On the Choice of Algebra for Quantization

Benjamin H. Feintzeig∗

Department of Philosophy
University of Washington

Savery Hall
Seattle, WA 98195-3550

Abstract

In this paper, I examine the relationship between physical quantities and
physical states in quantum theories. I argue against the claim made by
Arageorgis (1995) that the approach to interpreting quantum theories known
as Algebraic Imperialism allows for “too many states”. I prove a result
establishing that the Algebraic Imperialist has very general resources that
she can employ to change her abstract algebra of quantities in order to rule
out unphysical states.

∗I would like to thank Hans Halvorson Jim Weatherall, and two anonymous referees for helpful
comments.

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1 Introduction

To construct a quantum theory, one performs a procedure known as
“quantization”. Quantization can be thought of as having two steps: first,
one constructs an abstract C*-algebra to represent the physical quantities, or
observables, of the system, which must obey the canonical commutation
relations, and second, one finds a representation of that algebra in the
bounded operators on some Hilbert space. Much philosophical attention has
been directed at this second step, as it is now well known that in many cases
of physical interest, including quantum field theories and quantum statistical
theories in the thermodynamic limit, there is not a unique Hilbert space
representation of the algebra. This leads to two general interpretive options.
The Hilbert Space Conservative claims that we must pick one particular
Hilbert space representation of our algebra (out of the many competing
ones), and use this one as our quantum theory. On the other hand, the
Algebraic Imperialist asserts that we do not need any Hilbert space
representations to interpret our theory, and instead can think of our physical
theory as being comprised by the abstract algebra itself.

The purpose of this paper is to argue that the methods of the Algebraic
Imperialist have a particular virtue, which I call adaptability. By this, I mean
that one can make small, but systematic changes in the abstract algebraic
framework to deal with problems that arise.

The particular problem that I am concerned with confronting from the
algebraic perspective here is how to construct a quantum theory that allows
for the correct space of physically possible states. In many physical theories,
one can find models that appear pathological or unphysical. And indeed, it
has been claimed that one of the downfalls of the algebraic framework for
quantum theories is that it allows for “too many states”. And so, the
argument goes, one is forced to use Hilbert space methods to restrict the
physical state space under consideration.

I will show in this paper, however, that one need not appeal to Hilbert
space methods to perform this reduction of the state space. Instead, one can
reduce the physical state space using algebraic methods alone. I will present
a result that establishes precise conditions under which this reduction of the
state space can be performed. This result provides a general strategy
through which to use algebraic methods to construct quantum theories.

In slightly more detail, the Algebraic Imperialist advocates using the
collection of states on the abstract algebra of the canonical commutation
relations as the state space of our quantum theory. But there are states on
this algebra that some researchers describe as unphysical. These unphysical
states are often ones that cannot be represented by any density operator in

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the relevant Hilbert space; this means they are ruled out by the Hilbert
Space Conservative, who countenances as possible only states that appear in
her favored Hilbert space representation. So it seems that the Algebraic
Imperialist allows for extra states that Hilbert space methods can excise from
the theory.

The response I want to suggest here, on behalf of the Algebraic
Imperialist, is that we have been considering the wrong algebra, at least for
the purposes of assessing the space of physical states. There is often a
different algebra that one can use, which has exactly the physical state space
we were looking for. Moreover, I will show that there is a completely general
and systematic way of changing the abstract algebra—the one we use to
implement the canonical commutation relations—in order to reduce the
physical state space. Importantly, this general procedure is appropriate for
the Algebraic Imperialist because it does not use Hilbert space methods.
This allows the Algebraic Imperialist a new freedom in constructing quantum
theories, and it leads to a new perspective on the issues surrounding
quantization. In this sense, the argument of this paper can be thought of as
an argument in favor of Algebraic Imperialism because it shows how much
more one can do with algebraic methods than previously thought.

But the arguments that I present need not come attached to the
interpretive position of Imperialism; the results of this paper are important
for anyone who wants to use a quantization procedure to construct quantum
theories, including the Hilbert Space Conservative. Before one can even
begin asking about the necessity of Hilbert space representations, one must
grapple with the question of how to construct an appropriate C*-algebra in
the first step of the quantization procedure. According to the Imperialist,
this is where all of the interesting physical development happens in
quantization, but even the Conservative needs to have some algebra in mind
before she can take its representations. Different options for the abstract
algebra of observables appear in the physics literature, but so far only one of
these options has been discussed in the philosophical literature to my
knowledge. As such, there has been no systematic discussion of how to
choose between different algebras. One contribution of this paper,
independent of the dispute between Imperialists and Conservatives, will be to
provide tools for choosing the appropriate algebra.

I want to note that I will not in this paper make a particular
recommendation as to which algebra we ought to use as the algebra of
physical observables in any specific case. I will comment briefly on the
physical, mathematical, and methodological issues that lead us to the algebra
that is standardly used to implement the canonical commutation relations,
and why I think we ought to consider other algebras, too. But what I mostly

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hope to show is that the question of which algebra to use is inextricably
tangled with the question of what states are physical, in such a way that any
progress toward answering one question yields progress for answering the
other. I will have to save more detailed analysis of which physical states
spaces (and hence which algebras) are appropriate for future work.

The paper is organized as follows. Section 2 briefly presents mathematical
preliminaries concerning both abstract algebraic methods and Hilbert space
representations. Section 3 presents the objection that Imperialism allows for
“too many” states, which we take up in section 4. There, we present a
general result concerning the reduction of the state space of an abstract
algebra, and show that it specifically allows the Algebraic Imperialist at least
all of the same resources that the Hilbert Space Conservative has for state
space reduction. Section 5 concludes with a discussion of the significance of
the results and further open questions.

2 Mathematical Preliminaries

The bounded observables of a physical theory carry the structure of a
C*-algebra.1 This means that one may add and multiply observables, and
multiply observables by scalars. In addition, a C*-algebra carries an
operation of involution that is a generalization of complex conjugation. A
C*-algebra A comes equipped with a norm, which is required to satisfy the
C*-identity:

‖A∗A‖ = ‖A‖2

for all A ∈ A. The norm defines a topology, called the norm topology, which
is characterized by the following condition for convergence. A net {Ai}⊆ A
converges to A in the norm topology2 iff

‖Ai −A‖→ 0

where the convergence is now in the standard topology on R. The C*-algebra
A is required to be complete with respect to this topology in the sense that

1In this section, we present only the minimal technical background required to
state the results of section §4. For more on operator algebras, see Kadison and
Ringrose (1997), Sakai (1971), and Landsman (1998). For more on algebraic quantum
theory, see Haag (1992), Bratteli and Robinson (1996), Emch (1972), and Wald
(1994). For philosophical introductions, see Halvorson (2006) and Ruetsche (2011).

