We make explicit in terms of categories a number of statements from the theory of partial inner product spaces (-spaces) and operators on them. In particular, we construct sheaves and cosheaves of operators on certain -spaces of practical interest.

1. Motivation

Partial inner product spaces (-spaces) were introduced some time ago by Grossmann and one of us (JPA) as a structure unifying many constructions introduced in functional analysis, such as distributions or generalized functions, scales of Hilbert or Banach spaces, interpolation couples, and so forth [1–4]. Since these structures have regained a new interest in many aspects of mathematical physics and in modern signal processing, a comprehensive monograph was recently published by two of us [5], as well as a review paper [6].

Roughly speaking, a -space is a vector space equipped with a partial inner product, that is, an inner product which is not defined everywhere, but only for specific pairs of vectors. Given such an object, operators can be defined on it, which generalize the familiar notions of operators on a Hilbert space, while admitting extremely singular ones.

Now, in the previous work, many statements have a categorical “flavor”, but the corresponding technical language was not used; only some hints in that direction were given in [7]. Here we fill the gap and proceed systematically. We introduce the category PIP of (indexed) -spaces, with homomorphisms as arrows (they are defined precisely to play that role), as well as several other categories of -spaces.

In a second part, we consider a single -space V_I as a category by itself, called V_I, with natural embeddings as arrows. For this category V_I, we show, in Sections 4 and 5, respectively, that one can construct sheaves and cosheaves of operators. There are some restrictions on the -space V_I, but the cases covered by our results are the most useful ones for applications. Then, in Section 6, we describe the cohomology of these (co)sheaves and prove that, in many cases, the sheaves of operators are acyclic; that is, all cohomology groups of higher order are trivial.

Although sheaves are quite common in many areas of mathematics, the same cannot be said of cosheaves, the dual concept of sheaves, for which very few concrete examples are known. Actually, cosheaves were recently introduced in the context of nonclassical logic (see Section 7) and seem to be related to certain aspects of quantum gravity. Hence the interest of having at one′s disposal new, concrete examples of cosheaves, namely, cosheaves of operators on certain types of -spaces.

2. Preliminaries

2.1. Partial Inner Product Spaces

We begin by fixing the terminology and notations, following our monograph [5], to which we refer for a full information. For the convenience of the reader, we have collected in Appendix A the main features of partial inner product spaces and operators on them.

Throughout the paper, we will consider an indexed

-space V_I = {V_r, r ∈ I}, corresponding to the linear compatibility #. The assaying subspaces are denoted by V_p, V_r, …, p, r ∈ I. The set I indexes a generating involutive sublattice of the complete lattice ℱ(V, #) of all assaying subspaces defined by #; that is,

()

The lattice properties are the following:

(i)
involution: ,
(ii)
infimum: V_p∧q : = V_p∧V_q = V_p∩V_q, (p, q, r ∈ I),
(iii)
supremum: .

The smallest element of ℱ(V, #) is V^# = ⋂_rV_r, and the greatest element is V = ⋃_rV_r, but often they do not belong to V_I.

Each assaying subspace V_r carries its Mackey topology , and V^# is dense in every V_r, since the indexed -space V_I is assumed to be nondegenerate. In the sequel, we consider projective and additive indexed -spaces (see Appendix A) and, in particular, lattices of the Banach or Hilbert spaces (LBS/LHS).

Given two indexed -spaces V_I, Y_K, an operator A : V_I → Y_K may be identified with the coherent collection of its representatives A≃{A_ur}, where each A_ur : V_r → Y_u is a continuous operator from V_r into Y_u. We will also need the set . Every operator A has an adjoint A^×, and a partial multiplication between operators is defined.

A crucial role is played by homomorphisms, in particular, mono-, epi-, and isomorphisms. The set of all operators from V_I into Y_K is denoted by Op(V_I, Y_K) and the set of all homomorphisms by Hom(V_I, Y_K).

For more details and references to related work, see Appendix A or our monograph [5].

2.2. Categories

According to the standard terminology [8], a (small) category C is a collection of objects X, Y, Z, … and arrows or morphisms α, β, …, where we note α ∈ hom(X, Y) or

, satisfying the following axioms.

(i)
Identity: for any object X, there is a unique arrow 1_X : X → X.
(ii)
Composition: whenever , there is a unique arrow β ∘ α such that .
(iii)
Associativity: whenever , one has (γ∘β)∘α = γ∘(β∘α).
(iv)
Unit law: whenever , one has α∘1_X = α and 1_Y∘α = α.

In a category, an object S is initial if, for each object X, there is exactly one arrow S → X. An object T is final or terminal if, for each object X, there is exactly one arrow X → T. Two terminal objects are necessarily isomorphic (isomorphisms in categories are defined exactly as for indexed -spaces, see Appendix A).

Given two categories C and D, a covariant functor

is a morphism between the two categories. To each object X of C, it associates an object

of D, and, to each arrow α : X → Y in C, it associates an arrow

of D in such a way that

()

whenever the arrow β∘α is defined in C.

Given a category C, the opposite category C^op has the same objects as C and all arrows reversed: to each arrow α : X → Y, there is an arrow α^op : Y → X, so that α^op∘β^op = (β∘α) ^op.

A contravariant functor

may be defined as a functor

, or directly on C, by writing

. Thus, we have

()

Some standard examples of categories are

(i)
Set, the category of sets with functions as arrows.
(ii)
Top, the category of topological spaces with continuous functions as arrows.
(iii)
Grp, the category of groups with group homomorphisms as arrows.

For more details, we refer to standard texts, such as Mac Lane [8].

3. Categories of -Spaces

3.1. A Single -Space as Category

We begin by a trivial example.

A single-indexed

-space V_I = {V_r, r ∈ I} may be considered as a category V_I, where

(i)
the objects are the assaying subspaces {V_r, r ∈ I, V^#, V};
(ii)
the arrows are the natural embeddings {E_rs : V_r → V_s, r ⩽ s}, that is, the representatives of the identity operator on V_I.

The axioms of categories are readily checked as follows.

(i)
For every V_r, there exists an identity, E_rr : V_r → V_r, the identity map.
(ii)
For every V_r, V_s with r ⩽ s, one has E_ss∘E_sr = E_sr and E_sr∘E_rr = E_sr.
(iii)
For every V_r, V_s, V_t with r ⩽ s ⩽ t, one has E_ts∘E_sr = E_tr.
(iv)
For every V_r, V_s, V_t, V_u with r ⩽ s ⩽ t ⩽ u, one has (E_ut∘E_ts)∘E_sr = E_ut∘(E_ts∘E_sr).

