Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

  • Subhash J. Bhatt
Regular Articles


Given anm-tempered strongly continuous action α of ℝ by continuous*-automorphisms of a Frechet*-algebraA, it is shown that the enveloping ↡-C *-algebraE(S(ℝ, A, α)) of the smooth Schwartz crossed productS(ℝ,A , α) of the Frechet algebra A of C-elements ofA is isomorphic to the Σ-C *-crossed productC *(ℝ,E(A), α) of the enveloping Σ-C *-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK *(S(ℝ, A, α)) =K *(C *(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC *-algebra defined by densely defined differential seminorms is given.


Frechet*-algebra enveloping Σ-C*-algebra smooth crossed product m-tempered action K-theory differential structure inC*-algebras 


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  1. [1]
    Bhatt S J, Karia D J, Topological algebras withC *-enveloping algebras,Proc. Indian Acad. Sci. (Math. Sci.) 102 (1992) 201–215MATHMathSciNetGoogle Scholar
  2. [2]
    Bhatt S J, Toplogical*-algebras withC *-enveloping algebrasII,Proc. Indian Acad. Sci. (Math. Sci.) 111 (2001) 65–94MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Bhatt S J, Inoue A and Ogi H, UnboundedC *-seminorms and unboundedC *-spectral algebras,J. Operator Theory 45 (2001) 53–80MATHMathSciNetGoogle Scholar
  4. [4]
    Bhatt S J, Inoue A and Ogi H, Spectral invariance,K-theory isomorphism and an application to the differential structure ofC *-algebras,J. Operator Theory 49 (2003) 389–405MATHMathSciNetGoogle Scholar
  5. [5]
    Blackadar B and Cuntz J, Differential Banach algebra norms and smooth subalgebras ofC*-algebras,J. Operator Theory 26 (1991) 255–282MATHMathSciNetGoogle Scholar
  6. [6]
    Brooks R M, On representingF *-algebras,Pacific J. Math. 39 (1971) 51–69MATHMathSciNetGoogle Scholar
  7. [7]
    Connes A, An analogue of the Thom isomorphism for crossed products of aC *-algebra by an action ofR, Adv. Math. 39 (1981) 31–55MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Fragoulopoulou M, Symmetric topological*-algebras: Applications,Schriften Math. Inst. Uni. Munster, 3 Ser., Heft 9 (1993)Google Scholar
  9. [9]
    Phillips N C, Inverse limits of C*-algebras,J. Operator Theory 19 (1988) 159–195MATHMathSciNetGoogle Scholar
  10. [10]
    Phillips N C,K-theory for Frechet algebras,Int. J. Math. 2(1) (1991) 77–129MATHCrossRefGoogle Scholar
  11. [11]
    Phillips N C and Schweitzer L B, RepresentableK-theory for smooth crossed products byR andZ, Trans. Am. Math. Soc. 344 (1994) 173–201MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Pedersen G K,C *-algebras and their automorphism groups,London Math. Soc. Monograph No. 14 (London, New York, San Francisco: Academic Press) (1979)Google Scholar
  13. [13]
    Rickart C E, General theory of Banach algebras (D. Van Nostrand Publ. Co.) (1960)Google Scholar
  14. [14]
    Schweitzer L B, Densem-convex Frechet algebras of operator algebra crossed products by Lie groups,Int. J. Math. 4 (1993) 601–673MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Schweitzer L B, Special invariance of dense subalgebras of operator algebras,Int. J. Math. 4 (1993) 289–317MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Schweitzer L B, A short proof thatM n(A) is local ifA is local and Frechet,Int. J. Math. 3 (1992) 581–589MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2006

Authors and Affiliations

  1. 1.Department of MathematicsSardar Patel UniversityVallabh VidyanagarIndia

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