2One could restrict attention here to sequences because the norm topology is
second countable, but for the weak topologies considered later, which are not second
countable, one must work with arbitrary nets.

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for every Cauchy net {Ai}⊆ A, i.e. for every net such that

‖Ai −Aj‖→ 0

there is an A ∈ A such that Ai → A in the norm topology. Standard results
in the theory of normed vector spaces tell us that every normed vector space
has a unique completion.3

Since A is a vector space, we can also consider the dual space A∗ of
bounded (i.e. norm continuous) linear functionals ρ : A → C. A state on a
C*-algebra A is just a particular kind of element of the dual space
A∗—namely one that is positive and normalized.4

The dual space A∗ can be used to define an alternative to the norm
topology on A, called the weak topology, which is characterized by the
following condition for convergence. A net {Ai}⊆ A converges in the weak
topology to A ∈ A iff for every ρ ∈ A∗,

ρ(Ai) → ρ(A)

where the convergence is now in the standard topology on C. The weak
topology is the coarsest topology on A with respect to which all of the linear
functionals in A∗ are continuous.

A C*-algebra A need not be complete with respect to its weak topology;
there may be nets {Ai}⊆ A that are Cauchy in the sense that

ρ(Ai −Aj) → 0

for every ρ ∈ A∗ without the net having a limit point A ∈ A such that
Ai → A in the weak topology. However, a C*-algebra can always be
completed in its weak topology to form its bidual A∗∗, as follows.5

The bidual carries a topology known as the weak* topology, which is a
natural generalization of the weak topology on A. The weak* topology is
characterized by the following condition for convergence. A net {Ai}⊆ A∗∗
converges in the weak* topology to A ∈ A∗∗ iff for every ρ ∈ A∗,

Ai(ρ) → A(ρ)

3A complete normed vector space is called a Banach space. A C*-algebra is thus
a Banach algebra whose norm is, in a certain sense, compatible with multiplication
and involution.

4A functional ρ ∈ A∗ is positive if ρ(A∗A) ≥ 0 for all A ∈ A and normalized if
‖ρ‖ = 1.

5See Feintzeig (2017b) for more on the completion of a C*-algebra into its bidual.

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The weak* topology on the bidual A∗∗ corresponds precisely to the extension
of the condition of convergence for the weak topology on A to the larger
algebra A∗∗. In particular, the weak* topology on A∗∗ is the coarsest
topology on A∗∗ that makes every linear functional in A∗ continuous. One
can show that the bidual A∗∗ is complete with respect to the weak* topology.

Moreover, the original C*-algebra A is canonically embedded in its bidual
by A ∈ A 7→  ∈ A∗∗, with  defined by

Â(ρ) = ρ(A)

for all ρ ∈ A∗. With respect to this embedding, A is dense in A∗∗ in the weak*
topology, so the bidual A∗∗ can be understood as the completion of A in its
weak topology, which is the subspace topology of the weak* topology on A∗∗.

Importantly, the algebraic approach can be translated back into the
familiar Hilbert space formalism for quantum mechanics. A representation of
a C*-algebra A is a pair (π,H), where H is a Hilbert space and π : A →B(H)
is a *-homomorphism into the bounded linear operators on H.6

We can use the Hilbert space structure of a representation (π,H) to
induce a new topology on the algebra π(A). The ultraweak topology is
characterized through the following condition for convergence: a net π(Ai)
converges to π(A) in the ultraweak topology if for every density operator ρ
on H,

Tr(π(Ai)ρ) → Tr(π(A)ρ)

The ultraweak topology represents a notion of convergence of expectation
values by certain states, namely the density operator states.

A state ω ∈ A∗ has a density operator representative in the representation
(π,H) just in case there is a density operator ρω such that ω(A) = Tr(Aρω)
for all A ∈ A. In general, there may be states on A without density operator
representatives in a given representation (π,H). In other words, the density
operator states on a representation may not exhaust the states on the
abstract algebra A.7 However, we know that a state ω ∈ A∗ has a density
operator representative in a given representation (π,H) of A just in case ω is
ultraweakly continuous in that representation. We will use these facts in our
discussion of appropriate state spaces below.

6One of the most fundamental results in the theory of C*-algebras, known as the
GNS Theorem (See Kadison and Ringrose, 1997), tells us that every C*-algebra has
Hilbert space representations.

7This follows immediately from the presence of unitarily inequivalent representa-
tions of an algebra, as discussed in Ruetsche (2011).

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3 Interpretive Options

In this section, I will explicate the objection that Algebraic Imperialism
allows for “too many states”. I will show that this question is tangled with
the choice of algebra of quantum observables in §3.2, and I will briefly
illustrate these issues in a simple example in §3.3.

3.1 Physical States

The procedure of quantization can be understood as involving two steps.
First, one finds a quantum algebra of observables A. Second, one chooses a
representation (π,H) of A on some Hilbert space. The interpretive debate
between Algebraic Imperialism and Hilbert Space Conservatism concerns just
whether this second step is necessary.8 According to the Algebraic
Imperialist, it is not: a quantum theory (or, at least its kinematics) can be
captured by the abstract algebra A in the sense that the physical quantities
of a system can be adequately represented by the elements of A—or perhaps
by weak limits in A∗∗—and the physical states can be represented by states
on A. On the other hand, according to the Hilbert Space Conservative, we
need to pick a representation (π,H). Then we represent the physical
quantities of our system by elements of π(A)—or perhaps by ultraweak
limits—and the physical states by density operators on H.

The objection that has been leveled at Algebraic Imperialism that I want
to discuss is that the Imperialist allows for “too many” states, in the sense
that many states on the abstract algebra A are unphysical (See Arageorgis,
1995). Taking a Hilbert space representation allows us to focus on an
appropriate collection of physical states by focusing our attention on only the
states that have density operator representatives in our chosen
representation. So the fact that Hilbert space methods provide resources to
rule out unphysical states is supposed to count in favor of the Hilbert Space
Conservative.

Before we can see what these unphysical states are, we need to specify the
algebra of observables we are using—after all, different algebras will in
general have different state spaces. The algebra of observables that gives rise
to the unphysical states at issue is known as the Weyl Algebra. This algebra
puts the canonical commutation relations between position and momentum
observables in bounded form by considering only exponentiated forms of

8For more on Algebraic Imperialism and Hilbert Space Conservatism, see
Ruetsche (2002, 2003, 2006, 2011).

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those observables.9

The states that one might consider unphysical come in many varieties.
For example, the Weyl algebra allows for non-regular states (Halvorson, 2001,
2004; Beaume et al., 1974), ones which fail to satisfy a continuity condition
and in doing so fail to allow one to simultaneously define both position and
momentum observables from the Weyl operators. These non-regular states
do not have density operator representatives in the usual Hilbert space
representation (the Schrödinger representation) of the Weyl algebra. It is a
standard move in algebraic quantum theory to restrict attention only to
regular states on the Weyl algebra and their representations to rule out these
non-regular states.

