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In the present paper, we examine the perturbation of continuous frames and Riesz-type frames in Hilbert C * -modules. We extend the Casazza–Christensen general perturbation theorem for Hilbert space frames to continuous frames in Hilbert C * -modules. We obtain a necessary condition under which the perturbation of a Riesz-type frame of Hilbert C * -modules remains to be a Riesz-type frame. Also, we examine the effect of duality on the perturbation of continuous frames in Hilbert C * -modules, and we prove that if the operator frame of a continuous frame F is near to the combination of the synthesis operator of a continuous Bessel mapping G and the analysis operator of F, then G is a continuous frame. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is, in the general setting of the adjointable operators on Hilbert C ∗ -modules, to develop two new tools that can be applied to deal with the positive solutions of certain operator equations, the operator norm and related quantities. Among other things, the positivity of a 2 × 2 operator matrix is clarified without any preconditions on its entries, and a generalized version of the mixed Schwarz inequality with a parameter is derived. Numerical examples are provided to illustrate the non-triviality of this newly obtained inequality. [ABSTRACT FROM AUTHOR]

We introduce and study some new uniform structures for Hilbert C ∗ -modules over a C ∗ -algebra A. In particular, we prove that in some cases they have the same totally bounded sets. To define one of them, we introduce a new class of A -functionals: locally adjointable functionals, which have interesting properties in this context and seem to be of independent interest. A relation between these uniform structures and the theory of A -compact operators is established. [ABSTRACT FROM AUTHOR]

Let ℳ be a Hilbert C * -module. In this paper, we show that there is a one-to-one correspondence between all Hilbert C * -module higher derivations { φ n : ℳ → ℳ } n = 0 ∞ with φ 0 = I satisfying φ n (〈 x , y 〉 z) = ∑ i + j + k = n 〈 φ i (x) , φ j (y) 〉 φ k (z) (x , y , z ∈ ℳ , n ∈ ℕ ∪ { 0 }) and all Hilbert C * -module derivations { ψ n : ℳ → ℳ } n = 1 ∞ satisfying ψ n (〈 x , y 〉 z) = 〈 ψ n (x) , y 〉 z + 〈 x , ψ n (y) 〉 z + 〈 x , y 〉 ψ n (z) (x , y , z ∈ ℳ , n ∈ ℕ) , and we show that for every Hilbert C * -module higher derivation { φ n } n = 0 ∞ on ℳ , there exists a unique sequence of Hilbert C * -module derivations { ψ n } n = 1 ∞ on ℳ such that ψ n = ∑ k = 1 n (∑ ∑ j = 1 k r j = n (- 1) k - 1 ⁢ r 1 ⁢ φ r 1 ⁢ φ r 2 ⁢ ... ⁢ φ r k ) for all positive integers n, where the inner summation is taken over all positive integers r j with ∑ j = 1 k r j = n . [ABSTRACT FROM AUTHOR]