In the category V_I, we have:

(i)
is an initial object: for every V_r ∈ V_I, there is a unique arrow .
(ii)
V_∞ : = V = ∑_r∈I is a terminal object: for every V_r ∈ V_I, there is a unique arrow E_∞r : V_r → V.
(iii)
The compatibility defines a contravariant functor V_I → V_I.

Although this category seems rather trivial, it will allow us to define sheaves and cosheaves of operators, a highly nontrivial (and desirable) result.

3.2. A Category Generated by a Single Operator

In the indexed

-space V_I = {V_r, r ∈ I}, take a single totally regular operator, that is, an operator A that leaves every V_r invariant. Hence so does each power Aⁿ, n ∈ ℕ. Then this operator induces a category A(V_I), as follows.

(i)
The objects are the assaying subspaces {V_r, r ∈ I}.
(ii)
The arrows are the operators .

The axioms of categories are readily checked as follows.

(i)
For every V_r, there exists an identity, A_rr : V_r → V_r, since A is totally regular.
(ii)
For every V_r, V_s with r ⩽ s, one has and .
(iii)
For every V_r, V_s, V_t with r ⩽ s ⩽ t, one has .
(iv)
For every V_r, V_s, V_t, V_u with r ⩽ s ⩽ t ⩽ u, one has .

As for V_I, the space is an initial object in A(V_I) and V_∞ : = V is a terminal object.

The adjunction A ↦ A^# defines a contravariant functor from A(V_I) into , where the latter is the category induced by A^#. The proof is immediate.

3.3. The Category PIP of Indexed -Spaces

The collection of all indexed

-spaces constitutes a category, which we call PIP, where

(i)
objects are indexed -spaces {V_I};
(ii)
arrows are homomorphisms A : V_I → Y_K, where an operator A ∈ Op(V_I, Y_K) is called a homomorphism if
- (a)
  for every r ∈ I there exists u ∈ K such that both A_ur and exist;
- (b)
  for every u ∈ K there exists r ∈ I such that both A_ur and exist.

For making the notation less cumbersome (and more automatic), we will, henceforth, denote by A_KI an element A ∈ Hom(V_I, Y_K). Then the axioms of a category are obviously satisfied as follows.

(i)
For every V_I, there exists an identity, 1_I ∈ Hom(V_I, V_I), the identity operator on V_I.
(ii)
For every V_I, Y_K, one has 1_K∘A_KI = A_KI and A_KI∘1_I = A_KI.
(iii)
For every V_I, Y_K, W_L, one has B_LK∘A_KI = C_LI ∈ Hom(V_I, W_L).
(iv)
For every V_I, Y_K, W_L, Z_M, one has (C_ML∘B_LK)∘A_KI = C_ML∘(B_LK∘A_KI) ∈ Hom(V_I, Z_M).

The category PIP has no initial object and no terminal object; hence, it is not a topos.

One can define in the same way smaller categories LBS and LHS, whose objects are, respectively, lattices of the Banach spaces (LBS) and lattices of the Hilbert spaces (LHS), the arrows being still the corresponding homomorphisms.

3.3.1. Subobjects

We recall that a homomorphism M_KI ∈ Hom(V_I, Y_K) is a monomorphism if M_KLA_LI = M_KLB_LI implies A_LI = B_LI, for any pair A_LI, B_LI and any indexed -space W_L (a typical example is given in Appendix A). Two monomorphisms M_LI, N_LK with the same codomain W_L are equivalent if there exists an isomorphism U_KI such that N_LKU_KI = M_LI. Then a subobject of V_I is an equivalence class of monomorphisms into V_I. A -subspace W of an indexed -space V is defined as an orthocomplemented subspace of V, and this holds if and only if W is the range of an orthogonal projection, W_I = PV_I. Now the embedding M : W_I = PV_I ↦ V_I is a monomorphism; thus, orthocomplemented subspaces are subobjects of PIP.

However, the converse is not true, at least for a general indexed -space. Take the case where V is a noncomplete prehilbert space (i.e., V = V^#). Then every subspace is a subobject, but need not to be the range of a projection. To give a concrete example [5, Section 3.4.5], take V = 𝒮(ℝ), the Schwartz space of test functions. Let W = 𝒮₊ : = {φ ∈ 𝒮 : φ(x) = 0 for x ⩽ 0}. Then W^⊥ = 𝒮₋ : = {ψ ∈ 𝒮 : ψ(x) = 0 for x⩾0}; hence, W^⊥⊥ = W. However, W is not orthocomplemented, since every χ ∈ W + W^⊥ satisfies χ(0) = 0, so that W + W^⊥ ≠ 𝒮. Yet W is the range of a monomorphism (the injection), hence, a subobject. However, this example addresses an indexed -space which is not a LBS/LHS.

Take now a LBS/LHS V_I = {V_r, r ∈ I} and a vector subspace W. In order that W becomes a LBS/LHS W_I in its own right, we must require that, for every r ∈ I, W_r = W∩V_r and are a dual pair with respect to their respective Mackey topologies and that the intrinsic Mackey topology coincides with the norm topology induced by V_r. In other words, W must be topologically regular, which is equivalent that it be orthocomplemented [5, Section 3.4.2]. Now the injection M_I : W_I → V_I is clearly a monomorphism, and W_I is a subobject of V_I. Thus, we have shown that, in a LBS/LHS, the subobjects are precisely the orthocomplemented subspaces.

Coming back to the previous example of a noncomplete prehilbert space, we see that an arbitrary subspace W need not be orthocomplemented, because it may fail to be topologically regular. Indeed the intrinsic topology τ(W, W) does not coincide with the norm topology, unless W is orthocomplemented (see the discussion in [5, Section 3.4.5]). In the Schwartz example above, one has W^⊥⊥ = W, which means that W is τ(W, W) closed, hence, norm closed, but it is not orthocomplemented.

Remark 3.1. Homomorphisms are defined between arbitrary -spaces. However, when it comes to indexed -spaces, the discussion above shows that the notion of homomorphism is more natural between two indexed -spaces of the same type, for instance, two LBSs or two LHSs. This is true, in particular, when trying to identify subobjects. This suggests to define categories LBS and LHS, either directly as above or as subcategories within PIP, and then define properly subobjects in that context.