But there are many other states on the Weyl algebra that one might
consider ruling out as unphysical. Arageorgis (1995) mentions a proposal
that we ought to restrict attention to locally definite states, ones which
vanish on smaller and smaller regions of spacetime. Or perhaps we ought to
restrict attention to Hadamard states, ones which allow for an appropriately
well-defined stress-energy observable (See Wald, 1994). And Halvorson (2006)
takes up a suggestion by Doplicher, Haag, and Roberts that we use only
what he calls DHR states, ones which differ only locally from the vacuum. In
addition, we might want to restrict attention to states accessible from Fock
states, i.e. states that allow for an interpretation in terms of particle states
with creation and annihilation operators (See Petz, 1990, Ch. 4).

The results that I present below in §4 are immediately applicable to two
of the suggestions just considered: regular states and states accessible from
Fock states. The reason, as I will remark later, is that these states form the
folium of an irreducible Hilbert space representation of the Weyl algebra, and
I will show explicitly that the results of §4 apply to the folium of any
irreducible Hilbert space representation of any algebra. For the purposes of
this paper, however, I will not analyze any of the other suggestions in detail
because they require substantially more mathematical apparatus. Although
the results that I present in the next section are meant to be sufficiently
general to apply to any of these suggestions for appropriate physical state
spaces, and I hope further work can make these connections explicit for
locally definite states, Hadamard states, and DHR states.10

9The simplest Weyl operators take the form U(a) := eiaQ and V (b) := eibP for
position Q, momentum P , and constants a,b ∈ R (at least in the usual Schrödinger
representation). For more on the Weyl algebra, see Petz (1990) and Clifton and
Halvorson (2001).

10Specifically, the methods of DHR theory in some ways resemble the reduction
of the state space developed in §4, and the state space of the final theory is given

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One might already wonder whether the Algebraic Imperialist really needs
a response to the “too many states” objection. Given that there isn’t a clear
consensus as to what the physical states of any quantum theory are, one
might think that it is not worth putting stock in an objection that relies on
the assumption that at least some states are unphysical. Still, I think the
objection is worth considering for two reasons. First, I think there are
situations in which the Weyl algebra is not the natural choice of an algebra of
observables for the Algebraic Imperialist, and at least part of the motivation
for this claim is that there are states on the Weyl algebra that are viewed as
pathological and very rarely used (see §3.3). Addressing unphysical states in
general can help us understand this particular case. Second, I think that the
tools that one can develop in response, if one takes the “too many states”
objection seriously, have the potential to lead to progress in our
understanding of a wide variety of issues surrounding quantization. It may
very well prove useful for current and future physics to have tools for
systematically constructing, analyzing, and interpreting algebras and state
spaces. So the objection has at least instrumental value in that it leads to
new techniques and interesting questions, some of which I’ll be able to point
to at the end of this paper.

Furthermore, I should stress that it is not my purpose here to judge
whether any of the proposals cited above provide an adequate specification of
the states that we ought to deem physical. I mention these concrete
proposals only to show that others have expressed an interest in restricting
the space of quantum states. My goal in this paper is only to show that
however one wants to specify the collection of physical states, there is an
intimate relationship between this state space and the algebra of observables
of a quantum theory.

3.2 Physical Algebras

The fact that the abstract Weyl algebra allows for so many supposedly
unphysical states has been taken by some (e.g., Arageorgis, 1995) as an
argument against Algebraic Imperialism. However, I propose that this only
gives us reason to use a different algebra. Even if one particular algebra
allows for unphysical states, this does not imply that all abstract algebras
allow for unphysical states. There are other options for the abstract algebra
of observables.

To motivate this approach, I want to note that the Weyl algebra does not

in terms of the states on a particular Hilbert space representation. So prospects for
relating the DHR theory to the methods of this paper are hopeful.

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have a particularly stable priveleged status. To get the Weyl algebra, one
starts with the canonical commutation relations between position and
momentum observables, and then applies these to particular bounded
observables (namely, exponentiated position and momentum), and finally
completes the resulting algebra in norm. First, the approach of using
bounded observables to generate the Weyl algebra is largely for technical
convenience. One can approach quantization using unbounded operators
(Dubin et al., 2000), but one then needs to keep track of distinct domains for
these unbounded operators. Some (Wald, 1994; Kay and Wald, 1991) have
preferred to approach field quantization in terms of a *-algebra rather than a
C*-algebra, which corresponds to choosing a collection of possibly unbounded
observables. While there are still open questions concerning the relationship
between the unbounded approach following Wightman’s axiomatic
formulation (See Haag, 1992) and the bounded approach, these are beyond
the scope of this paper. All I wish to point out is that proponents of the
unbounded approach already reject the Weyl algebra. In what follows in this
paper, I will consider only bounded approaches in terms of C*-algebras
because one can demonstrate the kinds of choices one has to make in that
context already. It might also be of interest to investigate whether the results
of §4 can be generalized to arbitrary *-algebras.

Second, using the particular bounded observables defined as
exponentiated position and momentum and completing the algebra in norm
are also choices made largely to simplify technical machinery. Using the Weyl
unitaries—those exponentiated position and momentum operators—allows
one to put the canonical commutation relations in a particularly simple form
so that they fully define the multiplication relation for the algebra. As I’ll
show below, there are other bounded operators, namely the compact
operators, that one could also use to implement the canonical commutation
relations. Further, one might consider completing an algebra in other
topologies besides that defined by a norm, as Feintzeig (2017b) considers
completing a C*-algebra in its weak topology. So the Weyl algebra, which we
arrive at through norm completion, is not the only option.

Finally, it’s worth pointing out that the canonical commutation relations
we use to generate the Weyl algebra are not a priori principles we must hold
onto at all costs. Surely, they form some of the core tenets of quantum theory
by underlying the uncertainty relations, but in doing so they constitute
empirical hypotheses, which are open to revision just like any other principle
of quantum theory. Alternative commutation relations will not play any role
in the discussion that follows; as we will see, all of the algebras I consider in
this paper are modifications of the Weyl algebra so that they, in some sense,
implement the same commutation relations. I do not think this poses a

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problem for this paper as my goal is not to justify the commutation relations,
but instead to take them on as empirical hypotheses made by all quantum
theories. I do, however, think this point raises an important concern—in
addition to the considerations raised here concerning the choice of algebra,
there is also much philosophical work to be done to analyze the canonical
commutation relations (in their many forms), perhaps by applying the
approach of Alfsen and Shultz (2001) to explicate the physical content of the
algebraic structure in terms of the constraints it places on the state space of
an algebra. Such work on the significance of the canonical commutation
relations is beyond the scope of this paper. But noticing that one might have
occasion to use different commutation relations helps motivate the approach
of considering alternative algebras of observables.

