Also see a terse introduction to abstract algebra in [Appendix J]
A topological space is a set X together with a collection OMEGA, of subsets, such that:
i) X is in OMEGA and the null set {} is in OMEGA
ii) For all u and v in OMEGA,
u INTERSECTION v is in OMEGA
iii) If for all lambda in LAMBDA, a generalized index set,
u(lambda) is in LAMBDA, then
UNION u(lambada) is in OMEGA
OMEGA is the collection of "open" sets. A subset of X is said to be "closed" if its complement in X is open. Intuitively the open sets approximate their members and generalize the open balls of a metric space.
The notion of a topological space is frequently a bit too general. A system of 4 increasingly strong separation axioms, due to Alexandroff and Hopf, can then be sequentially assumed.
T_0: For each x in S there is a U in T which
contains x.
T_1: If U and V in T both contain x, then there is
a W in T such that W is a proper subset of U and
W is a proper subset of V.
T_2: If U(x) is a neighborhood of x, and y is in U(x),
then there exists a U(y) such that U(y) is a proper
subset of U(x).
T_3: For x not equal to y, there exist neighborhoods U(x),
U(y) such that U(x) INTERSECTION U(y) = {} (null set).
A topological space that satisfies all four axioms is called a Hausdorff Space.
A metric space is a set X equipped with a real-valued "distance" function d(., .) defined on the cartesian Crossproduct of X with itself so that d() has the properties:
For all x, y, z in X:
i) d(x, y) >= 0
ii) d(x, y) = 0 iff x = y
iii) d(x, y) = d(y, x)
iv) d(x, y) <= d(x, z) + d(z, y)
A norm on a vector space V is a real valued function N(.) such that for all x, y in V, alpha in the scalar field K:
i) N(x) >= 0
ii) N(x) = 0 iff x = 0
iii) N(alpha x) = |alpha| N(x)
|alpha| is the absolute value of alpha
iv) N(x + y) <= N(x) + N(y)
A norm determines the norm topology by the open ball neighborhoods of x with radius a.
N(x; a) = { y in V : N(x - y) < a }
A complex inner product space
is a vector space V over the
complex field C equipped with a complex valued
function <.|.>, the inner product
on V CROSS V such that,
for all x, y, z in V, alpha in C:
i) <x|x> >= 0 and <x|x> = 0 iff x = 0
ii) <x|y + z> = <x|y> + <x|z>
iii) <x|alpha y> = alpha <x|y>
iv) <x|y> = <y|x>^*
where "*" indicates complex conjugation.
An inner product induces a norm on V by
N(x) := sqrt( <x|x> )
and a norm induces a metric by
d(x, y) := N(x - y)
Regarding the relative strengths of hypotheses of introducing
into a space, a metric, an inner product and a norm, as implications:
inner product => normed & linear => metric
None of these implications are generally reversible.
An inner product requires that the space be isomorphic to its space
of linear functionals; an inner product can be introduced on any
finite dimensional vector space. This is not so for infinite dimensional
spaces. The classical L^p spaces are defined as spaces of measurable real
valued functions defined on a measure space S with measure m, so
that
|| f ||_p = [ integral |f(s)|^p dm(s) ]^(1/p)
s in S
the p-norm of f is bounded; these are normed linear spaces, complete
in their norm topology (Banach spaces) that are not inner product
spaces, unless p=2, the case of a Hilbert space.
A metric space need not be a linear space.
A normed space possesses an inner product iff the norm statisfies
(N(x - y))^2 + (N(x + y))^2 = 2 (N(x))^2 + 2 (N(y))^2
the parallelogram law.
A norm derived from an inner product by N(x) = sqrt(), can
recover the inner product by:
<x|y> = (1/4)(N(x + y)^2 - N(x - y)^2)
for a real vector space,
<x|y> = (1/4)(N(x + y)^2 + i N(x + i y)^2 - N(x - y)^2 - i N(x - i y)^2)
for a real complex vector space; these are known as polarization
formulas.
In particular then for A, B, C, in Alg some finite dimensional complex matric algebra with tacitly assumed Euclidean inner product, and for alpha in C the complex field, we have for the trace functional Tr(.):
Tr( A! (B + C) ) = Tr( A! B ) + Tr( A! C )
Tr( A! (alpha B) ) = alpha Tr( A! B )
Tr( (alpha A)! B ) = alpha* Tr( A! B )
Tr( A! ) = [Tr( A )]*
so
Tr( A! B ) = [Tr( B! A )]*
Tr( A! A ) >= 0
and
Tr( A! A ) = 0, iff A = 0
Therefore, the map from Alg CROSS Alg to C
(A, B) -> (1/n)Tr( A! B )
is an Euclidean inner product on Alg, and
N(A) := sqrt( (1/n)Tr( A! A ) )
is its induced norm.
Furthermore, this inner product is preserved under unitary
similarity transformations on the algebra:
If A' = U! A U, B' = U! B U, then
Tr( A'! B' ) = Tr( A! B )
Pseudoinner Products on Hilbert(n) and Alg(n)
Expressed in Terms of the Euclidean Inner Product
Technically, the so called Minkowski inner product is not an inner product at all since it violates condition i) and allows vectors of zero Minkowski length that are not identically zero, but that have vanishing norm. This is formally arranged by using a noneuclidean Hermitean ground form say G on the linear space with inner product being defined by
(x|y) = <x|G|y>
rather than
(x|y) = <x|y>
The ground form used for such an "indefinite inner product" on an Hilbert(n) can be extended to an indefinite inner product defined on the algebra of linear operators Alg(n) acting on Hilbert(n) by defining
(A, B)_G := (1/n)Tr( A! G B )
with G again being required to be only Hermitean and of rank n.
