Inequivalent representations of the dual space (original) (raw)
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Representation of certain homogeneous Hilbertian operator spaces and applications
Inventiones mathematicae, 2010
Following Grothendieck's characterization of Hilbert spaces we consider operator spaces F such that both F and F * completely embed into the dual of a C*-algebra. Due to Haagerup/Musat's improved version of Pisier/Shlyakhtenko's Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum C ⊕ R of the column and row spaces (the corresponding class being denoted by QS(C ⊕ R)). We first prove a representation theorem for homogeneous F ∈ QS(C ⊕ R) starting from the fundamental sequences * Partially supported by NSF DMS 05-56120.
Duality for Unbounded Operators, and Applications
arXiv: Functional Analysis, 2015
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise linking between such two Hilbert spaces when it is assumed that D is dense in one of the two; but generally not in the other. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and operator theory of reflection positivity.
Spectral theory of linear operators
Advances in Mathematics, 1983
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Spaces of compact operators and their dual spaces
Rendiconti del Circolo Matematico di Palermo, 2004
We show that (L w , ||| • |||) is a Banach ideal of operators and that the continuous dual space K (X, Y) * is complemented in (L w (X, Y), ||| • |||) *. This results in necessary and sufficient conditions for K (X, Y) to be reflexive, whereby the spaces X and Y need not satisfy the approximation property. Similar results follow when X and Y are locally convex spaces. * Financial support from the Potchefstroom University and Maseno University is greatly acknowledged. * * Financial support from the NRF and Potchefstroom University is greatly acknowledged.
Operator Hilbert spaces without the operator approximation property
Proceedings of the American Mathematical Society, 2002
We use a technique of Szankowski to construct operator Hilbert spaces that do not have the operator approximation property, including an example in a noncommutative Lp space for p = 2. 2-sum (as defined in [P2]) of row operator spaces has a subspace without the operator space version of the approximation property, or OAP. Since this subspace is a Hilbert space at the Banach space level, it has the Banach approximation property and even a basis. Thus, this is an example of an operator space with the AP but without the OAP. This answers a question of J. Kraus. Furthermore, this is also the first example of an operator space with a basis but without a complete basis. M. Junge suggested that a similar construction using rows in the Schatten p-class S p and Rademacher functions in L p [0, 1] could lead to an example of a Hilbertian subspace of L p [S p ] failing the OAP. In the third section we verify that this is indeed possible. These are new examples of Hilbertian subspaces of noncommutative L p spaces that are not completely complemented, even if p > 2. An operator space E is a Banach space E with an isometric embedding into B(H), the set of all bounded operators on a Hilbert space H. Or, equivalently, an
arXiv (Cornell University), 2010
These are the lecture notes I took from a topic course taught by Professor Jorgensen during the fall semester of 2009. The course started with elementary Hilbert space theory, and moved very fast to spectral theory, completely positive maps, Kadison-Singer conjecture, induced representations, self-adjoint extensions of operators, etc. It contains a lot of motivations and illuminating examples. I would like to thank Professor Jorgensen for teaching such a wonderful course. I hope students in other areas of mathematics would benefit from these lecture notes as well. Unfortunately, I have not been able to fill in all the details. The notes are undergoing editing. I take full responsibility for any errors and missing parts. Feng Tian 03. 2010 Contents Chapter 1. Elementary Facts 1.1. Transfinite induction 1.2. Dirac's notation 1.3. Operators in Hilbert space 1.4. Lattice structure of projections 1.5. Ideas in the spectral theorem 1.6. Spectral theorem for compact operators Chapter 2. GNS, Representations 2.1. Representations, GNS, primer of multiplicity 2.2. States, dual and pre-dual 2.3. Examples of representations, proof of GNS 2.4. GNS, spectral thoery 2.5. Choquet, Krein-Milman, decomposition of states 2.6. Beginning of multiplicity 2.7. Completely positive maps 2.8. Comments on Stinespring's theorem 2.9. More on the CP maps 2.10. Krien-Milman revisited 2.11. States and representation 2.12. Normal states 2.13. Kadison-Singer conjecture Chapter 3. Appliations to Groups 3.1. More on representations 3.2. Some examples 3.3. Induced representation 3.4. Example-Heisenberg group 3.5. Coadjoint orbits 3.6. Gaarding space 