Some of the choices made in defining an algebra of observables may affect
the physical content of a quantum theory, and some may not. I submit that
one of the ways (but not the only way) to probe the physical content—or
perhaps better, the physical consequences —of using a particular algebraic
structure is to analyze the collection of states that algebraic structure allows.
If two algebras differ in their state spaces, and if one wants to countenance
these differing states as physically significant and employ them for physical
purposes, then the choice of algebra becomes significant for physics as well.
By using the phrase ‘physical consequences’, I mean to signal that one may
not intend in choosing a particular algebra to assert that all of the states on
that algebra are to be regarded as physically significant. In practice, working
mathematical physicists use the conditions of the previous section to restrict
attention to certain states after they have chosen an algebra. But some (e.g.
Halvorson, 2004) have started from the premise that we do regard all states
on an algebra as physically significant. So it’s at least worth investigating to
what extent one can hold onto this premise and make a principled choice of
algebra that leads to an appropriate collection of physical states.

As an aside: given that one may have worries about whether there are
any principled approaches to choosing the collections of physical states and
physical quantities, I think it’s worth pointing out that there is hope on the
horizon. One can use tools developed recently for taking the classical limit of
a quantum theory via deformation quantization with continuous fields of
C*-algebras (Landsman, 1998, 2006) to try to provide a principled way of
picking out the physical algebra and physical state space. The principled
approach I have in mind is to abide by the following guideline: the classical
limit of a quantum theory should lead to a physical algebra and physical
state space for the classical theory. This provides substantive constraints on
the algebras and state spaces we use for our quantum theories. While not
providing a complete and ultimate solution, this guideline at least has the

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advantage of reducing questions about quantum theories to questions about
classical theories, with the hopes that we have a better understanding of how
to answer these questions in the classical case. I mention this proposal only to
point out future directions for research and also potential applications of the
tools developed in this paper. Further discussion of this principled approach
requires additional technical tools and is beyond the scope of this paper.

3.3 Example: Regular States and Compact
Operators

In the next section, I will present general algebraic results that allow us to
change the algebra implementing the canonical commutation relations in
order to restrict its physical state space. But before I present these results, I
want to note that we already have a procedure, at least in the case of a
simple system with finitely many degrees of freedom,11 for eliminating
unphysical states by choosing an appropriate algebra. For these simple
systems, it is standard to restrict attention to only regular states, in part
because a result known as the Stone-von Neumann Theorem tells us that
there is a unique irreducible representation12 of the Weyl algebra in which all
of the regular states (and, it turns out, only the regular states) have density
operator representatives. This representation is just the usual Schrödinger
representation, and in this representation the ultraweak closure of the Weyl
algebra is the algebra B(H). So this leads us back to the familiar setting for
non-relativistic quantum mechanics, where every (self-adjoint) element of
B(H) is considered as a physically significant observable and every physical
state has a density operator representative.13

However, for these simple systems with finitely many degrees of freedom,
there is another path one can take to get the same theory without using
Hilbert space methods.14 Instead of choosing the Weyl algebra, one can
directly use the algebra of compact operators on a separable Hilbert space
(i.e., L2(Rn)).15 Then one immediately finds that the state space of this
algebra is equivalent to the collection of regular states. One way to see this is

11Specifically, the procedure will work for quantizing a classical system with finitely
many degrees of freedom and simply connected phase space R2n.

12A representation (π,H) of a C*-algebra A is irreducible if the only subspaces of
H that π(A) leaves invariant are {0} and H.

13See Summers (1999) or Petz (1990) for more on the Stone-von Neumann theorem
and the Schrödinger representation of the Weyl algebra.

14For more on this algebraic approach to regular states, see Feintzeig (2017a).
15For example, one might arrive at this algebra through the prescription known

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to notice that the compact operators have a unique irreducible
representation,16 so one has an immediate analog of the Stone-von Neumann
Theorem. The bidual of the algebra of compact operators is its ultraweak
closure in this representation, the familiar algebra B(H). And there are no
non-regular or otherwise unphysical states on the algebra of compact
operators that cannot be represented as density operators on this
representation.

Thus, one can construct the same quantum theory that we get through
the Weyl algebra and the Stone von-Neumann theorem by instead using the
compact operators and forgoing the need to restrict attention to some
subspace of states. This procedure provides a route for the Algebraic
Imperialist to arrive at the same collection of physical states the Hilbert
Space Conservative arrives at. The key to this procedure is an auspicious
choice of algebra of observables for our quantum theory. So, I suggest that
one might pay closer attention to the choice of algebra in quantization
procedures in other cases when constructing quantum theories. The next
section shows that if one takes seriously the importance of this choice of
algebra, then one can develop powerful algebraic tools generalizing this
procedure for representing physical systems with specified state spaces.

4 Algebraic Adaptability

The purpose of this section is to develop a general response on behalf of the
Algebraic Imperialist to the objection that the abstract algebra allows for
“too many states”. First, in §4.1 we prove a general result providing
necessary and sufficient conditions under which one can find a C*-algebra
with a restricted state space. This allows the Algebraic Imperialist far more
flexibility than previously thought in constructing a theory with the
appropriate state space. Next, in §4.2 we illustrate how this result can be
applied to transform any algebra sufficiently similar to the Weyl algebra to
the algebra B(H) of all bounded operators on a separable Hilbert space. This
shows that the Algebraic Imperialist has at least as much power as the
Hilbert Space Conservative to limit the collection of states she deems
physical—the Algebraic Imperialist always has the option to choose the
collection of states with density operator representatives in a given Hilbert

as Berezin quantization (Landsman, 1998, 2006). If one is worried that one needs a
Hilbert space to define this algebra, note that Landsman (1990) provides a way to
understand this construction from a purely C*-algebraic point of view.

16See Thm. 10.4.6 of Kadison and Ringrose (1997, 751).

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space representation as her priveleged collection of states and then apply the
results here to find a new algebra with precisely that state space. As a
corollary, we show the results here apply when the space of physical states is
chosen to be the collection of regular states or the collection of particle states
accessible from a Fock state.

4.1 General Algebraic Results

Suppose that we have a C*-algebra A and some preferred subset of its dual
space V ⊆ A∗. This section shows necessary and sufficient conditions for the
existence of an algebra inheriting “the same” algebraic relations but with V
as its entire dual space.