The formal definitions of indefinite norm and metric then also follow.
N_G(A) := sqrt( (1/n)Tr( A! G A ) )
d_G(A, B) := N_G( A - B )
The indefinite inner product, norm and metric has
been expressed on both Hilbert(n) and Alg(n)
in terms of the standard Euclidean inner product, norm and metric
and the indefinite Hermitean form G(n).
Continuing, one defines also G(n)-Hermiticity, i.e., Hermiticity with
respect to G(n) as follows.
If Euclidean Hermiticity is expressed
A! = A
and if a scaling of Hilbert(n), hence Alg(n) is performed
so that the diagonal elements of G(n) consists of (n-1) +1s,
and one -1, then
G(n) = G^(-1)(n)
and G(n)-Hermiticty is expressed as
G(n) A! = A G(n)
Right G(n)-Hermiticity and Left G(n)-Hermiticity
are equivalent because G(n) = G^(-1)(n).
The G(n)-Hermitean conjugate operator of A is
A^# = G^(-1)(n) A! G(n)
If V is any operator that preserves G by similarity transformation,
V G V^(-1) = G
so
V^(-1)! G V! = G
and A and B transform relatively contragrediently,
A -> V^(-1) A V^(-1)!
B -> V! B V
then the indefinite inner product so defined is invariant under
the group of transformations that leave G invariant in the
similarity action defined above:
Tr( (V^(-1) A V^(-1)!)! G (V! B V) )
= Tr( V^(-1) A! V^(-1)! G V! B V )
= Tr( V^(-1) A! G B V )
= Tr( V V^(-1) A! G B )
= Tr( A! G B )
Therefore,
(1/n)Tr( (V^(-1) A V^(-1)!)! G (V! B V) )
= (1/n)Tr( A! G B )
A Hilbert space is an inner product space complete in its induced norm topology. (We consider only complex Hilbert spaces and therefore only complex inner product spaces, and include, of course, finite dimensional Hilbert spaces.)
An involution of a complex algebra is a mapping
A -> A~
onto the algebra such that:
i) (A~)~ = A
ii) (A + B)~ = A~ + B~
iii) (AB)~ = B~ A~
iv) (aA)~ = a* A~
where '*' denotes comnplex conjugation
A C*-algebra is an algebra Alg with involution that
is isomorphic to a norm closed algebra of bounded operators
on a Hilbert space. This definition includes finite dimensional algebras
acting on finite dimensional complex linear spaces with Euclidean
inner product.
With the isomorphism, the induced norm of the Hilbert space
induces in a natural way, a norm as well as an inner
product on Alg.
Let A be an element of Alg, x be an element of Hilbert.
N( A x )
N(A) := sup --------
x not 0 N( x )
The definition of the norm of an element of Alg
turns out to be equivalent to "spectral radius". Loosely
speaking, this is the absolute value of a maximal eigenvalue.
For finite dimensional algebras, this definition coincides
exactly with the norm defined by the trace.
To extend the notion of trace to operators acting on an infinite dimensional separable Hilbert space, for any orthonormal basis |f_k> define
infin
Tr( A ) = SUM <f_k| A |f_k>
k=1
An operator A is said to be of trace class iff
Tr( |A| ) is finite
where
|A| := sqrt( A! A )
The square root of the operator can and is taken
unambiguously positive since (A!A) is always a positive operator.
Not all bounded operators are of trace class.
But, they constitute
a *-ideal in the algebra of bounded operators on a Hilbert space.
N_1(A) := Tr( |A| )
the trace class operators are a Banach space with that norm and the
operators of finite rank are
N_1-dense in the *-ideal of trace class operators.
In the following, we review some definitions of, and relations between various topologies on Alg(Hilbert) the algebra of bounded operators on a Hilbert space.
The open sets of the norm (uniform) topology of Alg are once again, generated by the open balls of the induced norm.
A set M in Alg is open in
the strong operator topology iff
for all A in M
there exists {x_j: j = 1, ..., n }, a finite set in Hilbert
and there exists an epsilon > 0 such that M contains
{ B in Alg(Hilbert): N(Bx_j - Ax_j) <= epsilon, for
j = 1, ..., n.
This does not say anything about how B acts on the orthogonal complement of the subspace spanned by the x_j. Therefore, no non-empty set open in the strong operator topology is bounded in norm. The norm topology is stronger (= finer = "has more open sets" = "fewer convergent sequences" ) than the strong operator topology. Loosely, to approximate an operator convergently in the strong operator topology, you must be able to approximate it on any finite set of vectors simultaneously.
A set M in Alg(Hilbert) is open in the weak operator topology
(QFT jargon: convergence in transition amplitudes)
iff for all A in M,
there exists {x_j, y_j in Hilbert: j = 1, ..., n } and epsilon > 0
such that
| <x_j|By_j> - <x_j|Ay_j> | <= epsilon
implies that B is in M.
The strong operator topology is stronger than the weak operator topology. The involution mapping A -> A! is continuous in the weak operator topology but NOT in the strong operator topology. The mapping (A,B) -> AB is not continuous in either topology. That is, multiplication is not jointly continuous. Multiplication is, however, continuous separately in each variable, and jointly strongly continuous in both topologies on bounded sets in Alg(Hilbert).
A von Neumann algebra is a self-adjoint algebra of operators on a Hilbert space that contains the identity operator and is closed in the weak operator topology. The latter condition turns out to be equivalent to its being closed in the strong operator topology. Closure in the strong operator topology is, however, more restrictive than closure in the norm topology. Where a C*-algebra can be studied abstractly, a von Neumann algebra depends very much on the explicit Hilbert space realization upon which the operators act.
Email me, Bill Hammel at