Inner Products and Module Maps of Hilbert C∗-modules
نویسنده
چکیده
Let E and F be two Hilbert C∗-modules over C∗-algebras A and B, respectively. Let T be a surjective linear isometry from E onto F and φ a map from A into B. We will prove in this paper that if the C∗-algebras A and B are commutative, then T preserves the inner products and T is a module map, i.e., there exists a ∗-isomorphism φ between the C∗-algebras such that 〈Tx, Ty〉 = φ(〈x, y〉), and T (xa) = T (x)φ(a). In case A or B is noncommutative C∗-algebra, T may not satisfy the equations above in general. We will also give some condition such that T preserves the inner products and T is a module map.
منابع مشابه
$C^{*}$-semi-inner product spaces
In this paper, we introduce a generalization of Hilbert $C^*$-modules which are pre-Finsler modules, namely, $C^{*}$-semi-inner product spaces. Some properties and results of such spaces are investigated, specially the orthogonality in these spaces will be considered. We then study bounded linear operators on $C^{*}$-semi-inner product spaces.
متن کاملThe study on controlled g-frames and controlled fusion frames in Hilbert C*-modules
Controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces. Fusion frames and g-frames generalize frames. Hilbert C*-modules form a wide category between Hilbert spaces and Banach spaces. Hilbert C*-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a C*...
متن کاملFrames in super Hilbert modules
In this paper, we define super Hilbert module and investigate frames in this space. Super Hilbert modules are generalization of super Hilbert spaces in Hilbert C*-module setting. Also, we define frames in a super Hilbert module and characterize them by using of the concept of g-frames in a Hilbert C*-module. Finally, disjoint frames in Hilbert C*-modules are introduced and investigated.
متن کاملOn the 2-Adjointable Operators and Superstability of Them Between 2-Pre Hilbert $C^*$-module Spaces
In this paper, first, we introduce the new concept of 2-inner product on Banach modules over a $C^*$-algebra. Next, we present the concept of 2-linear operators over a $C^*$-algebra. Our result improve the main result of the paper Z. Lewandowska. In the final of this paper, we define the notions 2-adjointable mappings between 2-pre Hilbert C*-modules and prove supperstability of them ...
متن کاملArens regularity and derivations of Hilbert modules with the certain product
Let $A$ be a $C^*$-algebra and $E$ be a left Hilbert $A$-module. In this paper we define a product on $E$ that making it into a Banach algebra and show that under the certain conditions $E$ is Arens regular. We also study the relationship between derivations of $A$ and $E$.
متن کامل