Tomita–Takesaki theory
None
In the theory of von Neumann algebras , a part of the mathematical field of functional analysis , Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors , and has led to a good structure theory for these previously intractable objects.
The theory was introduced by Minoru Tomita ( 1967 ) , but his work was hard to follow and mostly unpublished, and little notice was taken of it until Masamichi Takesaki ( 1970 ) wrote an account of Tomita's theory. [1]
Modular automorphisms of a state
Suppose that
M
is a von Neumann algebra acting on a
Hilbert space
H
, and Ω is a
cyclic and separating vector
of
H
of norm 1. (
Cyclic
means that
MΩ
is dense in
H
, and
separating
means that the map from
M
to
MΩ
is injective.) We write
for the vector state
of
M
, so that
H
is constructed from
using the
Gelfand–Naimark–Segal construction
. Since Ω is separating,
is faithful.
We can define a (not necessarily bounded) antilinear operator
S
0
on
H
with dense domain
MΩ
by setting
for all
m
in
M
, and similarly we can define a (not necessarily bounded) antilinear operator
F
0
on
H
with dense domain
M'Ω
by setting
for
m
in
M
′
, where
M
′
is the
commutant
of
M
.
These operators are closable, and we denote their closures by S and F = S *. They have polar decompositions
where
is an antilinear isometry of
H
called the
modular conjugation
and
is a positive (hence, self-adjoint) and densely defined operator called the
modular operator
.
Commutation theorem
The main result of Tomita–Takesaki theory states that:
for all t and that
the commutant of M .
There is a 1-parameter group of
modular automorphisms
of
M
associated with the state
, defined by
.
The modular conjugation operator
J
and the 1-parameter unitary group
satisfy
and
The Connes cocycle
The modular automorphism group of a von Neumann algebra M depends on the choice of state φ. Connes discovered that changing the state does not change the image of the modular automorphism in the outer automorphism group of M . More precisely, given two faithful states φ and ψ of M , we can find unitary elements u t of M for all real t such that
so that the modular automorphisms differ by inner automorphisms, and moreover u t satisfies the 1-cocycle condition
In particular, there is a canonical homomorphism from the additive group of reals to the outer automorphism group of M , that is independent of the choice of faithful state.
KMS states
The term KMS state comes from the Kubo–Martin–Schwinger condition in quantum statistical mechanics .
A
KMS state
on a von Neumann algebra
M
with a given 1-parameter group of automorphisms α
t
is a state fixed by the automorphisms such that for every pair of elements
A
,
B
of
M
there is a bounded continuous function
F
in the strip
0 ≤ Im(
t
) ≤ 1
, holomorphic in the interior, such that
Takesaki and Winnink showed that any (faithful semi finite normal) state
is a KMS state for the 1-parameter group of modular automorphisms
. Moreover, this characterizes the modular automorphisms of
.
(There is often an extra parameter, denoted by β, used in the theory of KMS states. In the description above this has been normalized to be 1 by rescaling the 1-parameter family of automorphisms.)
Structure of type III factors
We have seen above that there is a canonical homomorphism δ from the group of reals to the outer automorphism group of a von Neumann algebra, given by modular automorphisms. The kernel of δ is an important invariant of the algebra. For simplicity assume that the von Neumann algebra is a factor. Then the possibilities for the kernel of δ are:
- The whole real line. In this case δ is trivial and the factor is type I or II.
- A proper dense subgroup of the real line. Then the factor is called a factor of type III 0 .
- A discrete subgroup generated by some x > 0. Then the factor is called a factor of type III λ with 0 < λ = exp(−2 π / x ) < 1, or sometimes a Powers factor.
- The trivial group 0. Then the factor is called a factor of type III 1 . (This is in some sense the generic case.)
Left Hilbert algebras
The main results of Tomita–Takesaki theory were proved using left and right Hilbert algebras. [2]
A
left Hilbert algebra
is an algebra
with involution
x
→
x
♯
and an inner product (·,·) such that
-
Left multiplication by a fixed
a
∈
is a bounded operator.