We introduce the notion of Hilbert C∗$C^*$‐module independence: Let A$\mathcal {A}$ be a unital C∗$C^*$‐algebra and let Ei⊆E,i=1,2$\mathcal {E}_i\subseteq \mathcal {E},\,\,i=1, 2$, be ternary subspaces of a Hilbert A$\mathcal {A}$‐module E$\mathcal {E}$. Then, E1$\mathcal {E}_1$ and E2$\mathcal {E}_2$ are said to be Hilbert C∗$C^*$‐module independent if there are positive constants m and M such that for every state φi$\varphi _i$ on ⟨Ei,Ei⟩,i=1,2$\langle \mathcal {E}_i,\mathcal {E}_i\rangle ,\,\,i=1, 2$, there exists a state φ on A$\mathcal {A}$ such that mφi(|x|)≤φ(|x|)≤Mφi(|x|2)12,for allx∈Ei,i=1,2.$$\begin{align*} m\varphi _i(|x|)\le \varphi (|x|) \le M\varphi _i{(|x|^2)}^{\frac{1}{2}},\qquad \mbox{for all\nobreakspace }x\in \mathcal {E}_i, i=1, 2. \end{align*}$$We show that it is a natural generalization of the notion of C∗$C^*$‐independence of C∗$C^*$‐algebras. Moreover, we demonstrate that even in the case of C∗$C^*$‐algebras, this concept of independence is new and has a nice characterization in terms of Hahn–Banach–type extensions. We show that if ⟨E1,E1⟩$\langle \mathcal {E}_1,\mathcal {E}_1\rangle$ has the quasi extension property and z∈E1∩E2$z\in \mathcal {E}_1\cap \mathcal {E}_2$ with ∥z∥=1$\Vert z\Vert =1$, then |z|=1$|z|=1$. Several characterizations of Hilbert C∗$C^*$‐module independence and a new characterization of C∗$C^*$‐independence are given. One of characterizations states that if z0∈E1∩E2$z_0\in \mathcal {E}_1\cap \mathcal {E}_2$ is such that ⟨z0,z0⟩=1$\langle z_0,z_0\rangle =1$, then E1$\mathcal {E}_1$ and E2$\mathcal {E}_2$ are Hilbert C∗$C^*$‐module independent if and only if ∥⟨x,z0⟩⟨y,z0⟩∥=∥⟨x,z0⟩∥∥⟨y,z0⟩∥$\Vert \langle x,z_0\rangle \langle y,z_0\rangle \Vert =\Vert \langle x,z_0\rangle \Vert \,\Vert \langle y,z_0\rangle \Vert$ for all x∈E1$x\in \mathcal {E}_1$ and y∈E2$y\in \mathcal {E}_2$. We also provide some technical examples and counterexamples to illustrate our results. [ABSTRACT FROM AUTHOR]

We show that full Hilbert C*-modules over a commutative unital C*-algebra A have fixed point property for nonexpansive mappings if and only if A is finite dimensional. We also show that the same is true for every Hilbert C*-module with unit vectors over an arbitrary unital C*-algebra. In particular, a classification of full Hilbert C*-modules with unit vectors over unital C*-algebras is given via fixed point property for nonexpansive mappings. [ABSTRACT FROM AUTHOR]

1In this paper, we show that the tensor product of a finite number of α-duals for standard gframes (resp. standard fusion frames) is an α-dual for the tensor product of the standard gframes (resp. the standard fusion frames) in the tensor product of Hilbert C*-modules. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the K-Riesz bases and the K-g-Riesz bases in Hilbert C*-modules; we characterize the concept of K-Riesz bases by a bounded below operator and the standard orthonormal basis for Hilbert C*-modules H. Also We give some properties and characterization of K-g-Riesz bases by a bounded surjective operator and g-orthonormal basis for H. Finally we consider the relationships between K-Riesz bases and K-g-Riesz bases. [ABSTRACT FROM AUTHOR]

We study Hilbert C ∗ -modules over a C ∗ -algebra A for which the Banach A-dual module carries a natural structure of Hilbert A-module. In this direction we prove that if A is monotone complete, M and N are Hilbert A-modules, M is self-dual, and both T : M → N and its Banach A-dual T ′ : N ′ → M ′ have trivial kernels and cokernels then M ≅ N ′ . With the help of this result, for a monotone complete C ∗ -algebra A, we prove that the index of any A-Fredholm operator can be calculated as the difference of its kernel and cokernel as in the Hilbert space case. [ABSTRACT FROM AUTHOR]

We introduce the notion of 2-Hilbert modules and give some characterizations. The key idea is to consider a 2-Hilbert space instead of a Hilbert space. We employ our results to obtain reverse Cauchy–Schwarz and triangle inequality in this setting. [ABSTRACT FROM AUTHOR]

In his seminal paper [On the XY-model on two-sided infinite chain, Publ. RIMS Kyoto Univ. 20 (1984) 277–296], Araki introduced an elegant extension of the Jordan–Wigner transformation which establishes a precise connection between quantum spin systems and Fermi lattice gases in one dimension in the so-called infinite system idealization of quantum statistical mechanics. His extension allows in particular for the rigorous study of numerous aspects of the prominent XY chain over the two-sided infinite discrete line without having to resort to a thermodynamic limit procedure at an intermediate or at the final stage. We rigorously review and elaborate this extension from scratch which makes the paper rather self-contained. In the course of the construction, we also present a simple and concrete realization of Araki's crossed product extension. [ABSTRACT FROM AUTHOR]

