PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 49–56 DOI: 10.2298/PIM141114001V

THE INDEX OF PRODUCT SYSTEMS OF HILBERT MODULES: TWO EQUIVALENT DEFINITIONS

Biljana Vujošević

Abstract. We prove that a conditionally completely positive definite kernel, as the generator of completely positive definite (CPD) semigroup associated with a continuous set of units for a product system over a C^{∗}-algebra B, allows a construction of a Hilbert B − B module. That construction is used to define the index of the initial product system. It is proved that such definition is equivalent to the one previously given by Kečkić and Vujošević [On the index of product systems of Hilbert modules, Filomat, to appear, ArXiv:1111.1935v1 [math.OA] 8 Nov 2011]. Also, it is pointed out that the new definition of the index corresponds to the one given earlier by Arveson (in the case B = ℂ).

1. Introduction

Product systems over ℂ have been studied during last several decades in connection with E_{0}-semigroups acting on a type I factor. Although the main problem of classification of all nonisomorphic product systems is still open, this theory is well developed. The reader is referred to [2] and references therein. In the present century there are some significant results that generalize this theory to product systems over a C^{∗}-algebra B, either in connection with E_{0} semigroups (see [8, 10]) or in connection with quantum probability dynamics (see [4, 3, 9]).

There are many difficulties in generalizing the notion of the index of a product system introduced in [1] to this more general concept. Up to our knowledge there are attempts in this direction done in [11] and recently in [5].

Here we give another definition of the index of product systems of Hilbert B–B modules and show that it is equivalent to the one previously given in [5]. Also, we point out that the new definition of index corresponds to the one given by Arveson (in the case B = ℂ).

Throughout the paper B will denote a unital C^{∗}-algebra and 1 will denote its unit.

The rest of Section 1 is devoted to basic definitions.

2010 Mathematics Subject Classification: 46L53, 46L55.

Key words and phrases: product system, Hilbert module, index.

Communicated by Stevan Pilipović.

Definition 1.1. a) A product system over C^{∗}-algebra B is a family (E_{t})_{t⩾0} of Hilbert B–B modules, with E_{0} ≅ B, and a family of (unitary) isomorphisms φ_{t,s} : E_{t} ⊗ E_{s} → E_{t+s} , where ⊗ stands for the so called inner tensor product obtained by identifications ub⊗v ∼ u⊗bv, u⊗vb ∼ (u⊗v)b, bu⊗v ∼ b(u⊗v), (u ∈ E_{t}, v ∈ E_{s}, b ∈ B) and then completing in the inner product ⟨u ⊗ v, u_{1} ⊗ v_{1}⟩ = ⟨v, ⟨u, u_{1}⟩v_{1}⟩;

b) Unit on E is a family u_{t} ∈ E_{t}, t ⩾ 0, such that u_{0} = 1 and φ_{t,s} (u_{t} ⊗ u_{s}) = u_{t+s} , which will be abbreviated to u_{t}⊗u_{s} = u_{t+s} . A unit u_{t} is unital if ⟨u_{t}, u_{t}⟩= 1. It is central if for all b ∈ B and all t ⩾ 0 there holds bu_{t} = u_{t}b;

Definition 1.2. Two units u_{t} and v_{t} give rise to the family of mappings K_{t}^{u,v} : B → B, given by K_{t}^{u,v} (b) = ⟨u_{t}, bv_{t}⟩. All K_{t}^{u,v} are bounded ℂ-linear operators on B, and this family forms a semigroup. The set of units S is continuous if the corresponding semigroup (K_{t}^{ξ,η})_{ξ,η∈S} (with respect to Schur multiplying) is uniformly continuous. A single unit u_{t} is uniformly continuous, or briefly just continuous, if the set {u} is continuous, that is, the corresponding family K_{t}^{u,u} is continuous in the norm of the space B(B) (the algebra of all bounded ℂ-linear operators on B).

As it can be seen in [3], for a (uniformly) continuous set of units U, there can be formed a uniformly continuous completely positive definite semigroup (CPDsemigroup further on) K = (K_{t})_{t∈ℝ_{+}} .

Denote by L = \frac{d}{dt}K |_{t=0} the generator of CPD-semigroup K. It is well known [3] that L is conditionally completely positive definite, that is, for all finite n-tuples x_{1}, …, x_{n} ∈ U and for all a_{j}, b_{j} ∈ B there holds

(1.1)  ∑_{j=1}^{n} a_{j} b_{j} = 0 ⟹ ∑_{i,j=1}^{n} b_{i}^{∗}L^{x_{i} ,x_{j}} (a_{i}^{∗}a_{j})b_{j} ⩾ 0.

