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1.
Let E be a Hilbert C*-module,and ■ be an orthogonally complemented closed submodule of E.The authors generalize the definitions of ■-complementability and ■-compatibility for general(adjointable) operators from Hilbert space to Hilbert C*-module,and discuss the relationship between each other.Several equivalent statements about ■-complementability and ■-compatibility,and several representations of Schur complements of ■-complementable operators(especially,of ■-compatible operators and of positive ■-compatib... 相似文献
2.
Let M be a full Hilbert C*-module over a C*-algebra A,and let End*A(M) be the algebra of adjointable operators on M.We show that if A is unital and commutative,then every derivation of End A(M) is an inner derivation,and that if A is σ-unital and commutative,then innerness of derivations on "compact" operators completely decides innerness of derivations on End*A(M).If A is unital(no commutativity is assumed) such that every derivation of A is inner,then it is proved that every derivation of End*A(Ln(A)) is also inner,where Ln(A) denotes the direct sum of n copies of A.In addition,in case A is unital,commutative and there exist x0,y0 ∈ M such that x0,y0 = 1,we characterize the linear A-module homomorphisms on End*A(M) which behave like derivations when acting on zero products. 相似文献
3.
The aim of the present paper is to solve some major open problems of Hilbert C*-module theory by applying various aspects of multiplier C*-theory. The key result is the equivalence established between positive invertible quasi-multipliers of the C*-algebra of compact operators on a Hilbert C*-module {, ., } and A-valued inner products on , inducing an equivalent norm to the given one. The problem of unitary isomorphism of C*-valued inner products on a Hilbert C*-module is considered and new criteria are formulated. Countably generated Hilbert C*-modules turn out to be unitarily isomorphic if they are isomorphic as Banach C*-modules. The property of bounded module operators on Hilbert C*-modules of being compact and/or adjointable is unambiguously connected to operators with respect to any choice of the C*-valued inner product on a fixed Hilbert C*-module if every bounded module operator possesses an adjoint operator on the module. Every bounded module operator on a given full Hilbert C*-module turns out to be adjointable if the Hilbert C*-module is orthogonally complementary. Moreover, if the unit ball of the Hilbert C*-module is complete with respect to a certain locally convex topology, then these two properties are shown to be equivalent to self-duality. 相似文献
4.
Dan LI 《数学学报(英文版)》2012,28(9):1845-1850
Extending the notion of property T of finite von Neumann algebras to general von Neumann algebras, we define and study in this paper property T** for (possibly non-unital) C* -algebras. We obtain several results of property T** parallel to those of property T for unital C* -algebras. Moreover, we show that a discrete group Γ has property T if and only if the group C* -algebra Cr* (Γ) (or equivalently, the reduced group C* -algebra Cr* (Γ)) has property T**. We also show that the compact operators K(l2 ) has property T** but c0 does not have property T**. 相似文献
5.
We obtain new representations for the general positive and real-positive solutions of the equation axa*=c in a C*-algebra using the characterization of positivity based on a matrix representation of an element and the generalized Schur complement. Applications to the equation AXA*=C for operators between Hilbert spaces and for finite matrices are given. 相似文献
6.
It is shown that if A is a stably finite C∗-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C∗-algebra that are not isomorphic. 相似文献
7.
Maria Joiţa 《Central European Journal of Mathematics》2009,7(1):73-83
We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C
*-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E
*(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E
*) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London
Math. Soc. 49(1984), 289–306].
相似文献
8.
9.
Hua Xin Lin 《数学学报(英文版)》2002,18(1):181-198
Let X be a connected finite CW complex and d
X
: K
0(C(X)) →ℤ be the dimension function. We show that, if A is a unital separable simple nuclear C*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K
0(A) = ℚ⊕ kerd
x
and K
1(A) = K
1(C(X)), then A is isomorphic to an inductive limit of M
n
!(C(X)).
Received April 19, 2001, Accepted April 27, 2001. 相似文献
10.
Kazuyuki Sait 《Journal of Mathematical Analysis and Applications》2009,360(2):369-376
Akemann showed that any von Neumann algebra with a weak* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C*-algebra has a weak* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C*-algebra has the weak* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C*-algebra almost separably representable. We say that a unital C*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C*-algebra are weak*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C*-algebras but the general question remains open. 相似文献
11.
We characterize A-linear symmetric and contraction module operator semigroup{Tt}t∈R+L(l2(A)),where A is a finite-dimensional C-algebra,and L(l2(A))is the C-algebra of all adjointable module maps on l2(A).Next,we introduce the concept of operator-valued quadratic forms,and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on l2(A)and the set of non-negative densely defined A-valued quadratic forms.In the end,we obtain that a real and strongly continuous symmetric semigroup{Tt}t∈R+L(l2(A))being Markovian if and only if the associated closed densely defined A-valued quadratic form is a Dirichlet form. 相似文献
12.
