## Admissibility of Weights on Non-normed $*$-Algebras

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- by S. J. Bhatt, A. Inoue and H. Ogi PDF
- Trans. Amer. Math. Soc.
**351**(1999), 4629-4656 Request permission

## Abstract:

The notion of weights on (topological) $*$-algebras is defined and studied. The primary purpose is to define the notions of admissibility and approximate admissibility of weights, and to investigate when a weight is admissible or approximately admissible. The results obtained are applied to vector weights and tracial weight on unbounded operator algebras, as well as to weights on smooth subalgebras of a C$^*$-algebra.## References

- G. R. Allan,
*A spectral theory for locally convex algebras*, Proc. London Math. Soc. (3)**15**(1965), 399–421. MR**176344**, DOI 10.1112/plms/s3-15.1.399 - G. R. Allan,
*On a class of locally convex algebras*, Proc. London Math. Soc. (3)**17**(1967), 91–114. MR**205102**, DOI 10.1112/plms/s3-17.1.91 - J.-P. Antoine, H. Ogi, A. Inoue, and C. Trapani,
*Standard generalized vectors in the space of Hilbert-Schmidt operators*, Ann. Inst. H. Poincaré Phys. Théor.**63**(1995), no. 2, 177–210 (English, with English and French summaries). MR**1357495** - A. R. Collar,
*On the reciprocation of certain matrices*, Proc. Roy. Soc. Edinburgh**59**(1939), 195–206. MR**8** - Bruce A. Barnes,
*The properties $^{\ast }$-regularity and uniqueness of $C^{\ast }$-norm in a general $^{\ast }$-algebra*, Trans. Amer. Math. Soc.**279**(1983), no. 2, 841–859. MR**709587**, DOI 10.1090/S0002-9947-1983-0709587-7 - Subhash J. Bhatt,
*Representability of positive functionals on abstract star algebras without identity with applications to locally convex $^{\ast }$-algebras*, Yokohama Math. J.**29**(1981), no. 1, 7–16. MR**649234** - Frank F. Bonsall and John Duncan,
*Complete normed algebras*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR**0423029** - A. H. Chamseddine, J. Fröhlich, and O. Grandjean,
*The gravitational sector in the Connes-Lott formulation of the standard model*, J. Math. Phys.**36**(1995), no. 11, 6255–6275. MR**1355907**, DOI 10.1063/1.531243 - Fabio Cipriani, Daniele Guido, and Sergio Scarlatti,
*A remark on trace properties of $K$-cycles*, J. Operator Theory**35**(1996), no. 1, 179–189. MR**1389649** - Alain Connes,
*Noncommutative geometry and reality*, J. Math. Phys.**36**(1995), no. 11, 6194–6231. MR**1355905**, DOI 10.1063/1.531241 - Jacques Dixmier,
*$C^*$-algebras*, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR**0458185** - P. G. Dixon,
*Generalized $B^{\ast }$-algebras*, Proc. London Math. Soc. (3)**21**(1970), 693–715. MR**278079**, DOI 10.1112/plms/s3-21.4.693 - J. M. G. Fell and R. S. Doran,
*Representations of $^*$-algebras, locally compact groups, and Banach $^*$-algebraic bundles. Vol. 1*, Pure and Applied Mathematics, vol. 125, Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups and algebras. MR**936628** - S. P. Gudder and R. L. Hudson,
*A noncommutative probability theory*, Trans. Amer. Math. Soc.**245**(1978), 1–41. MR**511398**, DOI 10.1090/S0002-9947-1978-0511398-0 - Atsushi Inoue,
*An unbounded generalization of the Tomita-Takesaki theory*, Publ. Res. Inst. Math. Sci.**22**(1986), no. 4, 725–765. MR**871264**, DOI 10.2977/prims/1195177629 - Atsushi Inoue,
*$O^*$-algebras in standard system*, Math. Nachr.**172**(1995), 171–190. MR**1330628**, DOI 10.1002/mana.19951720113 - Atsushi Inoue and Witold Karwowski,
*Cyclic generalized vectors for algebras of unbounded operators*, Publ. Res. Inst. Math. Sci.**30**(1994), no. 4, 577–601. MR**1308958**, DOI 10.2977/prims/1195165790 - A. Inoue and H. Ogi,
*Regular weights on algebras of unbounded operators,*J. Math. Soc. Japan,**50**(1998), 227-252. - Theodore W. Palmer,
*Banach algebras and the general theory of $^*$-algebras. Vol. I*, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR**1270014**, DOI 10.1017/CBO9781107325777 - Theodore W. Palmer,
*Spectral algebras*, Rocky Mountain J. Math.**22**(1992), no. 1, 293–328. MR**1159960**, DOI 10.1216/rmjm/1181072812 - N. Christopher Phillips,
*Inverse limits of $C^*$-algebras*, J. Operator Theory**19**(1988), no. 1, 159–195. MR**950831** - James D. Powell,
*Representations of locally convex $^{\ast }$-algebras*, Proc. Amer. Math. Soc.**44**(1974), 341–346. MR**361803**, DOI 10.1090/S0002-9939-1974-0361803-X - Robert T. Powers,
*Self-adjoint algebras of unbounded operators*, Comm. Math. Phys.**21**(1971), 85–124. MR**283580** - Charles E. Rickart,
*General theory of Banach algebras*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101** - Konrad Schmüdgen,
*Unbounded operator algebras and representation theory*, Operator Theory: Advances and Applications, vol. 37, Birkhäuser Verlag, Basel, 1990. MR**1056697**, DOI 10.1007/978-3-0348-7469-4 - Kunimichi Takesue,
*Standard representations induced by positive linear functionals*, Mem. Fac. Sci. Kyushu Univ. Ser. A**37**(1983), no. 2, 211–225. MR**722244**, DOI 10.2206/kyushumfs.37.211

## Additional Information

**S. J. Bhatt**- Affiliation: Department of Mathematics Sardar Patel University Vallabh Vidyanagar 388120 Gujarat, India
**A. Inoue**- Email: sm010888@ssat.fukuoka-u.ac.jp
**H. Ogi**- Affiliation: Department of Applied Mathematics, Fukuoka University, Nanakuma, Jonan-ku, Fukuoka, 814-80 Japan
- Email: sm037255@ssat.fukuoka-u.ac.jp
- Received by editor(s): February 23, 1997
- Published electronically: April 12, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 4629-4656 - MSC (1991): Primary 46K10, 47D40
- DOI: https://doi.org/10.1090/S0002-9947-99-02414-9
- MathSciNet review: 1637133