Let $A$ be a unital Banach algebra. We give a characterization of the left Banach $A$-modules $X$ for which there exists a commutative unital $C^*$-algebra $C(K)$, a linear isometry $i\colon X\to C(K)$, and a contractive unital homomorphism $θ\colon A\to C(K)$ such that $i(a\cdotp x) =θ(a)i(x)$ for any $a\in A, x\in X$. We then deduce a "commutative" version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a $w^*$-version of the latter characterization, which generalizes some previous work of Effros and Ruan.
In this paper the concept of unbounded Fredholm operators on Hilbert C*- modules over an arbitrary C*-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert C*-modules over C*-algebras of compact operators. In the framework of Hilbert C*-modules over C*-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same.
The parallel sum for adjoinable operators on Hilbert $C^*$-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert $C^*$-modules. It is shown that there exist a Hilbert $C^*$-module $H$ and two positive operators $A, B\in\mathcal{L}(H)$ such that the operator equation $A^{1/2}=(A+B)^{1/2}X, X\in \cal{L}(H)$ has no solution, where $\mathcal{L}(H)$ denotes the set of all adjointable operators on $H$.
We study closedness of the range, adjointability and generalized invertibility of modular operators between Hilbert modules over locally C*-algebras of coefficients. Our investigations and the recent results of M. Frank [Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules, {\it J. K-Theory} {\bf 2} (2008), 453-462] reveal a number of equivalence properties of the category of Hilbert modules over locally C*-algebras which characterize precisely the inverse limit of C*-algebras of the C*-algebra of compact operators.
In this paper we present results concerning orthogonality in Hilbert $C^*$-modules. Moreover, for a $C^*$-algebra $\mathscr{A}$, we prove theorems concerning the multi-$\mathscr{A}$-linearity and its preservation by $\mathscr{A}$-linear operators. New version of solution of the orthogonality equation on Hilbert $C^*$-modules and mappings preserving orthogonality are also investigated.
Let $t$ be a regular operator between Hilbert $C^*$-modules and $t^†$ be its Moore-Penrose inverse. We investigate the Moore-Penrose invertibility of the Gram operator $t^*t$. More precisely, we study some conditions ensuring that $t^† = (t^* t)^† t^*= t^* (t t^*)^†$ and $(t^*t)^†=t^†t^{* †}$ hold. As an application, we get some results for densely defined closed operators on Hilbert $C^*$-modules over $C^*$-algebras of compact operators.
We construct a topology on the standard Hilbert module $l^2(\mathcal A)$ over a unital $W^*$-algebra $\mathcal A$ such that any "compact" operator, (i.e.\ any operator in the norm closure of the linear span of the operators of the form $x\mapsto\left<y,x\right>z$) maps bounded sets into totally bounded sets.
In a previous paper we generalized the theory of W*-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel "Rigged modules II", giving many new results about weak*-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of $H^{\infty}(D)$, and prove that for a weak* cl…
We apply the inequality $\left|\left<x,y\right>\right|\le\|x\|\,\left<y,y\right>^{1/2}$ to give an easy and elementary proof of many operator inequalities for elementary operators and inner type product integral transformers obtained during last two decades, which also generalizes all of them.
Halmos' two projections theorem for Hilbert space operators is one of the fundamental results in operator theory. In this paper, we introduce the term of two harmonious projections in the context of adjointable operators on Hilbert $C^*$-modules, extend Halmos' two projections theorem to the case of two harmonious projections. We also give some new characterizations of the closed submodules and their associated projections. As an application, a norm equation associated to a characterization of the Friedrichs angle is proved to be true in the framework of Hilbert $C^*$-modules.
Normality of bounded and unbounded adjointable operators are discussed. Suppose $T$ is an adjointable operator between Hilbert C*-modules which has polar decomposition, then $T$ is normal if and only if there exists a unitary operator $ \mathcal{U}$ which commutes with $T$ and $T^*$ such that $T=\mathcal{U} \, T^*.$ Kaplansky's theorem for normality of the product of bounded operators is also reformulated in the framework of Hilbert C*-modules.