3.3.2. Superobjects

Dually, one may define superobjects in terms of epimorphisms. We recall that a homomorphism N_KL ∈ Hom(V_I, W_K) is an epimorphism if A_IKN_KL = B_IKN_KL implies A_IK = B_IK, for any pair A_IK, B_IK and any indexed -space Y_L. Then a superobject is an equivalence class of epimorphisms, where again equivalence means modulo isomorphisms.

Whereas monomorphisms are natural in the context of sheaves; epimorphisms are natural in the dual structure, that is, cosheaves.

4. Sheaves of Operators on -Spaces

4.1. Presheaves and Sheaves

Let X be a topological space, and let C be a (concrete) category. Usually C is the category of sets, the category of groups, the category of abelian groups, or the category of commutative rings. In the standard fashion [8, 9], we proceed in two steps.

Definition 4.1. A presheaf on X with values in C is a map defined on the family of open subsets of X such that

(PS₁)
for each open set U of X, there is an object in C;
(PS₂)
for each inclusion of open sets T⊆U, there is given a restriction morphism in the category C, such that is the identity for every open set U and whenever S⊆T⊆U.

Definition 4.2. Let be a presheaf on X, and let U be an open set of X. Every element is called a section of over U. A section over X is called a global section.

Example 4.3. Let X be a topological space; let C be the category of vector spaces. Let associate to each open set U the vector space of continuous functions on U with values in ℂ. If T⊆U, associates to each continuous function on U its restriction to T. This is a presheaf. Any continuous function on U is a section of on U.

Definition 4.4. Let be a presheaf on the topological space X. One says that is a sheaf if, for every open set U ⊂ X and for every open covering {U_i} _i∈I of U, the following conditions are fulfilled:

(S₁)
given such that , for every i ∈ I, then s = s′ (local identity);
(S₂)
given such that , for every i, j ∈ I, then there exists a section such that , for every i ∈ I (gluing).

The section s whose existence is guaranteed by axiom (S₂) is called the gluing, concatenation, or collation of the sections s_i. By axiom (S₁), it is unique. The sheaf may be seen as a contravariant functor from the category of open sets of X into .

4.2. A Sheaf of Operators on an Indexed -Space

Let V_I = {V_r, r ∈ I} be an indexed -space, and let V_I be the corresponding category defined in Section 3.1. If we put on I the discrete topology, then I defines an open covering of V. Each V_r carries its Mackey’s topology .

We define a sheaf on V_I by the contravariant functor

given by

()

This means that an element of Op_r is a representative A_wr from V_r into some V_w. In the sequel, we will use the notation A_*r whenever the dependence on the first index may be neglected without creating ambiguities. By analogy with functions, the elements of Op_r may be called germs of operators. Op_r is the restriction to V_r of the set L_r of left multipliers [5, Section 6.2.3]:

()

Then we have:

(i)
When V_q⊆V_p, define by , for A_*p ∈ Op_p. Clearly, and if V_r ⊂ V_q ⊂ V_p. Hence, is a presheaf.

(ii)
(S₁) is clearly satisfied. As for (S₂), if A_*r ∈ Op_r and are such that ; that is, , then these two operators are the (r∧s, *) representative of a unique operator A ∈ Op(V_I). It remains to prove that A extends to V_r∨s and that its representative A_*r∨s extends both A_*r and .

Proposition 4.5. Let the indexed -space V_I be additive; that is, V_r∨s = V_r + V_s, for all r, s ∈ I. Then the map given in (4.1) is a sheaf of operators on V_I.

Proof. By linearity, A_*r and A_*s can be extended to an operator A^(r∨s) on V_r + V_s as follows:

()

This operator is well defined. Let indeed

be two decompositions of f ∈ V_r + V_s, so that

. Then

. Hence,

. Taking the restriction to V_r∩V_s = V_r∧s, this relation becomes A_*r∧s(f−f′)_r∧s = A_*r∧s(f′−f)_r∧s = 0, so that the vector A^(r∨s)f is uniquely defined.

Next, by additivity, V_r + V_s, with its inductive topology, coincides with V_r∨s, and, thus, A_*r∨s is the (r∨s, *)-representative of the operator A ∈ Op(V_I). Therefore, is a sheaf.

We recall that the most interesting classes of indexed -spaces are additive, namely, the projective ones and, in particular, LBSs and LHSs. Thus the proposition just proven has a widely applicable range.

5. Cosheaves of Operators on -Spaces

5.1. Pre-Cosheaves and Cosheaves

Pre-cosheaves and cosheaves are the dual notions of presheaves and sheaves, respectively. Let again X be a topological space, with closed sets W_i so that X = ⋃_i∈IW_i, and let C be a (concrete) category.

Definition 5.1. A pre-cosheaf on X with values in C is a map defined on the family of closed subsets of X such that

(PC₁)
for each closed set W of X, there is an object in C;
(PC₂)
for each inclusion of closed sets Z⊇W, there is given a extension morphism in the category C, such that is the identity for every closed set W and whenever T⊇Z⊇W.

Definition 5.2. Let be a pre-cosheaf on the topological space X. We say that is a cosheaf if, for every nonempty closed set W = ⋂_j∈JW_j, J⊆I and for every family of (local) sections , the following conditions are fulfilled:

(CS₁)
given such that , for every j ∈ I such that W⊆W_j, then t = t′;
(CS₂)
if , for every j, k ∈ J, then there exists a unique section such that , for every j ∈ J.

The cosheaf may be seen as a covariant functor from the category of closed sets of X into .

Now, there are situations where extensions do not exist for all inclusions Z⊇W but only for certain pairs. This will be the case for operators on a -space, as will be seen below. Thus, we may generalize the notions of (pre-) cosheaf as follows (there could be several variants).

Definition 5.3. Let X be a topological space, let 𝒞l(X) be the family of closed subsets of X, and let ≼ a be coarsening of the set inclusion in 𝒞l(X); that is, W≼Z implies W⊆Z, but not necessarily the opposite. A partial pre-cosheaf with values in C is a map defined on 𝒞l(X), satisfying conditions (PC₁) and

(pPC₂)
if Z≽W, there is given an extension morphism in the category C, such that is the identity for every closed set W and whenever T≽Z≽W.

We may use the term “partial,” since here not all pairs w ⩽ z admit extensions, but only certain pairs, namely, those that satisfy w≼z. This is analogous to the familiar situation of a compatibility.