Also,

(1.2)  L^{y,x} (b) = L^{x,y} (b^{∗})^{∗}.

It is known that K is uniquely determined by L. More precisely, K can be recovered from L by K = e^{tL} using the Schur product, i.e., K_{t}^{x,y} (b) = ⟨x_{t}, by_{t}⟩ = (exptL^{x,y})(b).

Remark 1.1. One should distinguish the continuous set of units from the set of continuous units. In the second case only K_{t}^{ξ,ξ} should be uniformly continuous for ξ ∈ S, whereas in the first case all K_{t}^{ξ,η} should be uniformly continuous.

In Section 2 the auxiliary statements, that are necessary for the proofs of the main result, are listed. In Section 3 another definition of the index of product systems of Hilbert B − B modules is obtained and the equivalency with the one previously given in [5] is proved. Also, it is pointed out that the new definition of the index corresponds to the one given by Arveson (in the case B = ℂ).

2. Preliminary results

In [6], Liebscher and Skeide introduce an interesting way to obtain new units in a given product system. The results are stated in Lemma 3.1, Proposition 3.3 and Lemma 3.4 of the mentioned paper and here they are quoted as

Proposition 2.1. a) Suppose that a continuous set S of units generates a product system E. Let t ↦ y_{t} ∈ E_{t} be a mapping (not necessarily unit), with K and K_{ξ} ∈ B(B) (ξ ∈ S) such that for all b ∈ B

⟨y_{t}, by_{t}⟩= b + tK(b) + O(t^{2}) and ⟨y_{t}, bξ_{t}⟩= b + tK_{ξ}(b) + O(t^{2}).

Then there exists a product system F ⊇ E and a unit ζ such that S ∪ {ζ} is continuous and L^{ζ,ζ} = K, L^{ζ,ξ} = K_{ξ}.

b) The following three conditions are equivalent.

(1) ζ ∈ E;

(2) ζ can be obtained as the norm limit of the sequence (y_{t/n})^{⊗n} ;

(3) lim_{n→∞} ⟨ζ_{t}, (y_{t/n})^{⊗n} ⟩ = ⟨ζ_{t}, ζ_{t}⟩.

Remark 2.1. In [6], a more general limit over the filter of all partitions of segment [0, t] was considered instead of lim_{n→∞} (y_{t/n})^{⊗n} . However, such a general context is not necessary here.

The previous proposition is used in [5, Proposition 2.3] to obtain new units in a product system in the following way.

Suppose that a continuous set S of units generates a product system E. Let x^{j} ∈ S, κ_{j} ∈ B, j = 1, …, n such that ∑ κ_{j} = 1. Then the functions t ↦ ∑ _{j=1}^{n} κ_{j} x_{t}^{j} and t ↦ ∑ _{j=1}^{n} x_{t}^{j}κ_{j} satisfy all the assumptions of Proposition 2.1 and the resulting units, denoted by κ_{1}x^{1} ⊞ ⋯⊞ κ_{n}x^{n} and x^{1}κ_{1} ⊞ ⋯⊞ x^{n}κ_{n}, belong to E. For example, the kernels of ζ = κ_{1}x^{1} ⊞ κ_{2}x^{2} ⊞ κ_{3}x^{3} are

(2.1) \begin{aligned}L^{ζ, ζ} = L^{x^{1}, x^{1}} L_{κ_{1}^{∗}} R_{κ_{1}} & + L^{x^{1}, x^{2}} L_{κ_{1}^{∗}} R_{κ_{2}} + L^{x^{1}, x^{3}} L_{κ_{1}^{∗}} R_{κ_{3}} + L^{x^{2}, x^{1}} L_{κ_{2}^{∗}} R_{κ_{1}} \\ & + L^{x^{2}, x^{2}} L_{κ_{2}^{∗}} R_{κ_{2}} + L^{x^{2}, x^{3}} L_{κ_{2}^{∗}} R_{κ_{3}} + L^{x^{3}, x^{1}} L_{κ_{3}^{∗}} R_{κ_{1}} \\ & + L^{x^{3}, x^{2}} L_{κ_{3}^{∗}} R_{κ_{2}} + L^{x^{3}, x^{3}} L_{κ_{3}^{∗}} R_{κ_{3}}, \\ L^{ζ, ξ} = L^{x^{1}, ξ} & L_{κ_{1}^{∗}} + L^{x^{2}, ξ} L_{κ_{2}^{∗}} + L^{x^{3}, ξ} L_{κ_{3}^{∗}} , \end{aligned}

where L_{b}, R_{b} : B → B are the left and the right multiplication operators for b ∈ B. Proposition 3.1 from [5] is quoted here as

Proposition 2.2. Let U be the set of all continuous units on a product system E. The relation ρ on U defined by

x ρ y ⇔ {x, y} is a continuous set

is an equivalence relation.