In this paper, we define a new concept of numerical range $W_{o}(\cdot)$ and prove its basic results. We also define the numerical radius $ω_{o}(\cdot)$ and prove that $$ω_{o}(T)\leq||| T|||\leq 2ω_{o}(T).$$
Let $n$ be any natural number. The $n$-centered operator is introduced for adjointable operators on Hilbert $C^*$-modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, $n$-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator $T$, if $T$ is Moore-Penrose invertible and is $n$-centered, then its Moore-Penrose inverse is also $n$-centered. A Hilbert space operator $T$ is constructed such that $T…
We introduce a uniform structure on any Hilbert $C^*$-module $\mathcal N$ and prove the following theorem: suppose, $F:{\mathcal M}\to {\mathcal N}$ is a bounded adjointable morphism of Hilbert $C^*$-modules over $\mathcal A$ and $\mathcal N$ is countably generated. Then $F$ belongs to the Banach space generated by operators $θ_{x,y}$, $θ_{x,y}(z):=x\langle y,z\rangle$, $x\in {\mathcal N}$, $y,z\in {\mathcal M}$ (i.e. $F$ is ${\mathcal A}$-compact, or "compact") if and only if $F$ maps the unit ball of ${\mathcal M}$ to a totally bounded set with respect to this uniform structure (i.e. $F$ is …
It is proved that for adjointable operators $A$ and $B$ between Hilbert $C^*$-modules, certain majorization conditions are always equivalent without any assumptions on $\overline{\mathcal{R}(A^*)}$, where $A^*$ denotes the adjoint operator of $A$ and $\overline{\mathcal{R}(A^*)}$ is the norm closure of the range of $A^*$. In the case that $\overline{{\mathcal R}(A^*)}$ is not orthogonally complemented, it is proved that there always exists an adjointable operator $B$ whose range is contained in that of $A$, whereas the associated equation $AX=B$ for adjointable operators is unsolvable.
We generalize some results on compact operators on Hilbert spaces to "compact" operators on some Hilbert right W*-modules. We present in this frame the Schatten decomposition of the compact operators, the trace, the Banach Lp-spaces and their duals, the Hilbert-Schmitt operators, and the integral operators as an example of Hilbert-Schmitt operators.
In this paper we establish the semi-Fredholm theory on Hilbert C*-modules as a continuation of Fredholm theory on Hilbert C*-modules established by Mishchenko and Fomenko. We give a definition of a semi-Fredholm operator on Hilbert C*-module and prove that these semi-Fredholm operators are those that are one-sided invertible modulo compact operators, that the set of proper semi-Fredholm operators is open and many other results that generalize their classical counterparts.
In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orth…
The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C$^*$-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.
Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^*$-module $\mathscr{E}$. Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert $C^*$-modules, we present the general solution of the equation $AX+YB=C$ when the ranges of $A,B$ and $C$ are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert $C^*$-modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for $AXA^*+BYB^*=C$. Finally, we deduce that there exist nonzero operators $X, Y\geq 0$ and $Z$ such that $AXA^*+BYB^*=CZ…
The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C*-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C*-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C*-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The diffe…
Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra $\CAlg{A}$. New classes of (possibly unbounded) operators $t:E\to F$ are introduced and investigated. Instead of the density of the domain $\Def(t)$ we only assume that $t$ is essentially defined, that is, $\Def(t)^\bot=\{0\}$. Then $t$ has a well-defined adjoint. We call an essentially defined operator $t$ graph regular if its graph $\Graph(t)$ is orthogonally complemented in $E\oplus F$ and orthogonally closed if $\Graph(t)^{\bot\bot}=\Graph(t)$. A theory of these operators is developed. Various characterizations of graph regular …
We study and compare the gap and the Riesz topologies of the space of all unbounded regular operators on Hilbert C*-modules. We show that the space of all bounded adjointable operators on Hilbert C*-modules is an open dense subset of the space of all unbounded regular operators with respect to the gap topology. The restriction of the gap topology on the space of all bounded adjointable operators is equivalent with the topology which is generated by the usual operator norm. The space of regular selfadjoint Fredholm operators on Hilbert C*-modules over the C*-algebra of compact operators is path…
We provide a characterization for operator valued completely bounded linear maps on Hilbert $C^*$-modules in terms of $φ$-maps. Also, we show that for every operator valued completely positive map $φ$ on a $C^*$-algebra $\mathcal{A}$, there is a unique (up to multiplication by a unitary operator) non-degenerate $φ$-map on each Hilbert $\mathcal{A}$-module.