Definition 5.4. Let be a partial pre-cosheaf on the topological space X. We say that is a partial cosheaf if, for every nonempty closed set W = ⋂_j∈JW_j, J⊆I and for every family of sections , the following conditions are fulfilled:

(pCS₁)
given such that and exist and are equal, for every j ∈ J, then t = t′;
(pCS₂)
if and exist and are equal, for every j, k ∈ J, then exists for every j ∈ J and there exists a unique section such that , for every j ∈ J.

5.2. Cosheaves of Operators on Indexed -Spaces

Let again V_I = {V_r, r ∈ I} be an indexed

-space. If we put on I the discrete topology, then the assaying subspaces may be taken as closed sets in V. In order to build a cosheaf, we consider the same map as before in (4.1):

()

Then we immediately face a problem. Given V_q, there is no V_p, with V_q⊆V_p, such that

exists. One can always find an operator A such that

and

; in other words, A_*q exists, but it cannot be extended to V_p. This happens, for instance, each time

has a maximal element (see Figure 3.1 in [5]). There are three ways out of this situation.

5.2.1. General Operators: The Additive Case

We still consider the set of all operators Op(V_I). Take two assaying subspaces V_r, V_s and two operators A^(r) ∈ Op_r, B^(s) ∈ Op_s, and assume that they have a common extension C^(r∨s) to Op_r∨s. This means that, for any suitable w, C^(r∨s) : V_r∨s → V_w is the (r∨s, w)-representative C_w,r∨s of a unique operator C ∈ Op(V_I), and C = A = B. A fortiori, A, and B coincide on V_r∧s = V_r∩V_s; that is, condition (pCS₂) is satisfied.

Assume now that V_I is additive; that is, V_r∨s coincides with V_r + V_s, with its inductive topology. Then we can proceed as in the case of a sheaf, by defining the extension C of A and B by linearity. In other words, since C^(r∨s)f_r = A_*rf_r and C^(r∨s)f_s = B_*sf_s, we may write, for any f = f_r + f_s ∈ V_r + V_s,

()

Here too, this operator is well defined. Let indeed

be two decompositions of f ∈ V_r + V_s, so that

. Then necessarily

. Hence,

. Taking the restriction to V_r∩V_s = V_r∧s, this relation becomes A_*r∧s(f − f′) _r∧s = B_*r∧s(f′−f)_r∧s, and this equals 0, since A = B on V_r∧s, by the condition (pcS₂).

The conclusion is that, for any pair of assaying subspaces V_r, V_s, the extension maps and always exist, and the condition (pCS₂) is satisfied for that pair. However, this is not necessarily true for any comparable pair, and this motivates the coarsening of the order given in Proposition 5.5 below. Condition (pPC₂) is also satisfied, as one can see by taking supremums (= sums) of successive pairs within three assaying subspaces V_r, V_s, V_t (and using the associativity of ∨). Thus, me may state the following.

Proposition 5.5. Let V_I = {V_r, r ∈ I} be an additive indexed -space. Then the map given in (4.1) is a partial cosheaf with respect to the partial order on I:

()

5.2.2. Universal Left Multipliers

We consider only the operators A which are everywhere defined; that is,

. This is precisely the set of universal left multipliers:

()

where

. Correspondingly, we consider the map

()

Here again, elements of LOp_r are representatives A_*r. In that case, extensions always exist. When V_q⊆V_p, define

for A_*q ∈ LOp_q. Clearly,

and

if V_r ⊂ V_q ⊂ V_p. Hence,

is a pre-cosheaf. Moreover, if

; that is,

, these two operators are the (p∨q, *) representative of a unique operator A ∈ LOp(V_I). Then, of course, one has

and

; thus,

is a cosheaf. Therefore,

Proposition 5.6. Let V_I = {V_r, r ∈ I} be an arbitrary indexed -space. Then the map given in (5.5) is a cosheaf on V_I with values in LOp(V_I), the set of universal left multipliers, with extensions given by .

5.2.3. General Operators: The Projective Case

We consider again all operators in Op(V_I). The fact is that the set ⋃_r∈IOp_r is an initial set in Op(V_I), like every

; that is, it contains all predecessors of any of its elements, and this is natural for constructing sheaves into it. But it is not when considering cosheaves. By duality, one should rather take a final set, that is, containing all successors of any of its elements, just like any

. Hence, we define the map:

()

Now indeed

is a final set. Elements of

have been denoted as A_r*, but, for definiteness, we could also replace A_r* by

where

We claim that the map defines a cosheaf, with extensions , for , which exist whenever q ⩽ p.

As in the case of Section 5.2.1, assume that have a common extension belonging to ; that is, A_r∨s* : = E_r∨s,rA_r* = B_r∨s* : = E_r∨s,sB_s*. Call this operator C_r∨s*. Thus, for any suitable w, C_r∨s,w is the (w, r∨s) representative of a unique operator C ∈ Op(V_I), which extends A and B. Thus C = A = B ∈ Op(V_I). Assume now that V_I is projective; that is, V_r∧s = V_r∩V_s, for all r, s ∈ I, with the projective topology. Then, C_r∧s* = A_r∧s* = B_r∧s*; that is, condition (CS₂) is satisfied with extensions and . The rest is obvious. As in the case of sheaves, extensions from V_r, V_s to V_r∨s exist by linearity, since V_I is projective, hence, additive. Thus, we may state the following.

Proposition 5.7. Let the indexed -space V_I be projective that is, V_r∧s = V_r∩V_s, for all r, s ∈ I, with the projective topology. Then the map given in (5.6) is a cosheaf of operators on V_I.

Corollary 5.8. Let V_I = {V_r, r ∈ I} be a projective indexed -space. Then V_I generates in a natural fashion a sheaf and a cosheaf of operators. If V_I is additive, but not projective; it generates a sheaf and a partial cosheaf of operators.

6. Cohomology of an Indexed -Space

6.1. Cohomology of Operator Sheaves

It is possible to introduce a concept of sheaf cohomology group defined on an indexed -space according to the usual definition of sheaf cohomology [10]. Let V_I be an indexed -space. The set of its assaying spaces, endowed with the discrete topology, defines an open covering of V. Endowed with the Mackey topology, , each V_r is a Hausdorff vector subspace of V.

Definition 6.1. Given the indexed -space V_I, let {V_j, j ∈ I} be the open covering of V by its assaying subspaces, and let be the sheaf of operators defined in (4.1) above. We call p-cochain with values in the map which associates, to each intersection of open sets , an operator belonging to .