A number of differences exist between the structures of Hilbert spaces and Hilbert C ∗ -modules. This work aims at studying some properties and characterizations of c-K-g-frames in Hilbert C ∗ -modules. Moreover, the dual c-K-g-Bessel system for a given c-K-g-frame in a Hilbert C ∗ -module is introduced and all dual c-K-g-Bessel systems are characterized. Finally, we provide sufficient conditions under which the sum of two c-K-g-frames in a Hilbert C ∗ -module is a c-K-g-frame. [ABSTRACT FROM AUTHOR]

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of ℒ (H) , where ℒ (H) denotes the set of all adjointable operators on H. The C*-subalgebras of ℒ (H) generated by elements in {P − P ∧ Q, Q − P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π ˜ , X) of i(P, Q, H) such that π ˜ (P − P ∧ Q) = π (P) − π (P) ∧ π (Q) , π ˜ (Q − P ∧ Q) = π (Q) − π (P) ∧ π (Q). When (P, Q) is semi-harmonious, that is, ℛ (P + Q) ¯ and ℛ (2 I − P − Q) ¯ are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I − Q, I − P, H) are unitarily equivalent via a unitary operator in ℒ (H) . A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I − Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C*-modules are also provided. [ABSTRACT FROM AUTHOR]

Let ( X , ⟨ ⋅ , ⋅ ⟩) be a Hilbert C ∗ -module over a C ∗ -algebra A and let S (A) be the set of states on A . In this paper, we first compute the norm derivative for nonzero elements x and y of X as follows: lim t → 0 + ‖ x + t y ‖ − ‖ x ‖ t = 1 ‖ x ‖ max { Re φ (⟨ x , y ⟩) : φ ∈ S (A) , φ (⟨ x , x ⟩) = ‖ x ‖ 2 }. We then apply it to characterize different concepts of orthogonality in X . In particular, we present a simpler proof of the classical characterization of Birkhoff–James orthogonality in Hilbert C ∗ -modules. Moreover, some generalized Daugavet equation in the C ∗ -algebra B (H) of all bounded linear operators acting on a Hilbert space H is solved. [ABSTRACT FROM AUTHOR]

Generalized Weyl operators on Hilbert spaces have been introduced and studied by Djordjević (Proc Am Math Soc, 130:81–ll4, 2001). In this paper, we provide a generalization of his result in the setting of C ∗ -operators on Hilbert C ∗ -modules by giving sufficient conditions under which the sum of a generalized C ∗ -Weyl operator and a finitely generated C ∗ -operator is a generalized C ∗ -Weyl operator. Also, we obtain an extension of Djordjević’s results from the case of operators on Hilbert spaces to the case of operators on Banach spaces. Next, we consider semi- C ∗ -B-Fredholm operators on Hilbert C ∗ -modules and give sufficient conditions under which the composition of two mutually commuting semi- C ∗ -B-Fredholm operators is a semi- C ∗ -B-Fredholm operator, thus generalizing the result by Berkani regarding semi-B-Fredholm operators on Banach spaces. Finally, we consider Drazin invertible C ∗ -operators, and we give necessary and sufficient conditions for two mutually commuting C ∗ -operators to be Drazin invertible when their composition is Drazin invertible. [ABSTRACT FROM AUTHOR]

It is well known that an adjointable operator between Hilbert C*-modules admits a Moore-Penrose inverse if and only if it has closed range. In this paper, we give certain necessary and sufficient conditions for the existence of the reverse order law for the Moore-Penrose inverse of closed range adjointable operators in Hilbert C*-module settings. Some new related results are also derived, which can be used to establish connections with the reverse order law in Hilbert C*-modules. [ABSTRACT FROM AUTHOR]

