjugate-linear operator in
. Let
be a left Hilbert algebra in a Hilbert space
. For any
, let
denote the unique continuous linear operator on
such that
,
. The von Neumann algebra
generated by
is called the left von Neumann algebra of
. Let
be the closure of the mapping
and let
be the polar decomposition of
. Then
is an isometric involution and
is a non-singular positive self-adjoint operator in
satisfying
and
;
and
are called the modular operator and the modular conjugation operator of
, respectively. Let
denote the set of vectors
such that the mapping
is continuous. For any
, denote by
the unique continuous extension of
to
. Let
be the set of vectors
such that the mapping
is continuous. For any
, denote by
the unique continuous extension of
to
. Then
is a left Hilbert algebra in
, equipped with the multiplication
and the involution
, and
is equivalently contained in
, that is,
and they have the same modular (conjugation) operators. The set
is a left Hilbert algebra which is equivalently contained in
and
is a complex one-parameter group of automorphisms of
, called the modular automorphism group. It satisfies the conditions:
a) ,
,
;
b) ,
,
;
c) ,
;
d) ,
, is an analytic function on
. Such a left Hilbert algebra is called a modular Hilbert algebra (or Tomita algebra). Using the theory of modular Hilbert algebras, M. Tomita proved that
and
for all
. This theorem is called the Tomita fundamental theorem. M. Takesaki [a3] arranged and deepened this theory and connected this theory with the Haag–Hugenholtz–Winnink theory [a2] for equilibrium states for quantum statistical mechanics. After that, Tomita–Takesaki theory was developed by A. Connes [a1], H. Araki, U. Haagerup and the others, and has contributed to the advancement of the structure theory of von Neumann algebras, non-commutative integration theory, and quantum physics. Using an integral formula relating the resolvent of the modular operator
with the operators
, A. van Daele [a5] has simplified a discussion in the complicated Tomita–Takesaki theory.
References
[a1] | A. Connes, "Une classification des facteurs de type III" Ann. Sci. Ecole Norm. Sup. , 6 (1973) pp. 133–252 |
[a2] | R. Haag, N.M. Hugenholts, M. Winnink, "On the equilibrium states in quantum mechanics" Comm. Math. Phys. , 5 (1967) pp. 215–236 |
[a3] | M. Takesaki, "Tomita's theory of modular Hilbert algebras and its applications" , Lecture Notes Math. , 128 , Springer (1970) |
[a4] | M. Tomita, "Standard forms of von Neumann algebras" , The Vth Functional Analysis Symposium of Math. Soc. Japan, Sendai (1967) |
[a5] | A. Van Daele, "A new approach to the Tomita–Takesaki theory of generalized Hilbert algebras" J. Funct. Anal. , 15 (1974) pp. 378–393 |
Tomita-Takesaki theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tomita-Takesaki_theory&oldid=23089