- ♯ is the adjoint; in other words ( xy , z ) = ( y , x ♯ z ) .
- The involution ♯ is preclosed.
-
The subalgebra spanned by all products
xy
is dense in
w.r.t. the inner product.
A right Hilbert algebra is defined similarly (with an involution ♭) with left and right reversed in the conditions above.
A (unimodular) Hilbert algebra is a left Hilbert algebra for which ♯ is an isometry, in other words ( x , y ) = ( y ♯ , x ♯ ) . In this case the involution is denoted by x * instead of x ♯ and coincides with modular conjugation J . This is the special case of Hilbert algebras . The modular operator is trivial and the corresponding von Neumann algebra is a direct sum of type I and type II von Neumann algebras.
Examples:
-
If
M
is a von Neumann algebra acting on a Hilbert space
H
with a cyclic separating unit vector
v
, then put
= Mv and define ( xv )( yv ) = xyv and ( xv ) ♯ = x * v . The vector v is the identity of
, so
is a unital left Hilbert algebra. [3]
- If G is a locally compact group, then the vector space of all continuous complex functions on G with compact support is a right Hilbert algebra if multiplication is given by convolution, and x ♭ ( g ) = x ( g −1 )* . [3]
For a fixed left Hilbert algebra
, let
H
be its Hilbert space completion. Left multiplication by
x
yields a bounded operator λ(
x
) on
H
and hence a *-homomorphism λ of
into
B
(
H
). The *-algebra
generates the von Neumann algebra
Tomita's key discovery concerned the remarkable properties of the closure of the operator ♯ and its polar decomposition. If S denotes this closure (a conjugate-linear unbounded operator), let Δ = S * S , a positive unbounded operator. Let S = J Δ 1/2 denote its polar decomposition . Then J is a conjugate-linear isometry satisfying [4]
-
and
.
Δ is called the modular operator and J the modular conjugation .
In Takesaki (2003 , pp. 5–17) , there is a self-contained proof of the main commutation theorem of Tomita-Takesaki:
-
and
The proof hinges on evaluating the operator integral: [5]
By the
spectral theorem
,
[6]
that is equivalent to proving the equality with
e
x
replacing Δ; the identity for scalars follows by contour integration. It reflects the well-known fact that, with a suitable normalisation, the function
is its own Fourier transform.
Notes
- ↑ Takesaki 2003 , pp. 38–39
- ↑ Takesaki 2003 , pp. 1–39
- 1 2 Takesaki 2003 , p. 2
- ↑ Takesaki 2003 , p. 4
- ↑ Takesaki 2013 , pp. 15–16 harvnb error: no target: CITEREFTakesaki2013 ( help )
- ↑ Rudin 1991 .
References
- Borchers, H. J. (2000), "On revolutionizing quantum field theory with Tomita's modular theory", Journal of Mathematical Physics , 41 (6): 3604–3673, Bibcode : 2000JMP....41.3604B , doi : 10.1063/1.533323 , MR 1768633
- Bratteli, O. ; Robinson, D.W. (1987), Operator Algebras and Quantum Statistical Mechanics 1, Second Edition , Springer-Verlag, ISBN 3-540-17093-6
- Connes, Alain (1973), "Une classification des facteurs de type III" (PDF) , Annales Scientifiques de l'École Normale Supérieure , 4e série, 6 (2): 133–252, doi : 10.24033/asens.1247
- Connes, Alain (1994), Non-commutative geometry , Boston, MA: Academic Press , ISBN 978-0-12-185860-5 [ permanent dead link ]
- Dixmier, Jacques (1981), von Neumann algebras , North-Holland Mathematical Library, vol. 27, translated by F. Jellet, Amsterdam: North-Holland, ISBN 978-0-444-86308-9 , MR 0641217
- Inoue, A. (2001) [1994], "Tomita–Takesaki theory" , Encyclopedia of Mathematics , EMS Press
- Longo, Roberto (1978), "A simple proof of the existence of modular automorphisms in approximately finite-dimensional von Neumann algebras" , Pacific J. Math. , 75 : 199–205, doi : 10.2140/pjm.1978.75.199 , hdl : 2108/19146
- Nakano, Hidegorô (1950), "Hilbert algebras", The Tohoku Mathematical Journal , Second Series, 2 : 4–23, doi : 10.2748/tmj/1178245666 , MR 0041362
- Pedersen, G.K. (1979), C* algebras and their automorphism groups , London Mathematical Society Monographs, vol. 14, Academic Press, ISBN 0-12-549450-5
- Rieffel, M.A.; van Daele, A. (1977), "A bounded operator approach to Tomita–Takesaki theory", Pacific J. Math. , 69 : 187–221, doi : 10.2140/pjm.1977.69.187
- Rudin, Walter (1991). Functional Analysis . International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math . ISBN 978-0-07-054236-5 . OCLC 21163277 .