Let P and Q be projections on a Hilbert C * -module. This paper deals with the orthogonal complementarity of certain range closures associated with P and Q. Some necessary and sufficient conditions are derived under which (P, Q) is harmonious. [ABSTRACT FROM AUTHOR]

A bounded adjointable operator T ∈ L (H) in Hilbert C*-modules is called EP if ranges of T and T ∗ have the same closure. This definition is employed to achieve a new characterization of EP operators. We show that the anticommutator of EP operators is again an EP operator. It follows that the product of commuting EP operators is an EP operator. Some other conditions implying the product of EP operators to be an EP operator are presented. Finally, we prove that for EP operators S, T equality R (T + S) ¯ = R (T) + R (S) ¯ holds. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of continuous *-frames and *-C-controlled frames in Hilbert C*-modules. We present some results of continuous frames in the view of *-Ccontrolled frames in Hilbert C*-modules. Also we define *-(C,C')- frames and investigate multiplier operators for these frames. [ABSTRACT FROM AUTHOR]

In this paper, the K-woven frames in Hilbert C ∗ -modules are introduced, and several approaches to the construction of them are presented. Meanwhile, the weaving properties of K-frames in Hilbert C ∗ -modules under adjointable operators are investigated. Two different types of perturbation conditions such that, under them, two K-frames in Hilbert C ∗ -modules can be K-woven are also given. Finally, two sufficient conditions for the remainders to be still K-woven when deleting some elements from a pair of K-woven frames in Hilbert C ∗ -modules are obtained. [ABSTRACT FROM AUTHOR]

This article introduces the notion of L$_p$-C$^*$-semi-inner product space, a generalization of the concept of C$^*$-semi-inner product space introduced by Gamchi et al., where we consider H\"{o}lder's inequality instead of Cauchy Schwartz' inequality. We establish some basic results L$_p$-C$^*$-semi-inner product spaces, analogous to those valid for C$^*$-semi-inner product spaces and Hilbert C$^*$-modules.

Suppose that Z is a locally compact Hausdorff space and Ψ , Φ : E ⟶ F are C 0 (Z) -module maps between Hilbert C 0 (Z) -modules such that for every x , y ∈ E , x ⊥ y implies Ψ (x) ⊥ Φ (y). Then there exists a bounded complex-valued function φ on Z that is continuous on Z E = { z ∈ Z : ⟨ x , x ⟩ (z) ≠ 0 f o r s o m e x ∈ E } and satisfies ⟨ Ψ (x) , Φ (y) ⟩ = φ ⋅ ⟨ x , y ⟩ o n Z , f o r a l l x , y ∈ E. [ABSTRACT FROM AUTHOR]

Let (X , ⟨ · , · ⟩) be a Hilbert C ∗ -module over a C ∗ -algebra A and let S (A) be the set of states on A . For elements x and y of X , let ρ + (x , y) denote the right norm derivative of x in the direction of y. This paper gives another proof of the following result: ρ + (x , y) = max { Re φ (⟨ x , y ⟩) : φ ∈ S (A) , φ (⟨ x , x ⟩) = ‖ x ‖ 2 } , which has been obtained recently in Wójcik and Zamani (Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2022.2046688). [ABSTRACT FROM AUTHOR]

We introduce the notion of a tractable pair of operators as well as that of the generalized parallel sum in the setting of adjointable operators on Hilbert C ∗ -modules. Some significant results about the parallel sum known for matrices and Hilbert space operators are extended to the case of the generalized parallel sum. In particular, a factorization theorem on the parallel sum is proved, and a common upper bound of two positive operators is constructed in the Hilbert C ∗ -module case. The harmonic mean for positive operators on Hilbert C ∗ -modules is also dealt with. [ABSTRACT FROM AUTHOR]

