Frames on Hilbert C*-modules have been defined for unital C*algebras by Frank and Larson [5] and operator-valued frames on a Hilbert space have been studied in [8]. The goal of this paper is to introduce operatorvalued frames on a Hilbert C*-module for a σ-unital C*-algebra. Theorem 1.4 reformulates the definition given in [5] in terms of a series of rank-one operators converging in the strict ...

Let $(R,m)$ be a commutative Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d$, and let $I$ be an ideal of definition for $M$. In this paper, we extend cite[Corollary 10(4)]{P} and also we show that if $M$ is a Cohen-Macaulay $R$-module and $d=2$, then $lambda(frac{widetilde{I^nM}}{Jwidetilde{I^{n-1}M}})$ does not depend on $J$ for all $ngeq 1$, where $J$ is a minimal ...

In this paper, we state some results on product of operators with closed ranges and we solve the operator equation $TXS^*-SX^*T^*= A$ in the general setting of the adjointable operators between Hilbert $C^*$-modules, when $TS = 1$. Furthermore, by using some block operator matrix techniques, we nd explicit solution of the operator equation $TXS^*-SX^*T^*= A$.

We give a new Banach module characterization of W *-modules, also known as selfdual Hilbert C *-modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of W *-modules, to the setting where the operator algebras are σ-weakly closed algebras of operators on a Hilbert space. That is, we find the appropriate weak* topology variant of our earlier notion of r...

The goal of the present paper is a short introduction to a general module frame theory in C*-algebras and Hilbert C*modules. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and other bounded module operators with suitable ranges. We...

We show in this paper that the module structure and the orthogonality structure of a Hilbert C∗-module determine its inner product structure. Let A be a C∗-algebra, and E and F be Hilbert A-modules. Assume Φ : E → F is an A-module map satisfying 〈Φ(x),Φ(y)〉A = 0 whenever 〈x, y〉A = 0. Then Φ is automatically bounded. In case Φ is bijective, E is isomorphic to F . More precisely, let JE be the cl...

Let C denote the category of Hilbert modules which are similar to contractive Hilbert modules. It is proved that if H0, H ∈ C and if H1 is similar to an isometric Hilbert module, then the sequence 0 → H0 → H → H1 → 0 splits. Thus the isometric Hilbert modules are projective in C. It follows that ExtC (K, H) = 0, whenever n > 1, for H, K ∈ C. In addition, it is proved that (Hilbert modules simil...

We show that every infinite-dimensional commutative unital C∗-algebra has a Hilbert C∗-module admitting no frames. In particular, this shows that Kasparov’s stabilization theorem for countably generated Hilbert C∗-modules can not be extended to arbitrary Hilbert C∗-modules.