|
Operator-valued free Fisher information and modular frames
Author(s):
Bin
Meng;
Maozheng
Guo;
Xiaohong
Cao
Journal:
Proc. Amer. Math. Soc.
133
(2005),
3087-3096.
MSC (2000):
Primary 46L54, 42C15
Posted:
April 25, 2005
MathSciNet review:
2159789
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We introduce the operator-valued free Fisher information for a random variable in an operator-valued noncommutative probability space and point out its relations to the amalgamated freeness. Using M. Frank and D. Larson's modular frame notion we can construct the conjugate variable for an operator-valued semicircle variable with conditional expectation covariance. Then we obtain its free Fisher information and show it is equal to the index of the conditional expectation. At last the conjugate variable with respect to a modular frame operator for a semicircle variable is also constructed.
References:
-
- 1.
- M.
Baillet, Y. Denizeau and J. -F.Havet, Indice
d'une
espérance conditionelle. Compositio
Math. 66 (1988) 199-236.
MR
0945550 (90e:46050)
- 2.
-
T. M. Cover and Joy A. Thomas, Elements
of information
theory. John Wiley & Sons, Inc.
Chichester, 1976.
MR
1122806 (92g:94001)
- 3.
-
M. Frank, E. Kirchberg , On conditional
expectations of
finite index J. Operator Theory , 40
(1998), 87-111.
MR
1642530 (2000k:46080)
- 4.
- M.
Frank, D. R. Larson, A module frame
concept for
Hilbert C*-modules. in: Functional and
Harmonic Analysis of Wavelets (San
Antonio, TX, Jan. 1999), A.M.S., Providence, R.I.,
Contemp. Math. 247
(2000), 207-233.
MR
1738091 (2001b:46094)
- 5.
- M.
Frank, D. R. Larson , Frames in Hilbert
C*-modules
and C*-algebras. J. Operator Theory,
48 (2002), 273-314.
MR
1938798 (2003i:42040)
- 6.
- L.
Ge, Applications of free entropy to
finite von
Neumann algebras. II Annals of Mathematics,
147(1998), 143-157.
MR
1609522 (99c:46068)
- 7.
- V.
F. Jones, Index for subfactors.
Invent. Math.
72
(1983), 1-25.
MR
0696688 (84d:46097)
- 8.
-
C. Lance, Hilbert
 Modules. LMS note
series, 210, Cambridge University
Press, 1995.
MR
1325694 (96k:46100)
- 9.
-
A. Nica, D.Shlyakhtenko, R.Speicher, Operator-valued
distributions. 1. Characterizations of freeness.
IMRN 2002: 29 (2002)
1509-1538.
MR
1907203 (2003f:46105)
- 10.
- A.
Nica, D. Shlyakhtenko, R. Speicher, Some
minimization
problems for the free analogue of the Fisher information.
Advances in Mathematics 141
(1999) 282-321.
MR
1671766 (2000b:46120)
- 11.
-
S. Popa, Markov traces on universal Jones
algebras
and subfactors of finite index. Invent.Math
111 (1993),
375-405.
MR
1198815 (94c:46128)
- 12.
- S.
Popa, Classsification of subfactors and
their
endomorphisms. CMBS 86
AMS (1995).
MR
1339767 (96d:46085)
- 13.
-
R. Speicher, Combinatorial theory of
the free product
with amalgamation and operator-valued free probability
theory.
Memoirs of AMS 627 (1998).
MR
1407898 (98i:46071)
- 14.
-
D. Shlyakhtenko, Free entropy with respect
to a
completely positive map. Amer. J. Math
122 (2000), 45-81.
MR
1737257 (2001f:46102)
- 15.
-
V. S. Sunder, An invitation to von Neumann
algebras. Springer-Verlag, New York,
1987.
MR
0866671 (88f:46115)
- 16.
- Y.
Watatani, Index for
 subalgebras. Memoirs
AMS
vol. 83, 424 (1990).
MR
0996807 (90i:46104)
- 17.
- D.
Voiculescu, K. J. Dykema, A. Nica, Free
random
variables. CRM Monograph Series, Vol.
1, AMS, 1992.
MR
1217253 (94c:46133)
- 18.
-
D. Voiculescu, Operations on certain
Non-commutative
operator-valued random variables. Ast
 risque 232
(1995),
243-275.
MR
1372537 (97b:46081)
- 19.
-
D. Voiculescu, The analogues of entropy
and of Fisher's
information measure in free probability theory
III: The absence of Cartan
subalgebras, Geometric and Functional Analysis.
Vol.6. 1(1996).
MR
1371236 (96m:46119)
- 20.
-
D. Voiculescu, The analogues of entropy
and of Fisher's
information measure in free probability theory
V. Noncommutative Hibert
Transforms. Inventiones Mathematicae,
132(1998), 189-227.
MR
1618636 (99d:46087)
- 21.
-
D. Voiculescu, The analogues of entropy
and of Fisher's
information measure in free probability theory
VI. Liberation and mutual free
information. Advances in Mathematics
146(1999), 101-166.
MR
1711843 (2001a:46064)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
46L54, 42C15
Retrieve articles in all Journals with
MSC (2000):
46L54, 42C15
Additional Information:
Bin
Meng
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
Address at time of publication:
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic of China
Email:
b.meng@nuaa.edu.cn
Maozheng
Guo
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
Xiaohong
Cao
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
DOI:
10.1090/S0002-9939-05-08111-6
PII:
S 0002-9939(05)08111-6
Keywords:
Conjugate variable,
free Fisher information,
Hilbert $C^\ast$-module,
modular frame
Received by editor(s):
February 12, 2004
Received by editor(s) in revised form:
June 7, 2004 and June 8, 2004
Posted:
April 25, 2005
Communicated by:
David R. Larson
Copyright of article:
Copyright
2005,
American Mathematical Society
|