3.7. Decomposition of representation 3.8. Summary of induced reprep, d/dx example 3.9. Connection to Nelson's spectral theory Chapter 4. Unbounded Operators 4.1. Unbounded operators, definitions 4.2. Self-adjoint extensions Bibliography 98 CHAPTER 1 Elementary Facts 1.1. Transfinite induction Let (X, ≤) be a paritially ordered set. A sebset C of X is said to be a chain, or totally ordered, if x, y in C implies that either x ≤ y or y ≤ x. Zorn's lemma says that if every chain has a majorant then there exists a maximal element in X. Theorem 1.1. (Zorn) Let (X, ≤) be a paritially ordered set. If every chain C in X has a majorant (or upper bound), then there exists an element m in X so that x ≥ m implies x = m, for all x in X. An illuminating example of a partially ordered set is the binary tree model. Another example is when X is a family of subsets of a given set, partially ordered by inclusion. Zorn's lemma lies at the foundation of set theory. It is in fact an axiom and is equivalent to the axiom of choice and Hausdorff's maximality principle. Theorem 1.2. (Hausdorff Maximality Principle) Let (X, ≤) be a paritially ordered set, then there exists a maximal totally ordered subset L in X. The axiom of choice is equivalent to the following statement on infinite product, which itself is extensively used in functional analysis. Theorem 1.3. (axiom of choice) Let A α be a family of nonempty sets indexed by α ∈ I. Then the infinite Cartesian product Ω = α A α is nonempty. 1.1. TRANSFINITE INDUCTION However, the positivity condition may not be satisfied. Hence one has to pass to a quotient space by letting N = {f ∈ H 0 , f, f = 0}, andH 0 be the quotient space H 0 /N. The fact that N is really a subspace follows from the Cauchy-Schwartz inequality above. Therefore, •, • is an inner product onH 0. Finally, let H be the completion ofH 0 under •, • and H is a Hilbert space. Definition 1.8. Let H be a Hilbert space. A family of vectors {u α } in H is said to be an orthonormal basis of H if (1) u α , u β = δ αβ and (2) span{u α } = H. We are ready to prove the existance of an orthonormal basis of a Hilbert space, using transfinite induction. Again, the key idea is to cook up a partially ordered set satisfying all the requirments in the transfinite induction, so that the maximum elements turns out to be an orthonormal basis. Notice that all we have at hands are the abstract axioms of a Hilbert space, and nothing else. Everything will be developed out of these axioms. Theorem 1.9. Every Hilbert space H has an orthonormal basis. To start out, we need the following lemmas. Lemma 1.10. Let H be a Hilbert space and S ⊂ H. Then the following are equivalent: (1) x ⊥ S implies x = 0 (2) span{S} = H Lemma 1.11. (Gram-Schmidt) Let {u n } be a sequence of linearly independent vectors in H then there exists a sequence {v n } of unit vectors so that v i , v j = δ ij. Remark. The Gram-Schmidt orthogonalization process was developed a little earlier than Von Neumann's formuation of abstract Hilbert space. Proof. we now prove theorem (1.9). If H is empty then we are finished. Otherwise, let u 1 ∈ H. If u 1 = 1, we may consider u 1 / u 1 which is a normalized vector. Hence we may assume u 1 = 1. If span{u 1 } = H we are finished again, otherwise there exists u 2 / ∈ span{u 1 }. By lemma (1.11), we may assume u 2 = 1 and u 1 ⊥ u 2. By induction, we get a collection S of orthonormal vectors in H. Consider P(S) partially order by set inclusion. Let C ⊂ P(S) be a chain and let M = ∪ E∈C E. M is clearly a majorant of C. We claim that M is in the partially ordered system. In fact, for all x, y ∈ M there exist E x and E y in C so that x ∈ E x and y ∈ E y. Since C is a chain, we may assume E x ≤ E y. Hence x, y ∈ E 2 and x ⊥ y, which shows that M is in the partially ordered system. By Zorn's lemma, there exists a maximum element m ∈ S. It suffices to show that the closed span of m is H. Suppose this is false, then by lemma (1.10) there exists x ∈ H so that x ⊥ M. Since m ∪ {x} ≥ m and m is maximal, it follows that x ∈ m, which implies x ⊥ x. By the positivity axiom of the definition of Hilbert space, x = 0. Corollary 1.12. Let H be a Hilbert space, then H is isomorphic to the l 2 space of the index set of an ONB of H. Remark 1.13. There seems to be just one Hilbert space, which is true in terms of the Hilbert space structure. But this is misleading, because numerous
Representation Theorems for Operators on Free Banach Spaces of Countable Type
p-Adic Numbers, Ultrametric Analysis and Applications, 2019
This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.
On operators with bounded approximation property
It is known that any separable Banach space with BAP is a complemented subspace of a Banach space with a basis. We show that every operator with bounded approximation property, acting from a separable Banach space, can be factored through a Banach space with a basis.