In this paper, we show that full Hilbert C(Z)-modules, where Z is a compact Hausdorff space may fail to have unit vectors. We also show that while real Hilbert CR(Z)-modules may not have unit vectors, their complexifications as (complex) Hilbert C(Z)-modules may have unit vectors. In particular, we prove that: (i) the unit vectors in full Hilbert C(Z)-modules are precisely the extreme points of their unit balls; (ii) the extreme and the exposed points of the unit ball of full Hilbert C(Z)-modules with unit vectors coincide as Z has a diffuse measure; otherwise, their unit balls have no exposed points. [ABSTRACT FROM AUTHOR]

We establish some new inequalities for K-g-frames in Hilbert C ∗ -modules using the Moore–Penrose inverse of the adjointable operator K and a parameter λ . It turns out that the results which we obtained can lead to some known results if we choose particular values for K and λ . We also give a double inequality for K-g-frames in Hilbert C ∗ -modules with the help of two adjointable operators induced by dual K-g-frame pair. [ABSTRACT FROM AUTHOR]

In this paper we study the Banach weak topology on Hilbert C*-modules. For a Hilbert C*-module, we show that every sequence that is convergent in the weak module topology generated by the inner product is also convergent in the Banach weak topology generated by continuous linear functionals and the converse is true for the Hilbert C*-modules that their underlying C*-algebras are either of the form of a c0-direct sum of matrices or of the form of a finite-dimensional matrix algebra. [ABSTRACT FROM AUTHOR]

We give some examples of EP and non-EP operators to show that the main results of Mohammadzadeh Karizaki et al. [Some results about EP modular operators. Linear Multilinear Algebra. DOI:10.1080/03081087.2020.1844613] are not correct even in the case of Hilbert spaces. [ABSTRACT FROM AUTHOR]

Some characterizations of products of two projections on a Hilbert space are generalized to the case of products of a finite number of projections on a Hilbert C ∗ -module. An example is constructed to show that in the Hilbert C ∗ -module case, X and its subset X ⊥ can be different, where X denotes the set of all products of two projections on a Hilbert C ∗ -module, and X ⊥ consists of those elements in X that have the polar decomposition. Some new phenomena are revealed when an operator is taken in X instead of X ⊥ . Some refinements are made on norms of differences of two projections. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a unified framework in the setting of Hilbert C ∗ -modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and C ∗ -valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the C ∗ -algebra of adjointable operators acting on a Hilbert C ∗ -module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert C ∗ -modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and present an interpolation theorem. Moreover, we study some basic properties of the so-called relative reproducing kernel Hilbert C ∗ -modules and give a characterization of dual modules. Among other things, we prove that every conditionally negative definite kernel gives us a reproducing kernel Hilbert C ∗ -module and a certain map. Several examples illustrate our investigation. [ABSTRACT FROM AUTHOR]

Let H be a Hilbert C ∗ -module, and let H M be the indefinite inner space induced by a self-adjointable and invertible operator M on H. Given weighted projections P and Q on H M , let S λ , k = (P Q) k - λ (Q P) k for a pair (k , λ) , where k is a natural number and λ is a complex number. It is proved that P Q - Q P is weighted Moore–Penrose invertible if and only if S λ , k is weighted Moore–Penrose invertible for every pair (k , λ) . [ABSTRACT FROM AUTHOR]

Improving and extending the notion of controlled frames, in this paper, we introduce controlled frames with two operators for Hilbert C*-modules. This generalize controlled frames in Hilbert and Hilbert C*-module. We show, in Hilbert C*module setting, controlled frame with two operators is classical frame. Also, we investigate controlled multiplier operators and their invertibility in Hilbert C*-modules. We show that the inverse of controlled frame multiplier with two operators is controlled frame multiplier too. [ABSTRACT FROM AUTHOR]

In this paper, we define C ∗ -module algebras as a generalization of Hilbert algebras and study their properties. We describe some structures of a C ∗ -module algebra including bounded elements and left (right) module adjointable maps. Furthermore we establish some properties of Hilbert algebras for C ∗ -module algebras. [ABSTRACT FROM AUTHOR]

