Dynamical Systems Implemented by Isomorphic Groups of Unitaries

Mosadeq, Maysam

doi:10.22130/scma.2023.559758.1158

Document Type : Research Paper

Author

Maysam Mosadeq

Department of Mathematics, Larestan Branch, Islamic Azad University, Larestan, Iran.

https://doi.org/10.22130/scma.2023.559758.1158

Abstract

Let $\varphi:A\to B$ be an isomorphism of $C^*$-algebras and $I$ be an ideal of $A.$ Introducing the concepts of unitary equivalent and the implemented Finsler modules, we show that the $\frac{A}{I}$-module $\frac{E}{E_{I}}$ and the implemented $\frac{B}{\varphi(I)}$-module $\frac{F}{F_{\varphi(I)}}$ are unitary equivalent. We also, establish a one to one correspondence between the groups $U(E)$ and $U(F)$ of unitaries on full Finsler modules $E$ and $F,$ respectively. Finally, we explain regularized dynamical systems and apply the aforementioned one to one correspondence to prove that each regularized dynamical system in $U(E)$ implements a regularized dynamical system in $U(F).$

Keywords

20.1001.1.23225807.2023.20.3.4.0

References

[1] R. Abazari, A. Niknam and M. Hassani, Existence of ground states for approximately inner two parameter $C_0$-groups on $C^*$-algebras, Bull. Iranian Math. Soc., 42 (2016), pp. 435-446.

[2] Gh. Abbaspour, M.S. Moslehian and A. Niknam, Dynamical systems on Hilbert $C^*$-modules, Bull. Iranian Math. Soc., 31 (2005), pp. 25-35.

[3] Gh. Abbaspour and M. Skeide, Generators of dynamical systems on Hilbert modules, Commun. Stoch. Anal., 1 (2007), pp. 193-207.

[4] M. Amyari and A. Niknam, A note on Finsler modules, Bull. Iranian Math. Soc., 29 (2003), pp. 77-81.

[5] M. Amyari and A. Niknam, On homomorphisms of Finsler modules, Intern. Math. Journal., 3 (2003), pp. 277-281.

[6] D. Bakic and B. Guljas, On a class of module maps of Hilbert $C^*$-modules, Math. Commun., 7 (2002), pp. 177-192.

[7] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol 1, Springer-Verlag, 1997.

[8] E.B. Davies and M.M.H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. Lond. Math. Soc., 55 (1987), pp. 181-208.

[9] R. DeLaubenfels, Integrated semigroups, $C$-semigroups and the abstract Cauchy problem, Semigroup Forum., 12 (1990), pp. 83-95.

[10] K.J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000.

[11] R. Hassanniah, M. Amyari and M. Hassani, Imprimitivity Finsler $C^*$-bimodules, Nonlinear Funct. Anal. Appl.,
19 (2014), pp. 479-487.

[12] A. Hossein, M. Hassani and A. Niknam, Generalized $\sigma$-derivation on Banach algebras, Bull. Iranian Math. Soc., 37 (2011), pp. 81-94.

[13] I. Kaplansky, Modules over operator algebras, Amer. J. Math., 75 (1953), pp. 839-858.

[14] E.C. Lance, Hilbert $C^*$-Modules, Lecture Note Series 210, Cambridge Univ. Press, Cambridge, 1995.

[15] I. Miyadera and N. Tanaka, Exponentially bounded $C$-semigroups and generation of semigroups, J. Math. Anal. Appl , 143 (1989), pp. 358-378.

[16] M. Mirzavaziri and M.S. Moslehian, Automatic continuity of $\sigma$-derivations in $C^*$-algebras, Proc. Amer. Math. Soc., 134 (2006), pp. 3319-3327.

[17] M. Mosadeq, Conflating two kinds of derivations to construct the generator of a new dynamical system, in: Proc. 15th Seminar on Differential Equations and Dynamical Systems, Guilan, Iran, 2021, 172-175.

[18] M. Mosadeq, $\sigma$-$C^*$-dynamics of $K(H)$, J. Math. Ext., 16 (2022), pp. 1-22.

[19] M. Mosadeq, M. Hassani and A. Niknam, ($\sigma,\gamma$)-generalized dynamics on modules, J. Dyn. Syst. Geom.
Theor., 9 (2011), pp. 171-184.

[20] M. Mosadeq, M. Hassani and A. Niknam, Approximately inner $\sigma$-dynamics on $C^*$-algebras, J. Mahani. Math. Res. Cent., 1 (2012), pp. 55-63.

[21] M. Mosadeq, M. Hassani and A. Niknam, Approximately quasi inner generalized dynamics on modules, J. Sci. Islam. Repub. Iran., 23 (2012), pp. 245-250.

[22] M. Mosadeq, M. Hassani and A. Niknam, Closability of module $\sigma$-derivations, J. Math. Ext., 6 (2012), pp. 57-69.

[23] A. Niknam, Infinitesimal generators of $C^*$-algebras, Potential Anal., 6 (1997), pp. 1-9.

[24] W.L. Paschke, Inner product modules over $B^*$-algebras, Trans. Amer. Math. Soc.,
182 (1973), pp. 443-468.

[25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[26] J.K. Pedersen, Factorization in $C^*$-algebras, Expo. Math., 16 (1998), pp. 145-156.

[27] N. Phillips and N. Weaver, Modules with norms which take values in a $C^*$-algebra,
Pacific J. Math., 185 (1998) 163-181.

[28] I. Raeburn and D.P. Williams. Morita Equivalence and Continuous-Trace $C^*$-Algebra, Math. Surveys Monogr. 60, AMS, 1998.

[29] M.A. Rieffel. Morita equivalence for $C^*$-algebras and $W^*$-algebras, J. Pure Appl. Algebra., 5 (1974), pp. 51-96.

[30] S. Sakai, Operator Algebras in Dynamical Systems, Cambridge. Univ.
Press, 2010.

Sahand Communications in Mathematical Analysis

Dynamical Systems Implemented by Isomorphic Groups of Unitaries

References

References

Volume 20, Issue 3
April 2023
Pages 51-68

Dynamical Systems Implemented by Isomorphic Groups of Unitaries

References

References

Volume 20, Issue 3April 2023Pages 51-68

Volume 20, Issue 3
April 2023
Pages 51-68