# American Institute of Mathematical Sciences

2015, 2015(special): 965-973. doi: 10.3934/proc.2015.0965

## Exact lumping of feller semigroups: A $C^{\star}$-algebras approach

 1 Max-Planck-Institut for Mathematics in the Sciences, Inselstrasse 22, Leipzig, D-04103, Germany

Received  September 2014 Revised  December 2014 Published  November 2015

In this note we analyze a particular exact lumping of Feller semigroups in the context of $C^{\star}$-algebras, in order to pass from a space of functions defined on a locally compact Hausdorff space ${X}$ to a space of functions defined on a closed subspace ${\mathscr{C}}\subset X$. We want our reduction to preserve the essential properties of the Feller semigroup.
Citation: Lavinia Roncoroni. Exact lumping of feller semigroups: A $C^{\star}$-algebras approach. Conference Publications, 2015, 2015 (special) : 965-973. doi: 10.3934/proc.2015.0965
##### References:
 [1] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators,, Lecture Notes in Mathematics, 1184 (1986). [2] F. Atay and L. Roncoroni, Exact Lumpability of Linear Evolution Equations,, preprint, 109 (2013). [3] R. M. Blumenthal and R.k. Getoor, Markov processes and potential theory,, Pure and Applied Mathematics, 29 (1968). [4] H. Brezis, Analyse fonctionnelle,, (French) [Functional analysis] Thorie et applications. [Theory and applications] Collection Mathmatiques Appliques pour la Matrise. [Collection of Applied Mathematics for the Master's Degree] Masson, (1983). [5] N. L. Carothers, A Short Course on Banach Space Theory,, London Mathematical Society Student Texts, 64 (2005). [6] J. A. Van Casteren, Markov Processes, Feller Semigroups and Evolution Equations,, Series on Concrete and Applicable Mathematics, 12 (2011). [7] P. G. Coxson, Lumpability and Observability of Linear Systems,, Journal of Mathematical Analysis and Applications, 99 (1984), 435. [8] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolutions Equations,, With contributions by S. Brendle, 194 (2000). [9] L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach,, Linear algebra and its Applications 404 (2005), (2005), 85. [10] E. Kaniuth, A Course in Commutative Banach algebras,, Graduate Texts in Mathematics, 246 (2009). [11] G. Li and H. Rabitz, A general analysis of exact lumping in chemical kinetics,, Chemical Engineering Science, 44 (1989), 1413. [12] G. K. Pedersen, Analysis Now,, Graduate Texts in Mathematics, 118 (1989). [13] W. Rudin, Functional Analysis,, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., (1973). [14] J. Toth, G. Li, H. Rabitz and A. S. Tomlin, The effect of lumping and expanding on kinetic differential equations,, SIAM J. Appl. Math. 57 No.6 (1997), 57 (1997), 1531. [15] J. Wei and J. C. W. Kuo, A Lumping Analysis in Monomolecular Reaction Systems,, Ind. Eng. Chem. Fundamen., 8 (1969), 124. [16] Z. Rózsa and J. Tóth, Exact linear lumping in abstract spaces,, Proceedings of the 7th Colloquium on the Qualitative Theory of Differential Equations, (). [17] J. H. Zwart, Geometric Theory for Infinite Dimensional Systems,, Lecture Notes in Control and Information Sciences, 115 (1989).

show all references

##### References:
 [1] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators,, Lecture Notes in Mathematics, 1184 (1986). [2] F. Atay and L. Roncoroni, Exact Lumpability of Linear Evolution Equations,, preprint, 109 (2013). [3] R. M. Blumenthal and R.k. Getoor, Markov processes and potential theory,, Pure and Applied Mathematics, 29 (1968). [4] H. Brezis, Analyse fonctionnelle,, (French) [Functional analysis] Thorie et applications. [Theory and applications] Collection Mathmatiques Appliques pour la Matrise. [Collection of Applied Mathematics for the Master's Degree] Masson, (1983). [5] N. L. Carothers, A Short Course on Banach Space Theory,, London Mathematical Society Student Texts, 64 (2005). [6] J. A. Van Casteren, Markov Processes, Feller Semigroups and Evolution Equations,, Series on Concrete and Applicable Mathematics, 12 (2011). [7] P. G. Coxson, Lumpability and Observability of Linear Systems,, Journal of Mathematical Analysis and Applications, 99 (1984), 435. [8] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolutions Equations,, With contributions by S. Brendle, 194 (2000). [9] L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach,, Linear algebra and its Applications 404 (2005), (2005), 85. [10] E. Kaniuth, A Course in Commutative Banach algebras,, Graduate Texts in Mathematics, 246 (2009). [11] G. Li and H. Rabitz, A general analysis of exact lumping in chemical kinetics,, Chemical Engineering Science, 44 (1989), 1413. [12] G. K. Pedersen, Analysis Now,, Graduate Texts in Mathematics, 118 (1989). [13] W. Rudin, Functional Analysis,, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., (1973). [14] J. Toth, G. Li, H. Rabitz and A. S. Tomlin, The effect of lumping and expanding on kinetic differential equations,, SIAM J. Appl. Math. 57 No.6 (1997), 57 (1997), 1531. [15] J. Wei and J. C. W. Kuo, A Lumping Analysis in Monomolecular Reaction Systems,, Ind. Eng. Chem. Fundamen., 8 (1969), 124. [16] Z. Rózsa and J. Tóth, Exact linear lumping in abstract spaces,, Proceedings of the 7th Colloquium on the Qualitative Theory of Differential Equations, (). [17] J. H. Zwart, Geometric Theory for Infinite Dimensional Systems,, Lecture Notes in Control and Information Sciences, 115 (1989).
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