Thus, a p-cochain with value in is a set . The set of p-cochains is denoted by . For example, a 0-cochain is a set {A_j, j ∈ I}. A 1-cochain is a set .

Definition 6.2. Given the notion of p-cochain, the corresponding coboundary operator D is defined by

()

where the hat corresponds to the omission of the corresponding symbol. One checks that D is a coboundary operator; that is, DD = 0.

If we compute the action of the coboundary operator on 0-cochains, we get

()

This equation is nothing but the condition (S₁) in Definition 4.4, that is, the necessary condition for getting a sheaf.

We can also compute the action of the coboundary operator on 1-cochains. We get

()

Now, let us suppose that a 1-cochain is defined by 𝒜 = Dℬ, where ℬ is a 0-cochain on V. The previous formula becomes:

()

Using the properties of restrictions we get

()

which shows that indeed DDℬ = 0.

Now it is possible to define cohomology groups of the sheaf on a indexed -space V.

Definition 6.3. Let V_I be an indexed -space, endowed with the open covering of its assaying spaces, {V_j, j ∈ I}, and let be the corresponding sheaf of operators on V_I. Then the pth cohomology group is defined as , where is the set of p-cocycles, that is, p-cochains 𝒜 such that D𝒜 = 0, and is the set of p-coboundaries, that is, p-cochains ℬ for which there exists a (p − 1)-cochain 𝒞 with ℬ = D𝒞.

The definition of the group is motivated by the fact that the p-cocycle 𝒜 is given only up to a coboundary ℬ = D𝒞 : D𝒜 = 0 = D(𝒜 + ℬ) = D(𝒜 + D𝒞).

Our definition of cohomology groups of sheaves on indexed -spaces depends on the particular open covering we are choosing. So far we have used the open covering given by all assaying subspaces, but there might be other ones, typically consisting of unions of assaying subspaces. We say that an open covering {V_j, j ∈ J} of an indexed -space V_I is finer than another one, {U_k, k ∈ K}, if there exists an application t : J → K such that V_j ⊂ U_t(j), for all j ∈ J. For instance, U_t(j) could be a union ∪_lV_l containing V_j. According to the general theory of sheaf cohomology [9], this induces group homomorphisms . It is then possible to introduce cohomology groups that do not depend on the particular open covering, namely, by defining as the inductive limit of the groups with respect to the homomorphisms .

Using a famous result of Cartan and Leray and an unpublished work of J. Shabani, we give now a theorem which characterizes the cohomology of sheaves on indexed -spaces. We collect in Appendix B the definitions and the results needed for this discussion. We know that an indexed -space V endowed with the Mackey topology is a separated (Hausdorff) locally convex space, but it is not necessarily paracompact, unless V is metrizable, in particular, a Banach or a Hilbert space. In that case, it is possible to define a fine sheaf (see Definition B.3) of operators on V and apply the Cartan-Leray Theorem B.4. This justifies the restriction of metrizability in the following theorem.

Theorem 6.4. Given an indexed -space V_I, define the sheaf . Then, if V is metrizable for its Mackey topology, the sheaf is acyclic; that is, the cohomology groups are trivial for all p⩾1; that is, .

Proof. First of all, V is a paracompact space, since it is metrizable.

Next, we check that the sheaf on the paracompact space V is fine. Indeed, let {V_j, j ∈ J} be a locally finite open covering of V. We can associate to the latter a partition of unity {φ_j, j ∈ J}. Then we can use {φ_j, j ∈ J} to define the homomorphisms . Let V_r ⊂ ∪_s∈KV_s for some index set K. For each s ∈ K, we consider the set of operators Op_s = {A_*s : V_s → V}, and we define h_j : A_s ↦ h_j(A_s): = φ_jA_s. This defines a homomorphism Op_s → Op_s and then a homomorphism . One can check that satisfies conditions (1) and (2) in Definition B.3, and, thus, is a fine sheaf.

Then the result follows from Theorem B.4.

The crucial point of the proof here is the Cartan-Leray theorem. Let us give a flavor of the proof of a particular case of this classical result. Let [𝒞] be an element of

. We want to show that [𝒞] = 0. As an equivalence class, [𝒞] can be represented by an element

, such that D𝒞 = 0; that is,

()

Using the fact that

is a fine sheaf, we can choose the partition of unity {φ_i, i ∈ I} to define the 0-cochain ℰ = {E_j}, where E_j = −∑_k∈Iφ_k(C_jk). This sum is well defined since the covering is locally finite. Applying the coboundary operator, we get

()

Putting k = j₂ and using the equation

, we find

()

But since {φ_i, i ∈ I} is a partition of unity, we get

. This means that Dℰ = 𝒞, and; thus, [𝒞] = 0.

Of course, the restriction that V is to be metrizable is quite strong, but still the result applies to a significant number of interesting situations, for instance,

(1)
a finite chain of reflexive Banach spaces or Hilbert spaces, for instance, a triplet of Hilbert spaces or any of its refinements, as discussed in [5, Section 5.2.2],
(2)
an indexed -space V_I whose extreme space V is itself a Hilbert space, like the LHS of functions analytic in a sector described in [5, Section 4.6.3],
(3)
a Banach Gel′fand triple, in the sense of Feichtinger (see [11]), that is, an RHS (or LBS) in which the extreme spaces are (nonreflexive) Banach spaces. A nice example, extremely useful in Gabor analysis, is the so-called Feichtinger algebra 𝒮₀(ℝ^d), which generates the triplet
()
The latter can often replace the familiar Schwartz triplet of tempered distributions. Of course, one can design all sorts of LHSs of LBSs interpolating between the extreme spaces, as explained in [5, Sections 5.3 and 5.4].

In fact, what is really needed for the Cartan-Leray theorem is not that V becomes metrizable, but that it becomes paracompact. Indeed, without that condition, the situation becomes totally unmanageable. However, except for metrizable spaces, we could not find interesting examples of indexed -spaces with V paracompact.

6.2. Cohomology of Operator Cosheaves

It is also possible to get cohomological concepts on cosheaves defined from indexed -spaces. The assaying spaces can also be considered as closed sets. Let, thus, {W_j, j ∈ I} be such a closed covering of V, and let be a cosheaf of operators defined in (5.5), namely, together with maps as above. In the sequel, denotes the set of operators . Alternatively, one could consider the cosheaf defined in (5.6), and proceed in the same way.