First we will need a condition to ensure that V can be dual to a space
supporting an appropriate C*-norm. To that end, define
‖·‖V : A → R as follows for all A ∈ A:

‖A‖V = sup
ω∈V ;‖ω‖=1

|ω(A)|

With this definition, consider the following condition on V :

(i) For all A,B ∈ A,
‖AB‖V ≤‖A‖V‖B‖V

This condition is a minimal technical conditions to ensure V can be the dual
space to a C*-algebra: it ensures that V can be dual to an algebra whose
multiplication operation is norm continuous.

We need one more piece of background before we can present the main
result. The following piece of apparatus allows us to ensure that V has
enough structure to support algebraic operations inherited from A, as we will
make precise in a moment. For any ω ∈ A∗, define a relation ∼ω on A by

A ∼ω B iff for all C ∈ A, ω(AC) = ω(BC) and ω(CA) = ω(CB)

Two observables A and B satisfy this relation ∼ω just in case ω assigns the
same expectation value to A and B, and if whenever we concatenate another
measurement represented by the observable C, then ω yields the same
expectation value for a measurement of A and C as it does for B and C, at
least when the A-C and B-C measurements are performed in the same
relative order.

Now define a relation ∼V on A by

A ∼V B iff for all ω ∈ V , A ∼ω B

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Two observables A and B satisfy this relation ∼V just in case each state in V
assigns the same expectation values to A as to B, and similarly for
concatenating another measurement C, as above.

One easily checks that for any ω ∈ A∗, the relation ∼ω is an equivalence
relation, and similarly for any V ⊆ A∗, the relation ∼V is an equivalence
relation. Now consider the following condition:

(ii) V is maximal 17 in the sense that for all ω ∈ A∗,

if A ∼ω B whenever A ∼V B for A,B ∈ A, then ω ∈ V .

Condition (ii) guarantees that V is as big as possible, by containing all states
compatible with the states in V .

Here, a state ω ∈ A∗ is compatible with the states in V just in case it
judges equivalent any two observables that all of the states in V judge
equivalent. Intuitively, picking a collection of physical states tells us which
observables are really the same or distinct according to whether those
observables could possibly be assigned different expectation values by the
physical states. If two observables are always assigned the same expectation
value, then one is hard pressed to think of them as distinct observables, but
if two observables are at least sometimes assigned different expectation
values, then it is easier to see those observables as distinct. Now, there may
be a state ω that judges as distinct all the observables the states in V judge
distinct and judges equivalent all the observables the states in V judge
equivalent. In this case, we might have just as much reason to countenance
the state ω as physically significant as we have for the states in V . If one
believes this, then one is likely to say that we have just forgotten to include
ω in our specification of V . The maximality condition ensures that we don’t
forget to include any such states.

As we will see, condition (ii) is used to ensure that V is “big enough” to
support the algebraic operations of A. This can be seen as a technical
redundancy because we can generate a maximal subspace from any other
subspace V0 ⊆ A∗ by the following proposition.18

17If condition (i) is satisfied, then condition (ii) holds iff for all ω ∈ A∗,

if ω(A) = 0 whenever A ∈ I, then ω ∈ V

where I = {A ∈ A : ω(A) = 0 for all ω ∈ V}. In other words, V is the collection of
all functionals absolutely continuous with respect to all of the functionals in V . One
could take this statement as the definition of maximality instead, and the results in
this paper would hold exactly as stated.

18Proofs of all results are contained in the appendix.

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Proposition 1. Let A be a C*-algebra and let V0 ⊆ A∗. Let

V = {ω ∈ A∗: for all A,B ∈ A, if A ∼V0 B, then A ∼ω B}

Then V is maximal in the sense of condition (ii) above.

In this case, we will say that the vector space V is generated by V0. Given
a collection of states in V0, V is the smallest maximal collection of states
containing V0, and it is the collection of all states compatible with those in V0
in the sense above. Now we are ready to present our main result, which says
that the conditions listed above are necessary and sufficient for reducing the
state space of our algebra.

Theorem 1. Let A be a C*-algebra and let V ⊆ A∗. Then there exists a
C*-algebra B and a surjective *-homomorphism f : A → B such that
B∗ ∼= V with the isomorphism given by ω ∈ B∗ 7→ (ω ◦f) ∈ V iff conditions
(i) and (ii) are satisfied.

The main idea of the proof (contained in the appendix) is to construct the
C*-algebra B by taking the quotient of the original algebra A by the
equivalence relation ∼V ; in other words, B = A/ ∼V . Moreover, we can show
that the algebra we get through this construction is, in a certain sense,
unique.

Theorem 2. Let (B,f) be the pair given by Thm. 1. For any other
C*-algebra C and surjective *-homomorphism g : A → C such that C∗ ∼= V
with the isomorphism given by ω ∈ C∗ 7→ (ω ◦g) ∈ V , there is a
*-isomorphism α : B → C such that α◦f = g.

These theorems show that one can specify the physical state space
however one likes, as long as it satisfies conditions (i) and (ii). Whenever
these conditions are satisfied, and only when these conditions are satisfied,
we can find a new algebra A/ ∼V that inherits the algebraic relations of A
through the surjective *-homomorphism f, and this new algebra has precisely
the physical state space V as its entire state space.

4.2 Example: B(H)
In this section we deal with the same simple quantum theories of systems
with finitely many degrees of freedom considered in §3.3. Such a theory has
as physical states the density operators on a Hilbert space H with
observables given by all self-adjoint operators in B(H). But suppose that
we’ve chosen some other algebra of quantum observables. Let A be any

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C*-subalgebra of B(H) that contains the constants and separates density
operator states in the sense that for any two density operators ρ1,ρ2 on H,
there is an A ∈ A such that Tr(Aρ1) 6= Tr(Aρ2). The Schrödinger
representation of the Weyl algebra mentioned in the previous section is an
example of just such an algebra. Here we show that there is a natural way to
transform A into all of B(H) using purely algebraic methods.

Why do we need such a procedure, given that we can just take the
ultraweak closure of the algebra A in the natural inclusion representation on
H to get all of B(H)?19 Of course, we could obtain all of B(H) in this way,
but there is something unsatisfying about this approach from the algebraic
perspective. Why should we complete A in the ultraweak topology when
there may be states on A that are not ultraweakly continuous? There’s a
sense in which completing A in the ultraweak topology does not respect the
algebraic structure (or really the state space) of A, because the completion of
A in its abstract weak topology (as opposed to the ultraweak topology) to
form the bidual A∗∗ in fact leads us to a much larger algebra than B(H).20

I will show that Thm. 1 provides a general procedure to transform A into
B(H). To see this, notice that there is a sense in which A is too small and a
sense in which it is too large. A is too small in the sense that it may not
contain many elements of B(H) like projections (as is the case with the Weyl
algebra). But A is too large in the sense that it may allow for states that are
not ultraweakly continuous and so cannot be represented by density
operators on H. As such, we will first enlarge A to A∗∗ to obtain all of the
missing operators including the projections on H. Then we will restrict
attention to a collection of physical states by applying Thm. 1.