Thus, the set of all continuous units on some product system can be decomposed into mutually disjoint collection of maximal continuous sets.

Let E be a product system over a unital C^{∗}-algebra B with at least one continuous unit. (In view of [9, Definition 4.4] this means that E is non type III product system.) Further, let ω be an arbitrary continuous unit in E and let U = U_{ω} be the set of all uniformly continuous units that are equivalent to ω. (That refers to the equivalence relation ρ on U defined in Proposition 2.2.) As it can be seen in [5], the addition and multiplication by b ∈ B on U_{ω} are defined by

(2.2)  x + y = x ⊞ y ⊞ −ω, b · x = bx ⊞ (1 − b)ω, x · b = xb ⊞ ω(1 − b),

and the kernels of x + y, x · a, a · x are

(2.3) \begin{aligned}L^{x+y, x+y} & = L^{x, x} + L^{x, y} − L^{x, ω} + L^{y, x} + L^{y, y} − L^{y, ω} − L^{ω, x} − L^{ω, y} + L^{ω, ω} , \\ L^{x+y, ξ} & = L^{x, ξ} + L^{y, ξ} − L^{ω, ξ} , \end{aligned}

(2.4) \begin{aligned}L^{x·a,x·a} & = a^{∗}L^{x,x} a + (1 − a)^{∗}L^{ω,x} a + a^{∗}L^{x,ω} (1 − a) + (1 − a)^{∗}L^{ω,ω} (1 − a), \\ L^{x·a,ξ} & = a^{∗}L^{x,ξ} + (1 − a)^{∗}L^{ω,ξ} ,  ξ ∈ U, \end{aligned}

(2.5) \begin{aligned}L^{a·x,a·x} & = L^{x,x} L_{a^{∗}} R_{a} + L^{ω,x} L_{1−a^{∗}} R_{a} + L^{x,ω} L_{a^{∗}} R_{1−a} + L^{ω,ω} L_{1−a^{∗}} R_{1−a} , \\ L^{a·x,ξ} & = L^{x,ξ} L_{a^{∗}} + L^{ω,ξ} L_{1−a^{∗}} ,  ξ ∈ U, \end{aligned}

where L_{b}, R_{b} : B → B are the left and right multiplication operators for b ∈ B.

Remark 2.2. For x, y ∈ U_{ω}, x − y = x ⊞ (−y) ⊞ ω.

According to [5, Theorem 3.2], the set U with respect to the operations defined by (2.2) is a left–right B–B module.

In [5] it was proved that the mapping ⟨·, ·⟩_{1} : U × U → B given by

(2.6)  ⟨x, y⟩_{1} = (L^{x,y} − L^{x,ω} − L^{ω,y} + L^{ω,ω} )(1)

(ω is the same as in (2.2)) is a B-valued semi-inner product (in the sense that it can be degenerate, i.e., ⟨x, x⟩_{1} = 0 need not imply x = 0) and that it satisfies all the customary properties:

(a) For all x, y, z ∈ U, and α, β ∈ ℂ ⟨x, αy + βz⟩ _{1} = α ⟨x, y⟩_{1} + β ⟨x, z⟩_{1};

(b) For all x, y ∈ U, a ∈ B ⟨x, y · a⟩_{1} = ⟨x, y⟩_{1} a;
(c) For all x, y ∈ U ⟨x, y⟩_{1} = ⟨y, x⟩_{1}^{∗};
(d) For all x ∈ U ⟨x, x⟩_{1} ⩾ 0.

Also, the set N = {x ∈ U | ⟨x, x⟩_{1} = 0} is a submodule of U and U/N is a pre-Hilbert left–right B–B module.

3. The result

The definition of the index of the product system with at least one continuous unit, given in [5], is quoted here as

Definition 3.1. Let E be a product system, and let ω be a continuous unit on E. The index of a pair (E, ω) is the completion of pre-Hilbert left–right module U/_{∼} , where U = U_{ω} is the maximal continuous set of units containing ω, and ∼ is the equivalence relation defined by x ∼ y if and only if x − y ∈ N where N is the set mentioned at the end of Section 2. Naturally, the index will be denoted by ind(E, ω).