In this setup, we may introduce the necessary cohomological concepts, as in Definitions 6.1 and 6.2.

Definition 6.5. A p-cochain with values in the cosheaf is a map which associates to each union of closed sets , an operator of . A p-cochain is, thus, a set . The set of such p-cochains will be denoted by .

Definition 6.6. One can then introduce the coboundary operator as follows:

()

In view of the properties of the maps

, we check that

In the case of 0-cochains, a straightforward application of the formula leads to

()

And if we put this to zero, we get the constraint

, which has to be satisfied in order to build a cosheaf, according to the condition (CS₂). On 1-cochains, the coboundary action gives

()

The cohomology groups of the cosheaf

, with obvious notations, can then be defined in a natural way. Similarly for

and

Now it is tempting to proceed as in the case of sheaves and define the analog of a fine cosheaf, in such a way that one can apply a result similar to the Cartan-Leray theorem. But this is largely unexplored territory, so we will not venture into it.

7. Outcome

The analysis so far shows that several aspects of the theory of indexed -spaces and operators on them have a natural formulation in categorical terms. Of course, this is only a first step, many questions remain open. For instance, does there exist a simple characterisation of the dual category PIP^op? Could it be somehow linked to the category of partial *algebras, in the same way as Set^op is isomorphic to the category of complete atomic Boolean algebras (this is the so-called Lindenbaum-Tarski duality [12, Section VI.4.6]?

In addition, our constructions yield new concrete examples of sheaves and cosheaves, namely, (co)sheaves of operators on an indexed -space, and this is probably the most important result of this paper. Then, another open question concerns the cosheaf cohomology groups. Can one find conditions under which the cosheaf is acyclic, that is, , for all p⩾1, or, similarly, , for all p⩾1?

In guise of conclusion, let us note that cosheaf is a new concept which was introduced in a logical framework in order to dualize the sheaf concept [13]. In fact one knows that the category of sheaves (which is in fact a topos) is related to Intuitionistic logic and Heyting algebras, in the same way as the category of sets has deep relations with the classical proposition logic and Boolean algebras [12, Section I.1.10].

More precisely, Classical logic satisfies the noncontradiction principle ¬(p∧¬p) and the excluded middle principle (p∨¬p) (i.e., : = Not-(p And Not-p) and : = p Or Not-p.). Intuitionistic logic satisfies but not . Finally, we know that Paraconsistent logic, satisfying but not , is related to Brouwer algebras, also called co-Heyting algebras [14, 15]. Then, it is natural to wonder what is the category, if any, (mimicking the category of sets for the classical case and the topos of sheaves for the intuitionistic case), corresponding to Paraconsistent logic? The category of cosheaves can be a natural candidate for this. And this is the reason why it was tentatively introduced in a formal logic context. The category of closed sets of a topological space happens to be a cosheaf. But up to now we did not know any other examples of cosheaves in other areas of mathematics. Therefore, it is interesting to find here additional concrete examples of cosheaves in the field of functional analysis.

In a completely different field, the search for a quantum gravity theory, Shahn Majid [16, page 294] (see also the diagram in [17, page 122]) has proposed to unify quantum field theory and general relativity using a self-duality principle expressed in categorical terms. His approach shows deep connections, on the one hand, between quantum concepts and Heyting algebras (the relations between quantum physics and Intuitionistic logic are well known) and, on the other hand, following a suggestion of Lawvere [16], between general relativity (Riemannian geometry and uniform spaces) and co-Heyting algebras (Brouwer algebras). Therefore, it is very interesting to shed light on concepts arising naturally from Brouwer algebras, and this is precisely the case of cosheaves. Sheaves of operators on -spaces are connected to quantum physics. Is there any hope to connect cosheaves of operators on some -spaces to (pseudo-)Riemannian geometry, uniform spaces, or to theories describing gravitation? This is an open question suggested by Majid′s idea of a self-duality principle (let us note that classical logic is the prototype of a self-dual structure, self-duality being given by the de Morgan rule, which transforms into !).

Acknowledgments

J.-P. Antoine thanks Juma Shabani (UNESCO) for communicating his unpublished results. D. Lambert thanks Bertrand Hespel (FUNDP) for inspiring discussions. The authors all thank the anonymous referee for his constructive remarks that have definitely improved the paper.

Appendices

A. Partial Inner Product Spaces

A.1. -Spaces and Indexed -Spaces

For the convenience of the reader, we have collected here the main features of partial inner product spaces and operators on them, keeping only what is needed for reading the paper. Further information may be found in our review paper [6] or our monograph [5].

The general framework is that of a

-space V, corresponding to the linear compatibility #, that is, a symmetric binary relation f#g which preserves linearity. We call assaying subspace of V a subspace S such that S^## = S, and we denote by ℱ(V, #) the family of all assaying subspaces of V, ordered by inclusion. The assaying subspaces are denoted by V_r, V_q, …, and the index set is F. By definition, q ⩽ r if and only if V_q⊆V_r. Thus, we may write

()

General considerations imply that the family ℱ(V, #): = {V_r, r ∈ F}, ordered by inclusion, is a complete involutive lattice; that is, it is stable under the following operations, arbitrarily iterated:

(i)
involution:,
(ii)
infimum: V_p∧q : = V_p∧V_q = V_p∩V_q, (p, q, r ∈ F),
(iii)
supremum:.

The smallest element of ℱ(V, #) is V^# = ⋂_rV_r, and the greatest element is V = ⋃_rV_r.

By definition, the index set F is also a complete involutive lattice; for instance,

()

Given a vector space V equipped with a linear compatibility #, a partial inner product on (V, #) is a Hermitian form 〈·∣·〉 defined exactly on compatible pairs of vectors. A partial inner product space (-space) is a vector space V equipped with a linear compatibility and a partial inner product.

From now on, we will assume that our -space (V, #, 〈·∣·〉) is nondegenerate; that is, 〈f∣g〉 = 0 for all f ∈ V^# implies g = 0. As a consequence, (V^#, V) and every couple , are a dual pair in the sense of topological vector spaces [18]. Next, we assume that every V_r carries its Mackey topology , so that its conjugate dual is . Then, r < s implies V_r ⊂ V_s, and the embedding operator E_sr : V_r → V_s is continuous and has dense range. In particular, V^# is dense in every V_r.

As a matter of fact, the whole structure can be reconstructed from a fairly small subset of ℱ, namely, a generating involutive sublattice ℐ of ℱ(V, #), indexed by I, which means that

()

The resulting structure is called an indexed

-space and denoted simply by V_I : = (V, ℐ, 〈·∣·〉).