We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C$^*-$modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C$^*-$algebras of an operator space, which generalize the algebras of adjointable operators on a C$^*-$module, and the `imprimitivity C$^*-$algebra'. It also generalizes a classical Banach space notio…
Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of eleme…
We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.
We study the stability of Fredholm property for regular operators on Hilbert $C^*$-modules under some certain perturbations. We treat this problem when perturbing operators are (relatively) bounded or relatively compact. We also consider the perturbations of regular Fredholm operators in terms of the gap metric. In particular, we prove that the space of all regular Fredholm operators on a Hilbert $C^*$-module $E$ is open in the space of all regular operators on $E$ with respect to the gap metric. As an application, we construct some continuous paths of selfadjoint regular Fredholm operators wi…
Woronowicz introduced the functional calculus for normal operators in Hilbert C*-modules. The aim of this paper is to translate, if possible, some basic properties of the functional calculus in Hilbert spaces to the Hilbert C*-module framework.
We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a decisive role in this.
A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and g…
The two reference lists contain 54/22 references of papers and preprints concerned with the theory and/or various applications of Hilbert modules over Hilbert $*$-algebras and over (non-self-adjoint) operator algebras. They are far from being complete, but they give additional information about two research fields which are closely related to the theory of Hilbert C*-modules, i.e. they are complements to the reference guide about this circle of problems. Any additions, corrections and forthcoming information are welcome.
Let $T$ be an adjointable operator between two Hilbert $C^*$-modules and $T^*$ be the adjoint operator of $T$. The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac12$ and $\mathcal{R}(U^*)=\overline{\mathcal{R}(T^*)}$, where $U$ is a partial isometry, $\mathcal{R}(U^*)$ and $\overline{\mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$, respectively. Based on this new characterization of the polar decomposition, an application to the study of centered operators is carried out.
Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. The above module is usually called the Hilbert module over $\mathbb{C}[z_1, \ldots, z_n]$. Hilbert modules over $\mathbb{C}[z_1, \ldots, z_n]$ (or natural function algebras) …
We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect `nonselfadjoint operator algebra' with the C$^*-$algebraic framework. More particularly, we make use of the universal, or maximal, C$^*-$algebra generated by an operator algebra, and C$^*-$dilations. This technology is quite general, however it was developed to solve some problems arising in the theory of Morita equivalence of operator algebras, and as a result most of the applications given here (and in …
In this paper we give characterizations of essential left ideals of a C*-algebra $A$ in terms of their properties as operator $A$-modules. Conversely, we seek C*-algebraic characterizations of those ideals $J$ in $A$ such that $A$ is an essential extension of $J$ in various categories of operator modules. In the case of two-sided ideals, we prove that all the above concepts coincide. We obtain results, analogous to {M. Hamana's} results, which characterize the injective envelope of a C*-algebra as a maximal essential extension of the C*-algebra, but with completely positive maps replaced by co…
Rigged modules over an operator algebra are a generalization of Hilbert modules over a $C^{\star}$-algebra. We characterize the rigged modules over an operator algebra $\mathcal A$ which are orthogonally complemented in $C_\infty(\mathcal A),$ the space of infinite columns with entries in $\mathcal A.$ We show that every such rigged module `restricts' to a bimodule of Morita equivalence between appropriate stably isomorphic operator algebras.