- Shtern, A.I. (2001) [1994], "Hilbert algebra" , Encyclopedia of Mathematics , EMS Press
- Summers, S. J. (2006), "Tomita–Takesaki Modular Theory", in Françoise, Jean-Pierre; Naber, Gregory L.; Tsun, Tsou Sheung (eds.), Encyclopedia of mathematical physics , Academic Press/Elsevier Science, Oxford, arXiv : math-ph/0511034 , Bibcode : 2005math.ph..11034S , ISBN 978-0-12-512660-1 , MR 2238867
- Sunder, V. S. (1987), An Invitation to von Neumann Algebras , Universitext, Springer , doi : 10.1007/978-1-4613-8669-8 , ISBN 978-0-387-96356-3
- Strătilă, Şerban; Zsidó, László (1979), Lectures on von Neumann algebras. Revision of the 1975 original. , translated by Silviu Teleman, Tunbridge Wells: Abacus Press, ISBN 0-85626-109-2
- Strătilă, Şerban (1981), Modular theory in operator algebras , translated by Şerban Strătilă, Tunbridge Wells: Abacus Press, ISBN 0-85626-190-4
- Takesaki, M. (1970), Tomita's theory of modular Hilbert algebras and its applications , Lecture Notes Math., vol. 128, Springer, doi : 10.1007/BFb0065832 , ISBN 978-3-540-04917-3
- Takesaki, Masamichi (2003), Theory of operator algebras. II , Encyclopaedia of Mathematical Sciences, vol. 125, Berlin, New York: Springer-Verlag , ISBN 978-3-540-42914-2 , MR 1943006
- Tomita, Minoru (1967), "On canonical forms of von Neumann algebras", Fifth Functional Analysis Sympos. (Tôhoku Univ., Sendai, 1967) (in Japanese), Tôhoku Univ., Sendai: Math. Inst., pp. 101–102, MR 0284822
- Tomita, M. (1967), Quasi-standard von Neumann algebras , mimographed note, unpublished
Basic concepts | |
---|---|
Main results | |
Special Elements/Operators | |
Spectrum | |
Decomposition | |
Spectral Theorem | |
Special algebras | |
Finite-Dimensional | |
Generalizations | |
Miscellaneous | |
Examples | |
Applications |
|
Basic concepts | |
---|---|
Main results | |
Other results | |
Maps | |
Examples |
Banach space
topics
|
|
---|---|
Types of Banach spaces | |
Banach spaces are: | |
Function space Topologies | |
Linear operators | |
Operator theory | |
Theorems |
|
Analysis | |
Types of sets |
|
Subsets / set operations | |
Examples | |
Applications |
Spaces |
|
|||
---|---|---|---|---|
Theorems | ||||
Operators | ||||
Algebras | ||||
Open problems | ||||
Applications | ||||
Advanced topics | ||||