Then an indexed

-space V_I is said to be

(i)
additive if V_p∨q = V_p + V_q, for all p, q ∈ I,

(ii)
projective if ; here V_p∧q|_τ denotes V_p∧q equipped with the Mackey topology , the right-hand side denotes V_p∩V_q with the topology of the projective limit from V_p and V_q, and ≃ denotes an isomorphism of locally convex topological spaces.

For practical applications, it is essentially sufficient to restrict oneself to the case of an indexed

-space satisfying the following conditions:

(i)
every V_r, r ∈ I, is a Hilbert space or a reflexive Banach space, so that the Mackey topology coincides with the norm topology;

(ii)
there is a unique self-dual, Hilbert, assaying subspace .

In that case, the structure V_I : = (V, ℐ, 〈·∣·〉) is called, respectively, a lattice of Hilbert spaces (LHS) or a lattice of Banach spaces (LBS) (see [5] for more precise definitions, including explicit requirements on norms). The important facts here are that

(i)
every projective indexed -space is additive;

(ii)
an LBS or an LHS is projective if and only if it is additive.

Note that V^#, V themselves usually do not belong to the family {V_r, r ∈ I}, but they can be recovered as

()

A standard, albeit trivial, example is that of a Rigged Hilbert space (RHS) Φ ⊂ ℋ ⊂ Φ^# (it is trivial because the lattice ℱ contains only three elements). One should note that the construction of an RHS from a directed family of Hilbert spaces, via projective and inductive limits, has been investigated recently by Bellomonte and Trapani [19]. Similar constructions, in the language of categories, may be found in the work of Mityagin and Shvarts [20] and that of Semadeni and Zidenberg [21].

Let us give some concrete examples.

(i) Sequence Spaces Let V be the space ω of all complex sequences x = (x_n) and define on it (i) a compatibility relation by , (ii) a partial inner product . Then ω^# = φ, the space of finite sequences, and the complete lattice ℱ(ω, #) consists of Köthe′s perfect sequence spaces [18, § 30]. Among these, a nice example is the lattice of the so-called ℓ_ϕ spaces associated to symmetric norming functions or, more generally, the Banach sequence ideals discussed in [5, Section 4.3.2] and previously in [20, § 6] (in this example, the extreme spaces are, resp., ℓ¹ and ℓ^∞).

(ii) Spaces of Locally Integrable Functions Let V be , the space of Lebesgue measurable functions, integrable over compact subsets, and define a compatibility relation on it by f#g⇔∫_ℝ | f(x)g(x) | dx < ∞ and a partial inner product . Then , the space of bounded measurable functions of compact support. The complete lattice consists of the so-called Köthe function spaces. Here again, normed ideals of measurable functions in L¹([0,1], dx) are described in [20, § 8].

A.2. Operators on Indexed -Spaces

Let V_I and Y_K be two nondegenerate indexed

-spaces (in particular, two LHSs or LBSs). Then an operator from V_I to Y_K is a map A from a subset 𝒟(A) ⊂ V into Y, such that

(i)
, where is a nonempty subset of I.
(ii)
For every , there exists u ∈ K such that the restriction of A to V_r is a continuous linear map into Y_u (we denote this restriction by A_ur).
(iii)
A has no proper extension satisfying (i) and (ii).

We denote by Op(V_I, Y_K) the set of all operators from V_I to Y_K and, in particular, Op(V_I) : = Op(V_I, V_I). The continuous linear operator A_ur : V_r → Y_u is called a representative of A. Thus, the operator A may be identified with the collection of its representatives, A≃{A_ur : V_r → Y_u exists as a bounded operator}. We will also need the following sets:

()

The following properties are immediate:

(i)
is an initial subset of I: if and r′ < r, then , and A_ur′ = A_ur E_rr′, where E_rr′ is a representative of the unit operator.
(ii)
is a final subset of K: if and u′ > u, then , and A_u′r = E_u′u A_ur.

Although an operator may be identified with a separately continuous sesquilinear form on V^# × V^#, it is more useful to keep also the algebraic operations on operators, namely,

(i)
adjoint: every A ∈ Op(V_I, Y_K) has a unique adjoint A^× ∈ Op(Y_K, V_I) and one has A^×× = A, for every A ∈ Op(V_I, Y_K): no extension is allowed, by the maximality condition (iii) of the definition.
(ii)
partial multiplication: let V_I, W_L, and Y_K be nondegenerate indexed -spaces (some, or all, may coincide). Let A ∈ Op(V_I, W_L), and, B ∈ Op(W_L, Y_K). We say that the product BA is defined if and only if there is a ; that is, if and only if there is continuous factorization through some W_t:

()

Among operators on indexed

-spaces, a special role is played by morphisms.

An operator A ∈ Op(V_I, Y_K) is called a homomorphism if

(i)
for every r ∈ I, there exists u ∈ K such that both A_ur and exist;

(ii)
for every u ∈ K, there exists r ∈ I such that both A_ur and exist.

We denote by Hom(V_I, Y_K) the set of all homomorphisms from V_I into Y_K and by Hom(V_I) those from V_I into itself. The following properties are immediate.

Proposition A.1. Let V_I, Y_K, … be indexed -spaces. Then,

(i)
A ∈ Hom(V_I, Y_K) if and only if A^× ∈ Hom(Y_K, V_I);
(ii)
the product of any number of homomorphisms (between successive -spaces) is defined and is a homomorphism;
(iii)
if A ∈ Hom(V_I, Y_K), then f#g implies Af#Ag.

The definition of homomorphisms just given is tailored in such a way that one may consider the category PIP of all indexed

-spaces, with the homomorphisms as morphisms (arrows), as we have done in Section 3.3 above. In the same language, we may define particular classes of morphisms, such as monomorphisms, epimorphisms, and isomorphisms.

(i)
Let M ∈ Hom(W_L, Y_K). Then M is called a monomorphism if MA = MB implies A = B, for any two elements of A, B ∈ Hom(V_I, W_L), where V_I is any indexed -space.

(ii)
Let N ∈ Hom(W_L, Y_K). Then N is called an epimorphism if AN = BN implies A = B, for any two elements A, B ∈ Hom(Y_K, V_I), where V_I is any indexed -space.