Remark 3.1. If {ω, ω′} is a continuous set, then ind(E, ω) ≅ ind(E, ω′). Indeed, then U_{ω} = U_{ω′} and the isometric isomorphism is given by translation x ↦ x ⊞ −ω ⊞ ω′. Therefore, ind(E, ω) is independent on the choice of ω in the same continuous set of units.

The index of the product systems with at least one continuous unit may also be defined in a different way and we prove that these two definitions are equivalent. In detail, let E be a product system and let U be a continuous set of units in E. Consider the B-bimodule BUB where BUB is the set of all formal sums ∑ _{i} a_{i}x_{i}b_{i}, x_{i} ∈ U, a_{i}, b_{i} ∈ B with identification subject to the relations

(λa)xb ∼ ax(λb) (λ ∈ ℂ), (a_{1} + a_{2})xb ∼ a_{1}xb + a_{2}xb, ax(b_{1} + b_{2}) ∼ axb_{1} + axb_{2}.

For c ∈ B, (∑ _{i} a_{i}x_{i}b_{i})c = ∑ _{i} a_{i}x_{i}(b_{i}c) and c(∑ _{i} a_{i}x_{i}b_{i}) = ∑ _{i}(ca_{i})x_{i}b_{i}. Also consider B-subbimodule (BUB)_{0} = {∑ _{i} a_{i}x_{i}b_{i} ∈ BUB | ∑ _{i} a_{i}b_{i} = 0} and define the map ⟨·, ·⟩: (BUB)_{0} × (BUB)_{0} → B by

(3.1)  ⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{j} a′_{j}x′_{j}b′_{j}⟩ = ∑_{i,j} b_{i}^{∗}L^{x_{i} ,x′_{j}} (a_{i}^{∗}a′_{j})b′_{j}.

Lemma 3.1. The map (3.1) satisfies the following properties:

(a) For all a_{i}, b_{i}, c_{i}, c′_{i}, d_{i}, d′_{i} ∈ B, x_{i}, y_{i}, y′_{i} ∈ U, α, β ∈ ℂ

⟨ ∑_{i} a_{i}x_{i}b_{i}, α ∑_{i} c_{i}y_{i}d_{i} + β ∑_{i} c′_{i}y′_{i}d′_{i} ⟩

= α⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{i} c_{i}y_{i}d_{i}⟩ + β⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{i} c′_{i}y′_{i}d′_{i}⟩ ;

(b) For all a_{i}, a′_{i}, b_{i}, b′_{i} ∈ B, x_{i}, x′_{i} ∈ U, c ∈ B

⟨ ∑_{i} a_{i}x_{i}b_{i}, ( ∑_{i} a′_{i}x′_{i}b′_{i}) c⟩ = ⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{i} a′_{i}x′_{i}b′_{i}⟩ c;

(c) For all a_{i}, a′_{i}, b_{i}, b′_{i} ∈ B, x_{i}, x′_{i} ∈ U

⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{i} a′_{i}x′_{i}b′_{i}⟩ = ⟨ ∑_{i} a′_{i}x′_{i}b′_{i}, ∑_{i} a_{i}x_{i}b_{i}⟩^{∗};

(d) For all a_{i}, b_{i} ∈ B, x_{i} ∈ U ⟨∑ _{i} a_{i}x_{i}b_{i}, ∑ _{i} a_{i}x_{i}b_{i}⟩ ⩾ 0.

Proof. (a), (b) are easy to check. For (c) use (1.2) and (d) follows since L is conditionally CPD (1.1). □

From the previous lemma, the Causchy–Schwartz inequality can be derived (see [7, Proposition 1.2.4]):

⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{i} a′_{i}x′_{i}b′_{i}⟩ ⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{i} a′_{i}x′_{i}b′_{i}⟩^{∗}

⩽ ⟨ ∑_{i} a_{i}x_{i}b_{i}, ∑_{i} a_{i}x_{i}b_{i}⟩ ∥⟨ ∑_{i} a′_{i}x′_{i}b′_{i}, ∑_{i} a′_{i}x′_{i}b′_{i}⟩ ∥.