We study the Banach–Stone theorem in the framework of Hilbert C ∗ -modules and Finsler C ∗ -modules and show that each φ -morphism T between Hilbert C ∗ -modules and Finsler C ∗ -modules is a weighted composition operator of the form T f (y) = h (y) f (θ (y)) . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce the notion of *-T-controlled frames in Hilbert C * -modules. We present some results of frames in the view of *-T-controlled frames in Hilbert C * -modules. Also, we define *-w-frames and give their relations to *-T-controlled frames and multiplier operators. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability of the system of two operator equations A X = C and X B = D in the framework of Hilbert C ∗ -modules as well as the existence of Hermitian and positive solutions. We also provide formulae for the general forms of such solutions. [ABSTRACT FROM AUTHOR]

Let 𝒜 and 𝔄 be two C ∗ -algebras such that 𝒜 is a Banach 𝔄 -bimodule with the left and right compatible action of 𝔄 on 𝒜. We define 𝒜 ⋊ 𝔄 as a C ∗ -algebra, where it is a strongly splitting C ∗ -algebra extension of 𝔄 by 𝒜. Normal, self-adjoint, unitary, invertible and projection elements of 𝒜 ⋊ 𝔄 are characterized; sufficient and necessary conditions for existing unit and bounded approximate identity of 𝒜 ⋊ 𝔄 as a Banach algebra and as a C ∗ -algebra are given. We characterize ∗-automorphisms on 𝒜 ⋊ 𝔄 and give some results related to ∗-homomorphisms, ∗-representations and completely bounded maps on this C ∗ -algebra. Also, we have constructed a new Hilbert C ∗ -module X ⋊ Y over 𝒜 ⋊ 𝔄 , where X is a Hilbert C ∗ -module over 𝒜 and Y is a Hilbert C ∗ -module over 𝔄. [ABSTRACT FROM AUTHOR]

In this paper, we first provide an necessary and sufficient condition for vectors to be K-frame generators for unitary systems in Hilbert C ∗ -modules. We then characterize the K-dual Bessel generators of K-frame generators for unitary systems in Hilbert C ∗ -modules from several aspects. We also examine the properties of K-frame generators for unitary systems in Hilbert C ∗ -modules under adjointable operators, and some methods to construct new K-frame generators from given ones are presented. The results we obtained improve one corresponding result in Hilbert spaces setting and reflect, to a certain extent, the different behaviors of K-frame generators for unitary systems shown respectively in Hilbert C ∗ -modules and Hilbert spaces. [ABSTRACT FROM AUTHOR]

Let ℌ be a Hilbert space, and let K(ℌ) be the C*-algebra of compact operators on ℌ. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert K(ℌ)-module by employing the Birkhoff-James orthogonality. Among other things, we present a characterization of transitive relation of the norm-parallelism for elements in a certain Hilbert K(ℌ)-module. We also give some characterizations of the Schatten p-norms and the operator norm-parallelism for matrices. [ABSTRACT FROM AUTHOR]

The block operator matrix decomposition is used to get new conditions under which the product of two operators is EP while none of them is an EP operator, and the same technique is employed to achieve a new characterization of EP operators. The communication of classes idempotent and normal operators with other common classes of special operators, such as EP, star dagger, bi-EP, and bi dagger operators, are presented, which are derived through matrix operators and Moore–Penrose inverses of operators in the Hilbert C ∗ -module setting. [ABSTRACT FROM AUTHOR]

We give an extension of Hua's inequality for C*-valued norm in pre-Hilbert C*-modules setting. As its applications, some known and new generalizations of Hua's inequality are deduced. In particular, we establish the connection between C*-valued norm triangle inequality and C*-valued norm Hua's inequality. We also present some operator versions of Hua's inequality on Hilbert spaces. As a consequence, we obtain various forms of Hua's inequality. Moreover, we indicate that for our operator, Hua's inequality is equivalent to operator convexity of a given continuous real function. [ABSTRACT FROM AUTHOR]

We introduce a partial order relation on the set of all completely positive pairs of invariant symmetric and completely bounded multilinear maps on Hilbert C ∗ -modules and C ∗ -algebras and establish a Radon–Nikodým type theorem for these pairs in terms of their Stinespring type representations. [ABSTRACT FROM AUTHOR]