Ruy Exel 《K-Theory》1993,7(3):285-308
GivenC*-algebrasA andB and an imprimitivityA-B-bimoduleX, we construct an explicit isomorphismX
*:K
i
(A)K
i
(B), whereK
i
denotes the complexK-theory functors fori=0, 1. Our techniques do not require separability nor the existence of countable approximate identities. We thus extend to generalC*-algebras the result of Brown, Green, and Rieffel according to which, strongly Morita equivalentC*-algebras have isomorphicK-groups. The method employed includes a study of Fredholm operators on Hilbert modules.On leave from the University of São Paulo, Brazil. 相似文献
13.
In this article, we study tensor product of Hilbert C*-modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F, and we get more results.
For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K. 相似文献
14.
A Hilbert C*-module is a generalization of a Hilbert space for which the inner
product takes its values in a C*-algebra instead of the complex numbers. We use the bracket
product to construct some Hilbert C*-modules over a group C*-algebra which is generated by the
group of translations associated with a wavelet. We shall investigate bracket products and their
Fourier transform in the space of square integrable functions in Euclidean space. We will also show
that some wavelets are associated with Hilbert C*-modules over the space of essentially bounded
functions over higher dimensional tori. 相似文献
15.
Michael Skeide 《Proceedings Mathematical Sciences》2006,116(4):429-442
After discussing some basic facts about generalized module maps, we use the representation theory of the algebra ℬa(E) of adjointable operators on a HilbertB-moduleE to show that the quotient of the group of generalized unitaries onE and its normal subgroup of unitaries onE is a subgroup of the group of automorphisms of the range idealB
E
ofE inB. We determine the kernel of the canonical mapping into the Picard group ofB
E
in terms of the group of quasi inner automorphisms ofB
E
. As a by-product we identify the group of bistrict automorphisms of the algebra of adjointable operators onE modulo inner automorphisms as a subgroup of the (opposite of the) Picard group. 相似文献
16.
V. M. Manuilov 《Annals of Global Analysis and Geometry》1995,13(3):207-226
It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH*
A dual toH
A under slight restrictions one can find a set of eigenvectorsx
i H*
A and a non-increasing sequence of eigenvalues
i
A such thatK x
i=x
i
i
and the selfdual HilbertA-module generated by these eigenvectors is the wholeH*
A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA
and discuss the possibility of its diagonalization. 相似文献
17.
Subhash J. Bhatt 《Proceedings Mathematical Sciences》2006,116(2):161-173
Given anm-tempered strongly continuous action α of ℝ by continuous*-automorphisms of a Frechet*-algebraA, it is shown that the enveloping ↡-C
*-algebraE(S(ℝ, A∞, α)) of the smooth Schwartz crossed productS(ℝ,A
∞, α) of the Frechet algebra A∞ of C∞-elements ofA is isomorphic to the Σ-C
*-crossed productC
*(ℝ,E(A), α) of the enveloping Σ-C
*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK
*(S(ℝ, A∞, α)) =K
*(C
*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC
*-algebra defined by densely defined differential seminorms is given. 相似文献
18.
Maria Joiţa 《Czechoslovak Mathematical Journal》2004,54(3):727-737
In this paper the tensor products of Hilbert modules over locally C
*-algebras are defined and their properties are studied. Thus we show that most of the basic properties of the tensor products of Hilbert C
*-modules are also valid in the context of Hilbert modules over locally C
*-algebras. 相似文献
19.
We generalize the setting of the Stone-Weierstrass problem to the cone CP(A, H) of all completely positive linear maps of a C*-algebra A into the C*-algebra B(H) of all bounded linear operators on a Hilbert space H, and give some equivalent conditions for the problem. As a consequence, a new generalized spectrum, containing pure states and irreducible representations, will be introduced. 相似文献
20.
Michael Frank 《Annals of Global Analysis and Geometry》1985,3(2):155-172
Let A be a C*-algebra, K be a compact space, A(K) be the C*-algebra of all continuous maps from K into A, 12(A) be the standard countably generated Hilbert A-module. We investigate a set of maps from K into EndA(12(A)), which is isomorphic to EndA(K)(12(A(K))). We describe the subsets which are isomorphic to EndfA(K)
*(12(A(K))). GLA(K)(12(A(K))) and GLfA(K)
*(12(A(K))), respectively. As an application we deduce a criterion for the self-duality of 12(A) in the commutative case. 相似文献