It follows that the set N = {∑ _{i} a_{i}x_{i}b_{i} ∈ (BUB)_{0} | ⟨∑ _{i} a_{i}x_{i}b_{i}, ∑ _{i} a_{i}x_{i}b_{i}⟩ = 0} is equal to {∑ _{i} a_{i}x_{i}b_{i} ∈ (BUB)_{0} | ∀ ∑ _{i} a′_{i}x′_{i}b′_{i} ∈ (BUB)_{0}, ⟨∑ _{i} a_{i}x_{i}b_{i}, ∑ _{i} a′_{i}x′_{i}b′_{i}⟩ = 0}. So, N is a submodule of (BUB)_{0} and (BUB)_{0}/N is a pre-Hilbert left–right B–B module.

Theorem 3.1. Let E be a product system over a unital C^{∗}-algebra B. Let ω be an arbitrary continuous unit in E and let U be the maximal continuous set of units containing ω. The mapping f : U/_{∼} → (BUB)_{0}/N defined by f([y]) = y − ω + N is an isomorphism between pre-Hilbert B–B module U/_{∼} introduced in Definition 3.1 and pre-Hilbert B–B module (BUB)_{0}/N.

Proof. Let y, y′ ∈ U and y ∼ y′, i.e., ⟨y − y′, y − y′⟩ _{1} = 0 (the substraction is as in Remark 2.2). For 1y1 + (−1)y′1 ∈ (BUB)_{0} we also write y − y′ ∈ (BUB)_{0}. By (3.1) there holds

⟨y − y′, y − y′⟩ = L^{y,y}(1) − L^{y,y′}(1) − L^{y′,y}(1) + L^{y′,y′}(1),

and also, by (2.6) and (2.1),

⟨y − y′, y − y′⟩ _{1} = ⟨y ⊞ (−y′) ⊞ ω, y ⊞ (−y′) ⊞ ω⟩ _{1}
= L^{y,y}(1) − L^{y,y′}(1) − L^{y′,y}(1) + L^{y′,y′}(1).

Therefore, y − y′ ∈ N which means that f is well defined. Let [y], [z] ∈ U/_{∼} .

⟨f([y]), f([z])⟩_{(BUB)_{0} /N} = ⟨y − ω + N , z − ω + N ⟩ _{(BUB)_{0} /N} = ⟨y − ω, z − ω⟩
= L^{y,z} (1) − L^{ω,z} (1) − L^{y,ω} (1) + L^{ω,ω} (1)
= ⟨y, z⟩_{1} = ⟨[y], [z]⟩_{U/_{∼}} ,

so f is an isometry. For the surjectivity of f, it needs to be proved that for all ∑ _{i} a_{i}x_{i}b_{i}+N in (BUB)_{0}/N there exists [y] ∈ U/_{∼} such that ∑ _{i} a_{i}x_{i}b_{i}−y+ω ∈ N. The mapping t ↦ ω_{t} + ∑ _{i} a_{i}x_{i,t} b_{i} satisfies all the assumptions of Proposition 2.1 and let us denote the resulting unit by ζ. The kernels of ζ are given by

L^{ζ,ζ}(b) = L^{ω,ω} (b) + ∑_{i} b_{i}^{∗}L^{x_{i},ω} (a_{i}^{∗}b) + ∑_{i} L^{ω,x_{i}} (ba_{i})b_{i} + ∑_{i,j} b_{i}^{∗}L^{x_{i},x_{j}} (a_{i}^{∗}ba_{j})b_{j} ,

L^{ζ,ξ}(b) = L^{ω,ξ} (b) + ∑_{i} b_{i}^{∗}L^{x_{i},ξ} (a_{i}^{∗}b), ξ ∈ U, b ∈ B. (3.2)

By (3.1), (3.2), (1.2) it follows ⟨∑ _{i} a_{i}x_{i}b_{i} − ζ + ω , ∑ _{i} a_{i}x_{i}b_{i} − ζ + ω ⟩ = 0. Therefore, ∑ _{i} a_{i}x_{i}b_{i} − ζ + ω ∈ N and f([ζ]) = ∑ _{i} a_{i}x_{i}b_{i} + N. Let [x], [y] ∈ U/_{∼} . Denote ζ = x + y ∈ U (the addition is as in (2.2)). By (3.1), (2.3), (1.2) it follows that ⟨ζ − x − y + ω, ζ − x − y + ω⟩ = 0 which means ζ − x − y + ω ∈ N. Therefore,

f([x] + [y]) = f([x + y]) = ζ − ω + N = x − ω + y − ω + N = f([x]) + f([y]).