Let V
Q
0 be the collection of bounded linear functionals on A that are

ultraweakly continuous on H. The following proposition shows that reducing
A∗∗ by Thm. 1 with this preferred collection of states brings us back to the
usual setting of B(H).

Proposition 2. Let VQ be the vector subspace of bounded linear functionals

on A∗∗ generated by V
Q
0 . Then the C*-algebra A

∗∗/ ∼VQ in Thm. 1 is
*-isomorphic to B(H).

This shows that one can generate B(H) by first adding to A the weak limit
observables in A∗∗ and then reducing the algebra by Thm. 1 to restrict
attention to only states with density operator representatives on H. Again,

19Notice that the natural inclusion representation of the algebra A on H is irre-
ducible and employ Prop. 1.21.9 of Sakai (1971, 52).

20More precisely, the algebra B(H) will in general be properly embedded in A∗∗,
which one can see directly in the universal representation of A (See Feintzeig, 2017b).

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the first step of enlarging the algebra to A∗∗ is necessary because there may
be many bounded operators outside A. In particular, we cannot reduce A
directly to the compact operators (recall that B(H) is the bidual and hence
ultraweak closure of the compact operators) because there may be compact
operators that are not in A, as in the case of representations of the Weyl
algebra.

This shows that the Algebraic Imperialist always has the resources to
reduce the physical state space of her algebra in at least all of the ways the
Hilbert Space Conservative can. Where the Hilbert Space Conservative
would take an irreducible representation of the algebra and restrict attention
to density operator states in that representation, the Algebraic Imperialist
can just choose the vector subspace of A∗ generated by those states (i.e., the
density operator states in that representation) as her priveleged subspace and
directly apply Thm. 1 to obtain a new algebra with precisely the right state
space.

Moreover, we can immediately apply Prop. 2 to two of the candidates for
physical state spaces considered in §3.1. First, consider the Weyl algebra
W(R2n) over a finite-dimensional and simply connected phase space R2n with
the standard symplectic form. Let V R0 be the vector space of linear
combinations of regular states on W(R2n), where the linear combinations are
taken according to the vector space operations in W(R2n)∗ (not according to
the Hilbert space operations of some particular representation). Since the
Schrödinger representation on the Hilbert space HS is irreducible and
faithful, it follows that W(R2n) is *-isomorphic to a C*-subalgebra of B(HS)
that separates density operator states. And, since the Schrödinger
representation is regular, every regular state on W(R2n) is ultraweakly
continuous on HS (See Petz, 1990), which means V R0 is the collection of
bounded linear functionals on W(R2n) that are ultraweakly continuous in the
Schrödinger representation on HS. In other words, V R0 is just the physical
state space V

Q
0 defined above for the particular case of the Weyl algebra

W(R2n) in the Schrödinger representation on HS. Thus, we have the
following immediate corollary of Prop. 2.

Corollary 1. Let VR be the vector subspace of bounded linear functionals on
W(R2n)∗∗ generated by V R0 . Then the C*-algebra W(R

2n)∗∗/ ∼VR in Thm. 1
is *-isomorphic to B(HS).

This shows that Thm. 1 can be applied to the Weyl algebra with the
collection of physical states as the regular states.

Next, consider the Weyl algebra for a possibly infinite dimensional phase
space M (e.g. a field system), which we will denote W(M). We will say that
a state is accessible from a Fock state ω (See Petz, 1990, Ch. 4) if it is a

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vector state in the Fock representation21 for ω that is obtained by some
linear combination of creation operators acting on the vacuum vector
representing ω. Intuitively, a Fock state is one that can be represented as a
superposition of particle configurations obtained by creating particles in the
vacuum state ω. As is well known, this collection of vectors spans the Fock
space, i.e. the Hilbert space HF of the Fock representation (Petz, 1990, 34).
Since the Fock representation for ω is faithful and irreducible (Petz, 1990,
Thm. 4.7, 34), we know that W(M) is *-isomorphic to a C*-subalgebra of
B(HF ) that separates density operator states.

Now, in order to apply Thm. 1, we need to find a closed vector subspace
of W(M)∗ containing the physical states. Let V F0 be the smallest closed
vector subspace of W(M)∗ containing the states accessible from the Fock
state ω. To generate this vector space V F0 , we need to allow for mixtures and
arbitrary linear combinations of the states accessible from the Fock state ω,
and then close the resulting vector subspace in the norm topology on
W(M)∗. Allowing for mixtures (convex combinations) of states accessible
from the Fock state ω yields that all finite rank density operator states on
the Fock space HF are in V F0 , while allowing for arbitrary linear
combinations yields that all finite rank operators on HF are in V F0 . From
basic facts in the theory of Hilbert space operators (See Reed and Simon,
1980), we know that the finite rank operators are dense in the collection of
all trace class operators on HF with the trace norm, which is identical to the
standard supremum norm on those operators, considered as linear functionals
in W(M)∗ by the usual trace prescription. Thus, V F0 is the collection of all
trace class operators on HF , and hence is the collection of all bounded linear
functionals on W(M) that are ultraweakly continuous on the Fock space HF .
In other words, V F0 is just the physical state space V

Q
0 defined above for the

particular case of the Weyl algebra W(M) in the Fock representation on HF
Thus, we have the following immediate corollary of Prop. 2.

Corollary 2. Let VF be the vector subspace of bounded linear functionals on
W(M) generated by V F0 . Then the C*-algebra W(M)

∗∗/ ∼VF in Thm. 1 is
*-isomorphic to B(HF ).

This shows that Thm. 1 can be applied to the Weyl algebra with the
collection of physical states as those accessible from a Fock state.

One might worry at this point about the status of the collection of states
V we deem physically significant in order to apply Thm. 1. The goal of this
paper was to show that one can find purely algebraic resources for choosing

21The Fock representation is just the GNS representation for ω. For more on the
GNS representation and vector states, see Kadison and Ringrose (1997).

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an appropriate algebra of physical quantities. But it seems that the
specification of a space of physical states V is an extra resource we need to
avail ourselves of. And in the previous examples the state space that we use
consists of the folium of a Hilbert space representation, so one might doubt
that the methods of this section provide purely algebraic tools.

