In this paper we define and study chain conditions for Hilbert C＊-modules through their C＊-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that thes...In this paper we define and study chain conditions for Hilbert C＊-modules through their C＊-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C＊-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C＊-algebra with a faithful conditional expectation.展开更多
Let H be a separable Hilbert space, BH（I）, B（H） and K（H） the sets of all Bessel sequences {fi}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and i...Let H be a separable Hilbert space, BH（I）, B（H） and K（H） the sets of all Bessel sequences {fi}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms αH： BH（I）→ B（l2）,β ： BH（I） → B（H）, respectively, so that BH（I） becomes a unital C＊-algebra under each kind of multiplication and involution. It is proved that the two C＊-algebras （BH（I）, o, #） and （BH（I）, ., ＊） are ＊-isomorphic. It is also proved that the set FH （I） of all frames for H is a unital multiplicative semi-group and the set RH（I） of all Riesz bases for H is a self-adjoint multiplicative group, as well as the set KH （I） ：=β-1 （K（H）） is the unique proper closed self-adjoint ideal of the C＊-algebra BH （I）.展开更多
文摘In this paper we define and study chain conditions for Hilbert C＊-modules through their C＊-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C＊-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C＊-algebra with a faithful conditional expectation.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1140135911371012+2 种基金11301318)China Postdoctoral Science Foundation(Grant No.2014M552405)the Natural Science Research Program of Shaanxi Province(Grant No.2014JQ1010)
文摘Let H be a separable Hilbert space, BH（I）, B（H） and K（H） the sets of all Bessel sequences {fi}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms αH： BH（I）→ B（l2）,β ： BH（I） → B（H）, respectively, so that BH（I） becomes a unital C＊-algebra under each kind of multiplication and involution. It is proved that the two C＊-algebras （BH（I）, o, #） and （BH（I）, ., ＊） are ＊-isomorphic. It is also proved that the set FH （I） of all frames for H is a unital multiplicative semi-group and the set RH（I） of all Riesz bases for H is a self-adjoint multiplicative group, as well as the set KH （I） ：=β-1 （K（H）） is the unique proper closed self-adjoint ideal of the C＊-algebra BH （I）.