We consider projectivity and injectivity of Hilbert C*-modules in the categories of Hilbert C*-(bi-)modules over a fixed C*-algebra of coefficients (and another fixed C*-algebra represented as bounded module operators) and bounded (bi-)module morphisms, either necessarily adjointable or arbitrary ones. As a consequence of these investigations, we obtain a set of equivalent conditions characterizing C*-subalgebras of C*-algebras of compact operators on Hilbert spaces in terms of general properties of Hilbert C*-modules over them. Our results complement results recently obtained by B. Magajna, J…
Suppose $T$ and $S$ are bounded adjointable operators between Hilbert C*-modules admitting bounded Moore-Penrose inverse operators. Some necessary and sufficient conditions are given for the reverse order law $(TS)^{ †} =S^{ †} T^{ †}$ to hold. In particular, we show that the equality holds if and only if $Ran(T^{\ast}TS) \subseteq Ran(S)$ and $Ran(SS^{\ast}T^{\ast}) \subseteq Ran(T^{\ast}),$ which was studied first by Greville [{\it SIAM Rev. 8 (1966) 518--521}] for matrices.
A regular operator T on a Hilbert C^*-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of (I+T^*T) is dense. Semiregular operators are a slightly larger class of operators that may not have this property. It is shown that, like in the case of regular operators, one can, without any loss in generality, restrict oneself to semiregular operators on C^*-algebras. We then prove that for abelian C^*-algebras as well as for subalgebras of the algebra of compact operators, any closed semiregular operator is automatically regular. We also determi…
It is known that a continuous family of compact operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators in Hilbert modules over a commutative W*-algebra. The aim of the present paper is to generalize this fact for a finite W*-algebra $A$ not necessarily commutative. We prove that for a compact operator $K$ acting in the right Hilbert $A$-module $H^*_A$ dual to $H_A$ under slight restrictions one can find a set of "eigenvectors" $x_i\in H^*_A$ and a non-increasing sequence of "eigenvalues" $λ_i\in A$ such that $K\,x_i = x_…
Suppose $T$ and $S$ are bounded adjointable operators with close range between Hilbert C*-modules, then $TS$ has closed range if and only if $Ker(T)+Ran(S)$ is an orthogonal summand, if and only if $Ker(S^*)+Ran(T^*)$ is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between $Ran(S)$ and $Ker(T) \cap [Ker(T) \cap Ran(S)]^{\perp}$ is positive and $ \bar{Ker(S^*)+Ran(T^*)} $ is an orthogonal summand then $TS$ has closed range.
Let $X$ be a right Hilbert module over a $C^*$-algebra $A$ equipped with the canonical operator space structure. We define an elementary operator on $X$ as a map $ϕ: X \to X$ for which there exists a finite number of elements $u_i$ in the $C^*$-algebra $\mathbb{B}(X)$ of adjointable operators on $X$ and $v_i$ in the multiplier algebra $M(A)$ of $A$ such that $ϕ(x)=\sum_i u_i xv_i$ for $x \in X$. If $X=A$ this notion agrees with the standard notion of an elementary operator on $A$. In this paper we extend Mathieu's theorem for elementary operators on prime $C^*$-algebras by showing that the com…
In this paper, we characterize a class of additive maps on Hilbert $C^\ast$-modules which maps a "rank one" adjointable operators to another rank one operators.
By computing the completely bounded norm of the flip map on the Haagerup tensor product $C_0 Y_1\otimes_{C_0 X} C_0 Y_2$ associated to a pair of continuous mappings of locally compact Hausdorff spaces $Y_1\rightarrow X\leftarrow Y_2$, we establish a simple characterisation of the Beck-Chevalley condition for base change of operator modules over commutative $C^*$-algebras, and a descent theorem for continuous fields of Hilbert spaces.
It is known that the classical Hilbert--Schmidt theorem can be generalized to the case of compact operators in Hilbert $A$-modules $H_A^*$ over a $W^*$-algebra of finite type, i.e. compact operators in $H_A^*$ under slight restrictions can be diagonalized over $A$. We show that if $B$ is a weakly dense $C^*$-subalgebra of real rank zero in $A$ with some additional property then the natural extension of a compact operator from $H_B$ to $H_A^*\supset H_B$ can be diagonalized with diagonal entries being from the $C^*$-algebra $B$.