(iii)
An operator A ∈ Op(V_I, Y_K) is an isomorphism if A ∈ Hom(V_I, Y_K), and there is a homomorphism B ∈ Hom(Y_K, V_I) such that BA = 1_V, AB = 1_Y, the identity operators on V, Y, respectively.

Typical examples of monomorphisms are the inclusion maps resulting from the restriction of a support, for instance, the natural injection , where ℝ = Ω ∪ Ω′ is the partition of ℝ in two measurable subsets of nonzero measure. More examples and further properties of morphisms may be found in [5, Section 3.3] and in [7].

Finally, an orthogonal projection on a nondegenerate indexed -space V_I, in particular, an LBS or an LHS, is a homomorphism P ∈ Hom(V_I) such that P² = P^× = P.

-subspace W of a

-space V is defined in [5, Section 3.4.2] as an orthocomplemented subspace of V, that is, a vector subspace W for which there exists a vector subspace Z⊆V such that V = W ⊕ Z and

(i)
for every f ∈ V, where f = f_W + f_Z, f_W ∈ W, f_Z ∈ Z;
(ii)
if f ∈ W, g ∈ Z and f#g, then 〈f∣g〉 = 0.

In the same Section 3.4.2 of [5], it is shown that a vector subspace W of a nondegenerate

-space is orthocomplemented if and only if it is topologically regular, which means that it satisfies the following two conditions:

(i)
for every assaying subset V_r⊆V, the intersections W_r = W∩V_r and are a dual pair in V;
(ii)
the intrinsic Mackey topology coincides with the Mackey topology induced by V_r.

Then the fundamental result, which is the analogue to the similar statement for a Hilbert space, says that a vector subspace W of the nondegenerate

-space V is orthocomplemented if and only if it is the range of an orthogonal projection:

()

Clearly, this raises the question, discussed in Section 3.3.1, of identifying the subobjects of any category consisting of

-spaces.

B. Fine Sheaves

We collect here some classical notions and results used in Section 6. We recall first that the support of the continuous function φ : X → ℝ on the topological space X is the closed set supp φ : = closure{x ∈ X : φ ≠ 0}, which is the smallest closed set outside which φ is zero. The same definition applies to a distribution. Then we recall the standard notion of a partition of unity.

Definition B.1. Let U = {U_i, i ∈ I} be an open covering of the topological space X. A set of real and continuous functions {φ_i, i ∈ I} defined on X is called a partition of unity with respect to U if one has the following.

(i)
φ_i(x)⩾0, for all x ∈ X.
(ii)
supp φ_i ⊂ U_i, for all i ∈ I.
(iii)
Each point x ∈ X has an open neighborhood that meets supp φ for a finite number of i ∈ I only.
(iv)
∑_i∈Iφ_i(x) = 1, for all i ∈ I. This sum is well defined by (iii).

We recall that a topological space is paracompact if it is separated (Hausdorff), and every open covering admits a locally finite open covering that is finer [22, § 6]. Every metrizable locally convex space is paracompact, but there are nonmetrizable paracompact spaces as well. The following result is standard.

Theorem B.2. X is paracompact if and only if X is a separated topological space and each open covering of X admits a partition of unity.

The main use of paracompact spaces is for the definition of a fine sheaf, to which the Cartan-Leray theorem applies (see below).

Definition B.3. Let be a sheaf on a paracompact topological space X. One says that is fine if, for every locally finite open covering U = {U_i, i ∈ I} of X, there exists a set of homomorphisms such that

(1)
for each i ∈ I, there exists a closed set M_i of X such that M_i ⊂ U_i and for x∉M_i, where is the stalk of the sheaf at the point x (the stalk is defined by the inductive limit );
(2)
∑_i∈Ih_i = 1. This sum is well defined since the covering U is locally finite.

Then the basic result is the following standard theorem. (A good introduction to the cohomology of sheaves and to the Cartan-Leray theorem can be found in [23, pages 166–197].)

Theorem B.4 (Cartan-Leray). Let be a fine sheaf on a paracompact topological space X. Then is acyclic; that is, the higher-order sheaf cohomology groups are trivial; for all p ≥ 1.

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Partial Inner Product Spaces: Some Categorical Aspects

Abstract

1. Motivation

2. Preliminaries

2.1. Partial Inner Product Spaces

2.2. Categories

3. Categories of -Spaces

3.1. A Single -Space as Category

3.2. A Category Generated by a Single Operator

3.3. The Category PIP of Indexed -Spaces

3.3.1. Subobjects

3.3.2. Superobjects

4. Sheaves of Operators on -Spaces

4.1. Presheaves and Sheaves

4.2. A Sheaf of Operators on an Indexed -Space

5. Cosheaves of Operators on -Spaces

5.1. Pre-Cosheaves and Cosheaves

5.2. Cosheaves of Operators on Indexed -Spaces

5.2.1. General Operators: The Additive Case

5.2.2. Universal Left Multipliers

5.2.3. General Operators: The Projective Case

6. Cohomology of an Indexed -Space

6.1. Cohomology of Operator Sheaves

6.2. Cohomology of Operator Cosheaves

7. Outcome

Acknowledgments

Appendices

A. Partial Inner Product Spaces

A.1. -Spaces and Indexed -Spaces

A.2. Operators on Indexed -Spaces

B. Fine Sheaves

References

Citing Literature

References

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Partial Inner Product Spaces: Some Categorical Aspects

Abstract

1. Motivation

2. Preliminaries

2.1. Partial Inner Product Spaces

2.2. Categories

3. Categories of -Spaces

3.1. A Single -Space as Category

3.2. A Category Generated by a Single Operator

3.3. The Category PIP of Indexed -Spaces

3.3.1. Subobjects

3.3.2. Superobjects

4. Sheaves of Operators on -Spaces

4.1. Presheaves and Sheaves

4.2. A Sheaf of Operators on an Indexed -Space

5. Cosheaves of Operators on -Spaces

5.1. Pre-Cosheaves and Cosheaves

5.2. Cosheaves of Operators on Indexed -Spaces

5.2.1. General Operators: The Additive Case

5.2.2. Universal Left Multipliers

5.2.3. General Operators: The Projective Case

6. Cohomology of an Indexed -Space

6.1. Cohomology of Operator Sheaves

6.2. Cohomology of Operator Cosheaves

7. Outcome

Acknowledgments

Appendices

A. Partial Inner Product Spaces

A.1. -Spaces and Indexed -Spaces

A.2. Operators on Indexed -Spaces

B. Fine Sheaves

References

Citing Literature

References

Related

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