## Banach Journal of Mathematical Analysis

### Index computation for amalgamated products of product systems

Mithun Mukherjee

#### Abstract

The notion of amalgamation of product systems has been introduced in [7] which generalizes the concept of Skeide product, introduced by Skeide, of two product systems via a pair of normalized units. In this paper we show that amalgamation leads to a setup where a product system is generated by two subsystems and conversely whenever a product system is generated by two subsystems, it could be realized as an amalgamated product. We parameterize all contractive morphism from a Type I product system to another Type I product system and compute index of amalgamated product through contractive morphisms.

#### Article information

Source
Banach J. Math. Anal., Volume 5, Number 1 (2011), 148-166.

Dates
First available in Project Euclid: 14 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1313362987

Digital Object Identifier
doi:10.15352/bjma/1313362987

Mathematical Reviews number (MathSciNet)
MR2738527

Zentralblatt MATH identifier
1223.46061

#### Citation

Mukherjee, Mithun. Index computation for amalgamated products of product systems. Banach J. Math. Anal. 5 (2011), no. 1, 148--166. doi:10.15352/bjma/1313362987. https://projecteuclid.org/euclid.bjma/1313362987

#### References

• W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (1989), no. 409, iv+66 pp.
• S.D. Barreto, B.V.R. Bhat, V. Liebscher and M. Skeide, Michael Type I product systems of Hilbert modules, J. Funct. Anal. 212 (2004), no. 1, 121–181.
• B.V.R. Bhat, An index theory for quantum dynamical semigroups, Trans. of Amer. Math. Soc. 348 (1996), 561–583.
• B.V.. Bhat, Minimal dilations of quantum dynamical semigroups to semigroups of endomorphisms of $C^{\ast}$ algebras, J. Ramanujan Math. Soc. 14 (1999), 109–124.
• B.V.R. Bhat, Cocycles of CCR flows, Mem. Amer. Math. Soc. 149 (2001), no. 709, x+114 pp.
• B.V.R. Bhat, V. Liebscher and M. Skeide, A problem of Powers and the product of spatial product systems, Quantum probability and related topics, 93–106, QPPQ: Quantum Probab. White Noise Anal., 23, World Sci. Publ., Hackensack, NJ, 2008.
• B.V.R. Bhat and M. Mukherjee, Inclusion systems and amalgamated products of product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 1, 1–26.
• B.V.R. Bhat and M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 519–575.
• A. Guichardet, Symmetric Hilbert spaces and related topics, Springer Lecture Notes in Math. 261, Berlin, 1972.
• R.T. Powers, Addition of spatial $E\sb 0$-semigroups, Operator algebras, quantization, and non commutative geometry, 281–298, Contemp. Math. 365, Amer. Math. Soc., Providence, RI, 2004.
• K.R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products and Central Limit Theorems of Probability Theory, Springer Lecture Notes in Math. 272, Berlin, 1972.
• O.M. Shalit and B. Solel, Subproduct systems, Documenta Math. 14 (2009), 801–868.
• M. Skeide, Commutants of von Neumann modules, representations of $B^a(E)$ and other topics related to product systems of Hilbert modules, Advances in quantum dynamics (South Hadley, MA, 2002), 253–262, Contemp. Math., 335, Amer. Math. Soc., Providence, RI, 2003.
• M. Skeide, The index of (white) noises and their product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 617–655.
• M. Skeide, The Powers sum of spatial CDP-semigroups and CP-semigroups, preprint, arXiv: 0812.0077v1.