In response, I want to note that my purpose in this paper is only to show
that however one specifies the collection of physically significant states, we
can find an algebra with that collection of states as its full state space (as
long as it satisfies a few constraints). It is a natural question whether one
needs a Hilbert space representation to specify V , but that is not a question I
can deal with fully in this paper. It may be helpful, however, to make a
number of remarks concerning the above examples. First, one can define the
regular states in a way completely free of reference to any Hilbert space
representation by simply looking at the expectation values assigned by states
to one-parameter families of Weyl unitaries. It turns out this definition
specifies the folium of a particular Hilbert space representation, but this can
be understood as a consequence of the continuity assumptions. In the same
vein, although we have used a Fock representation to specify the states
accessible from a Fock state, this does not imply that one needs a Hilbert
space representation to define this collection of states. It is a further
substantive question whether one can give an algebraic specification of these
particle-like states, especially given that the definition of Fock states on their
own does not require the use of any Hilbert space representations. For
example can one understand these particle-like states to constitute the
smallest collection of states containing a Fock state and satisfying conditions
(i) and (ii)?

As may be clear by now, I am not interested in specifying a rigid set of
rules that the Algebraic Imperialist must abide by when specifying a
collection of physical states. I think asking for such a set of rules in some
sense misses the point. The point is that the use of Hilbert space methods
has been distracting and obscuring in some cases by confusing conditions of
technical convenience with conditions that have physical content. The tools
developed in this paper constitute just one step toward untangling these
conditions. The point is not that one has reason for rejecting Hilbert space
methods altogether, but that too much focus on the use of Hilbert space
methods or even debate about whether Hilbert space methods are useful can
distract from the application of alternative algebraic methods that have
promise for clarifying issues of physical interpretation that have confused
researchers working in the Hilbert space approach.

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5 Discussion

I have argued that Thm. 1 gives a general tool for the Algebraic Imperialist
to use to respond to the “too many states” objection. I have shown that
there is a very general procedure the Algebraic Imperialist can use to rid
herself of unphysical states. The result in Prop. 2 shows that this procedure
can apply to any collection of physical states that consists of all and only the
density operator states on some irreducible Hilbert space representation.
Specifically, this procedure applies to the regular states on the Weyl algebra,
which exhaust the density operator states in the Schrödinger representation,
and to the states accessible from a Fock state, which can be thought of as
particle states obtained through particle creation in a particular vacuum
state. But it is still an open question whether the Algebraic Imperialist can
apply this procedure to other existing candidates for physical state spaces. In
other words, it is still left to be shown that the rest of the standard proposals
for physical state spaces (e.g., the Hadamard states, locally definite states,
and DHR states) actually satisfy conditions (i) and (ii) of Thm. 1.
Answering this question would inform us about precisely how much freedom
the Imperialist has in responding to the objection that she allows for “too
many states”.

The results of this paper show that the Algebraic Imperialist has at least
as much flexibility for restricting the state space of a quantum theory as the
Hilbert Space Conservative does. Prop. 2 shows that when the physical
states themselves form the space of density operator states on an irreducible
Hilbert space representation, then it is possible to reduce the algebra to one
with an appropriate state space. But I have also claimed that the flexibility
or adaptability the Algebraic Imperialist gains through Thm. 1 is a virtue of
the algebraic point of view. As such, one ought to ask whether Thm. 1 gives
us more freedom for reducing the state space than the Hilbert Space
Conservative has. In other words, are there any subspaces of states V
satisfying (i) and (ii) that do not form the space of density operator states of
some Hilbert space representation? If not, then the procedure I have outlined
for reducing the state space of an abstract algebra works in exactly the same
cases that the Conservative’s procedure would work. Thus, I have not yet
made the case that this virtue I have brought to our attention—adaptability
of the algebra and state space—is a virtue of Imperialism over Conservatism.
My results do show that the Imperialist can deal with the objection that she
allows for “too many” states, but I admit that this may just bring the
Imperialist in line with the Hilbert Space Conservative.

Even with these open questions, I believe the results of this paper have
significance for the interpretation of algebraic quantum theories. They show

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that considerations of the physical content of the abstract algebra, its states,
and topologies, can lead us to new technical and conceptual tools beyond
those of Hilbert space representations. I think the tools developed here have
the potential to prove useful for understanding and interpreting quantum
theories. I think the fact that the perspective taken in this paper leads to
precise technical and conceptual questions whose answers would appear to
have philosophical significance shows the promise inherent in this approach. I
can only hope that others find these tools to be as useful as I do.

Appendix: Proofs of Results

In this appendix, we prove the results of §4. The arguments rely on some
technical notions not defined in the body of the paper; for explicit definitions,
see Kadison and Ringrose (1997). First, we prove the results of §4.1.

Proposition 1. Let A be a C*-algebra and let V0 ⊆ A∗. Define V by

V = {ω ∈ A∗: for all A,B ∈ A, if A ∼V0 B, then A ∼ω B}

Then V is maximal in the sense of condition (ii), i.e. for all ω ∈ A∗,

if A ∼ω B whenever A ∼V B for A,B ∈ A, then ω ∈ V .

Proof. (⊇) Suppose ω ∈ A∗ is such that A ∼ω B whenever A ∼V B for all
A,B ∈ A. Let A,B ∈ A be such that A ∼V0 B. Let ρ ∈ V . Then, by the
definition of V , A ∼ρ B. Since this holds for all ρ ∈ V , it follows that
A ∼V B. This implies by the assumption on ω that A ∼ω B. Again, by the
definition of V , it follows that ω ∈ V .

To prove Thm. 1, we will need the following lemma.

Lemma 1. Suppose V satisfies (i) and let

I = {A ∈ A : ω(A) = 0 for all ω ∈ V}

Then I is a closed two-sided ideal.

Proof. First, I is an additive subgroup of A since for all ω ∈ V and all
A,B ∈ I,

ω(A + B) = ω(A) + ω(B) = 0

Next, I is a two-sided ideal since for all A ∈ I, C ∈ A, and ω ∈ V , it follows
from condition (i) and the fact that ‖A‖V = 0 that

ω(AC) ≤‖ω‖‖A‖V‖C‖V = 0

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ω(CA) ≤‖ω‖‖C‖V‖A‖V = 0

Finally, we show I is closed. Suppose An ∈ I and ‖A−An‖→ 0 for A ∈ A.
Then

ω(A−An) ≤‖ω‖‖A−An‖→ 0

But we also know that for all n,

ω(A−An) = ω(A) −ω(An) = ω(A)

and hence the sequence ω(A−An) does not depend on n, from which it
follows that ω(A) = 0.

Theorem 1. Let A be a C*-algebra and let V ⊆ A∗. Then there exists a
C*-algebra B and a surjective *-homomorphism f : A → B such that
B∗ ∼= V with the isomorphism given by ω ∈ B∗ 7→ (ω ◦f) ∈ V iff conditions
(i) and (ii) are satisfied.