Let [x] ∈ U/_{∼} and b ∈ B. Denote η = x·b ∈ U and μ = b·x ∈ U (the multiplication is as in (2.2)). By (3.1), (2.4), (2.5), (1.2) it follows that

⟨η − ω − xb + ωb, η − ω − xb + ωb⟩ = 0,
⟨μ − ω − bx + bω, μ − ω − bx + bω⟩ = 0,

hence η − ω − xb + ωb ∈ N and μ − ω − bx + bω ∈ N. Therefore,

f([x] · b)= f([x · b]) = η − ω + N = xb − ωb + N = f([x])b,
f(b · [x])= f([b · x]) = μ − ω + N = bx − bω + N = bf([x]).  □

Corollary 3.1. Let E be a product system over a unital C^{∗}-algebra B. Let ω be an arbitrary continuous unit in E and let U be the maximal continuous set of units containing ω. The index of E may also be defined as the completion of pre-Hilbert left-right B − B module (BUB)_{0}/N.

Remark 3.2. Let E be an Arveson product system, i.e., E is a product system with B = ℂ, and let U be the set of its units. As it can be found in [2], for x, y ∈ U there exists a unique complex number c(x, y) satisfying ⟨x_{t}, y_{t}⟩ = e^{tc(x,y)} . The function c : U × U → ℂ is the covariance function of E. It is conditionally positive definite and there holds

(3.3)  L^{x,y} (1) = lim_{t→0} \frac{⟨x_{t}, y_{t}⟩ − 1}{t} = lim_{t→0} \frac{e^{tc(x,y)} − 1}{t} = c(x, y).

Since all L^{x,y} are ℂ-linear, the B-bimodule BUB is reduced to the complex vector space ℂU consisting of all formal sums ∑ _{i} a_{i}x_{i} with a_{i} ∈ ℂ, x_{i} ∈ U and its B-subbimodule (BUB)_{0} is reduced to (ℂU)_{0} = {∑ _{i} a_{i}x_{i} ∈ ℂU | ∑ _{i} a_{i} = 0}. Using (3.3), it follows

(3.4)  ⟨ ∑_{i} a_{i}x_{i}, ∑_{i} a′_{i}x′_{i}⟩ = ∑_{i,j} L^{x_{i} ,x′_{j}} (\overline{a_{i}}a′_{j})

= ∑_{i,j} L^{x_{i} ,x′_{j}} (1)\overline{a_{i}}a′_{j} = ∑_{i,j} c(x_{i}, x′_{j})\overline{a_{i}}a′_{j}.

According to Corollary 3.1, the index of E is the completion of the inner product space (ℂU)_{0}/N where N = {∑ _{i} a_{i}x_{i} ∈ (ℂU)_{0} | ⟨∑ _{i} a_{i}x_{i}, ∑ _{i} a_{i}x_{i}⟩ = 0}. That definition of the index corresponds to the one previously given by Arveson in [2]. In detail, following the notation in [2], ℂ_{0}U is the complex vector space consisting of all finitely nonzero functions f : U → ℂ satisfying ∑ _{x} f(x) = 0. There is a mapping ⟨·, ·⟩: ℂ_{0}U × ℂ_{0}U → ℂ defined by

(3.5)  ⟨f, g⟩ = ∑_{x,y∈U} c(x, y)\overline{f(x)} g(y).

If N = \{f | ⟨f, f⟩ = ∑ _{x,y} c(x, y)\overline{f(x)} f(y) = 0\}, the mapping (3.5) is an inner product on (ℂ_{0}U)/N and the index of E is defined as dimension of the completion of (ℂ_{0}U)/N. A basis for ℂ_{0}U is given by the set {δ_{x} | x ∈ U} where δ_{x}(x) = 1 and δ_{x}(y) = 0, ∀y ≠ x. The mapping x ↦ δ_{x} is a bijection between U and the basis vectors {δ_{x} | x ∈ U}, hence U may be considered as a linearly independent basis for ℂ_{0}U. Therefore, every f ∈ ℂ_{0}U may be written in the form f = ∑ _{i} a_{i}x_{i} where a_{i} = f(x_{i}) ∈ ℂ, x_{i} ∈ U. Consequently, we may identify (ℂU)_{0} and ℂ_{0}U, N and N and the mappings in (3.4) and (3.5).

The final conclusion is that, according to Corollary 3.1, the index of Arveson product system E may also be defined as the completion of the inner product space U/_{∼} , where ∼ is the equivalence relation introduced in Definition 3.1.

References

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Faculty of Mathematics (Received 19 07 2014) University of Belgrade Belgrade Serbia

   bvujosevic@matf.bg.ac.rs