Representations of $C^{\ast}$-ternary rings
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
It is proved that there is a one to one correspondence between representations of $C^{\ast}$-ternary ring $M$ and $C^{\ast}$-algebra $\mathcal{A}(M)$. We discuss primitive and modular ideals of a $C^{\ast}$-ternary ring and prove that a closed ideal $I$ is primitive or modular if and only if so is the ideal $\mathcal{A}(I)$ of $\mathcal{A}(M)$. We also show that a closed ideal in $M$ is primitive if and only if it is the kernel of some irreducible representation of $M$. Lastly, we obtain approximate identity characterization of strongly quasi-central $C^{\ast}$-ternary ring and the ideal structure of the TRO $ V\otimes^{\text{\rm tmin}} B$ for a $C^{\ast}$-algebra $B$.
Abstract
It is proved that there is a one to one correspondence between representations of $C^{\ast}$-ternary ring $M$ and $C^{\ast}$-algebra $\mathcal{A}(M)$. We discuss primitive and modular ideals of a $C^{\ast}$-ternary ring and prove that a closed ideal $I$ is primitive or modular if and only if so is the ideal $\mathcal{A}(I)$ of $\mathcal{A}(M)$. We also show that a closed ideal in $M$ is primitive if and only if it is the kernel of some irreducible representation of $M$. Lastly, we obtain approximate identity characterization of strongly quasi-central $C^{\ast}$-ternary ring and the ideal structure of the TRO $ V\otimes^{\text{\rm tmin}} B$ for a $C^{\ast}$-algebra $B$.
참고문헌 (23)
- F. Abadie, "Applications of ternary rings to C∗-algebras", Adv. Oper. Theory, 2(3), pp.293~317 (2017)[DOI]
- L. Arambaˇsi´c, "Irreducible representations of Hilbert C∗-modules", Math. Proc. R. Ir. Acad, 105A, pp.11~24 (2005)[DOI]
- E. Blanchard, "Non-simple purely infinite C∗-algebras : the Hausdorff case", J. Funct. Anal, 207(2), pp.461~513 (2004)[DOI]
- K. R. Davidson, "C∗-algebras by example, Fields Institute Monographs, 6" (1996), American Mathematical Society
- E. G. Effros, "On injectivity and nuclearity for operator spaces", Duke Math. J, 110(3), pp.489~521 (2001)[DOI]
- E. G. Effros, "Operator spaces, London Mathematical Society Monographs. New Series, 23" (2000), The Clarendon Press, Oxford University Press
- M. Hamana, "Injective envelopes of dynamical systems, in Operator algebras and operator theory (Craiova, 1989), Pitman Res. Notes Math. Ser., 271", pp.69~77 (1992), Longman Sci. Tech
- M. Hamana, "Triple envelopes and Silov boundaries of operator spaces ˇ", Math. J. Toyama Univ, 22, pp.77~93 (1999)
- L. A. Harris, "A generalization of C∗-algebras", Proc. London Math. Soc. (3), 42(2), pp.331~361 (1981)[DOI]
- M. R. Hestenes, "A ternary algebra with applications to matrices and linear transformations", Arch. Rational Mech. Anal, 11, pp.138~194 (1962)[DOI]
- Y. F. Jin, "Hom-derivations in C*-ternary algebras, Acta Math. Sin. (Engl. Ser.)", Acta Math. Sin. (Engl. Ser. ), 36(9), pp.1025~1038 (2020)[DOI]
- E. Kaniuth, "A course in commutative Banach algebras, Graduate Texts in Mathematics, 246" (2009), Springer
- A. Kansal, "Inductive limit in the category of TRO", Ann. Funct. Anal, 11(3), pp.748~760 (2020)[DOI]
- M. Kaur, "Local properties of ternary rings of operators and their linking C∗-algebras", J. Funct. Anal, 195(2), pp.262~305 (2002)[DOI]
- A. J. Lazar, "The space of ideals in the minimal tensor product of C∗-algebras", Math. Proc. Cambridge Philos. Soc, 148(2), pp.243~252 (2010)[DOI]
- M. S. Moslehian, "Almost derivations on C∗-ternary rings", Bull. Belg. Math. Soc. Simon Stevin, 14(1), pp.135~142 (2007)[DOI]
- G. J. Murphy, "C∗-Algebras and Operator Theory" (1990), Academic Press, Inc.
- C. Park, "Additive functional equations and partial multipliers in C∗-algebras", Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 113(3), pp.2261~2275 (2019)[DOI]
- R. Pluta, "Ternary operator categories", J. Math. Anal. Appl, 505(2) (2021)[DOI]
- M. Skeide, "Ideal submodules versus ternary ideals versus linking ideals", Algebr. Represent. Theory, 25(2), pp.359~386 (2022)[DOI]
- M. Takesaki, "Theory of Operator Algebras. I" (1979), Springer-Verlag
- D. P. Williams, "Crossed products of C∗-algebras, Mathematical Surveys and Monographs, 134" (2007), American Mathematical Society
- H. Zettl, "A characterization of ternary rings of operators", Adv. in Math, 48(2), pp.117~143 (1983)[DOI]