Proof. (⇐) First, we show that the conditions (i) and (ii) are sufficient. Let
I = {A ∈ A : A ∼V 0} and define a relation ∼I on A by

A ∼I B iff A = B + C for some C ∈ I

We show that A ∼I B iff A ∼V B. If A ∼V B, then one can show
(B −A) ∈ I, so B = A + (B −A) and hence A ∼I B. If A ∼I B, then there
is a C ∈ I such that A = B + C. For any ω ∈ V and any D ∈ A

ω(AD) = ω(BD) + ω(CD) = ω(BD)

ω(DA) = ω(DB) + ω(DC) = ω(DB)

Hence, A ∼V B. We will denote by [A] the equivalence class of all B ∈ A
such that A ∼V B and A ∼I B. By Thm. 1.8.2 of Dixmier (1977, 20), we
know that B := A/I = A/ ∼V is a C*-algebra and f : A → B defined by
f(A) = [A] for all A ∈ A is a surjective *-homomorphism.

Now suppose that ω ∈ A∗ is such that ω(C) = 0 for all C ∈ I. Then for
any A,B ∈ A such that A ∼V B, we know that A = B + C for some C ∈ I
and hence for all D ∈ A,

ω(AD) = ω(BD) + ω(CD) = ω(BD)

ω(DA) = ω(DB) + ω(DC) = ω(DB)

from which it follows that A ∼ω B. Since V is maximal, ω ∈ V . Since all
ω ∈ V assign ω(C) = 0 for all C ∈ I, it follows that

V = {ω ∈ A∗ : ω(C) = 0 for all C ∈ I}

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and by Prop. 2.11.8 of Dixmier (1977, 63), we know that B∗ ∼= V with the
isomorphism given by ω ∈ B∗ 7→ (ω ◦f) ∈ V .

(⇒) Suppose f : A → B is a surjective *-homomorphism such that
B∗ ∼= V with the isomorphism given by ω ∈ B∗ 7→ (ω ◦f) ∈ V . We
immediately know that V satisfies condition (i) because multiplication in the
C*-algebra B is norm continuous.

Let
I = {C ∈ A : f(C) = 0}

Then A/I ∼= B by Cor. 1.8.3 of Dixmier (1977, 21). Again, by Prop. 2.11.8
of Dixmier (1977, 63) we know that

V = {ω ∈ A∗ : ω(C) = 0 for all C ∈ I}

Suppose ω ∈ A∗ is such that for all A,B ∈ A, if A ∼V B, then A ∼ω B. For
any C ∈ I, we know C ∼V 0, which implies C ∼ω 0 and hence
ω(C) = ω(0) = 0. It follows that ω ∈ V , which shows that V satisfies
condition (ii).

Theorem 2. Let (B,f) be the pair given by Thm. 1. For any other
C*-algebra C and surjective *-homomorphism g : A → C such that C∗ ∼= V
with the isomorphism given by ω ∈ C∗ 7→ (ω ◦g) ∈ V , there is a
*-isomorphism α : B → C such that α◦f = g.

Proof. Suppose C is a C*-algebra with surjective *-homomorphism g : A → C
such that C∗ ∼= V with the isomorphism given by ω ∈ C∗ 7→ (ω ◦g) ∈ V .
Then define α : B → C by α([A]) = g(A) for all [A] ∈ B (one requires the
axiom of choice to choose a representative A for each [A] ∈ B). One easily
checks that α is well-defined (because A ∼V B implies ‖g(A) −g(B)‖ = 0,
which implies g(A) = g(B)) and a *-isomorphism. Furthermore, it follows
immediately that α◦f = g.

Now we prove the results of §4.2. Here, A is a C*-subalgebra of B(H)
containing the constants and separating density operator states. Let π
denote the representation π(A) = πU (A)P of A on HU , where (πU,HU ) is the
universal representation and P is the projection determined through Thm.
10.1.12 of Kadison and Ringrose (1997) associated with the inclusion
mapping of A on H. To prove Prop. 2, we will need the following lemma
characterizing P .

Lemma 2. P is the projection onto the span of all subspaces of HU carrying
a subrepresentation of πU quasi-equivalent to π.

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Proof. Let (π1,H1) and (π2,H2) be representations of A. Let P1,P2 ∈ πU (A)
be their associated central projections. First, we show that if π1,π2 are
quasi-equivalent, then P1 = P2. The representation

ϕ1 : A 7→ πU (A)P1 is quasi-equivalent to π1 and ϕ2 : A 7→ πU (A)P2 is
quasi-equivalent to π2. So if π1,π2 are quasi-equivalent, then so are ϕ1 and
ϕ2. By Thm 10.3.3 (ii) of (Kadison and Ringrose, 1997, 736), the central
carriers of P1 and P2 are equal, i.e. CP1 = CP2 . Since P1,P2 are central
projections, P1 = CP1 = CP2 = P2.

Next, we show that if π1,π2 are disjoint, then P1P2 = 0. If π1,π2 are
disjoint, then so are ϕ1 and ϕ2—for suppose there were a subrepresentation
of ϕ1 quasi-equivalent to a subrepresentation of ϕ2. Then we would have a
subrepresentation of π1 quasi-equivalent to a subrepresentation of π2 (by
composition of the relevant *-isomorphisms, see Kadison & Ringrose 10.3.4,
737). Hence, by Kadison & Ringrose 10.3.3, P1P2 = CP1CP2 = 0.

Proposition 2. Let VQ be the vector subspace of A
∗∗∗ generated by

V
Q
0 = {ω ∈ A

∗: ω is ultraweakly continuous on H}

Then the C*-algebra A∗∗/ ∼VQ in Thm. 1 is *-isomorphic to B(H).

Proof. First, notice that by Prop. 10.1.14 of (Kadison and Ringrose, 1997,
722), V

Q
0 is the collection of functionals ω ∈ A

∗ such that ω = Pω, where Pω
is defined as in Kadison and Ringrose (1997, 721-2). Let π(A) denote the
ultraweak closure of π(A).

Define j : (A∗∗/ ∼VQ ) → π(A) by

j([A]) = π̃U (A)P

for any A ∈ A∗∗, where π̃U is the unique weakly continuous extension of πU
to A∗∗. This map j is well-defined because for any A,B ∈ A∗∗, A ∼VQ B
implies π̃U (A)P = π̃U (B)P .

j is onto: for every  ∈ π(A),  = j([A]) for A = π̃−1U (α
−1(Â)), where

α : πU (A)P → π(A) is the *-isomorphism provided by Thm. 10.1.12 of
Kadison and Ringrose (1997). Furthermore, j is one-to-one: if π̃U (A)P = 0
for A ∈ A∗∗, then for all ω ∈ A∗ such that ω = Pω, we know that
ω(A) = ω(π̃U (A)P) = 0, and hence [A] = [0].

Since j obviously preserves algebraic operations, it follows that j is a
*-isomorphism, and since π is quasi-equivalent to the inclusion mapping of A
on H by Thm. 10.1.12 of Kadison and Ringrose (1997), it follows that
A∗∗/ ∼VQ is *-isomorphic to B(H).

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