-algebras are a type of involutive Banach algebras which arise in the study of operators on Hilbert spaces, Lie group representations, locally compact topological spaces, knots, noncommutative , among other topics in mathematics and theoretical physics . Their study was initiated in the 1930’s with the purpose of axiomatizing quantum mechanics, and still today, -algebras play a decisive role in formulations of quantum statistical mechanics and quantum .
The defining property of these algebras is that the norm and the involution are related in a very special way.
Definition 1 - A -algebra is a Banach *-algebra such that for all .
The equality in Definition 1 is sometimes called the axiom. It turns out that one can weaken this condition and still specify the same of algebras.
Definition 2 - A -algebra is a Banach algebra with an antilinear involution such that for all .
Definition 3 - A -algebra is a Banach algebra with an antilinear involution such that
2 C* Norm
This also implies that the norm in a -algebra is unique, in the sense that there is no other norm in the algebra that satisfies that axiom, i.e. that turns the algebra into a -algebra. This is a stark contrast to the case of general normed algebras, where one may find many norms which are with the algebraic structure.
Moreover, the norm occupies a unique amongst the possible norms for an involutive algebra. Suppose that is a algebra with norm . If is any other norm for which is a Banach *-algebra, then we must have
Hence we see that the norm enjoys an extremal property — it is the smallest possible norm for which is a Banach *-algebra.
3 Elements of a C*-algebra
Like in involutory rings, there are some special elements in -algebras that deserve some attention. We recall some definitions here:
Let be a -algebra with identity element . An element is said to be
It is many times useful to have some interpretation for this elements. One of this interpretations comes from complex analysis: we regard the elements of a -algebra as functions with values in and the involution as complex conjugation.
In this frame, self-adjoint elements correspond to real functions, unitary elements correspond to functions whose values lie in the unit circle in and positive elements correspond to positive functions (functions with values in ).
It is easily seen that self-adjoint elements are closed under addition, multiplication and multiplication by real numbers. It can be proven the same for positive elements (with multiplication by positive numbers).
There are some decompositions of elements in a -algebra analogous to some decompositions in complex analysis. For instance, every element in a -algebra has a unique decomposition of the form
where are self-adjoint. This is similar to the decomposition of a complex valued function in its real and imaginary parts.
Moreover, every self-adjoint element is of the form
where are positive elements. This is similar to the decomposition of real valued functions in its positive and negative parts.
There are many other aspects of the theory of -algebras for which this kind of interpretation proves to be very insightful.
For example, -algebras happen to have a natural partial ordering. One can define an ordering by declaring that when is positive. Given this ordering, one can then speak of such things as monotonic functions, monotonic sequences, and positive linear functionals on the algebra. These notions, in turn, prove to be extremely useful in the study of -algebras.
Having discussed these algebras in general terms, it is high time that we illustrate the definition with some examples.
As our first class of examples, we consider algebras of functions. Let be a compact Hausdorff topological space and let be the algebra of continuous functions from to . For the involution operation, we take pointwise complex conjugation and for the norm we take the norm of uniform convergence:
It is a routine matter to check that the norm and involution satisfy the appropriate algebraic requirements. Completeness under this norm follows from the fact that the uniform limit of continuous functions on a locally compact Hausdorff topological space is continuous.
More generally, instead of a compact space, we can take a locally compact Hausdorff space and consider the algebra of continuous functions that vanish at infinity, endowed with the same norm and involution. These are important examples of -algebras.
As our second class of examples, we consider operator algebras. Let be a complex Hilbert space with inner product and let be the algebra of bounded operators on . For the involution, we take the adjoint operation and as a norm we take the usual operator norm:
Again, it is straightforward to verify that the norm and involution satisfy the appropriate algebraic requirements, as is done in an attachment to this entry. Completeness under the norm follows from a well-known theorem of functional analysis.
5 Commutative and noncommutative C*-algebras
The algebras in Example 1 above are more than just an example. In fact, all commutative -algebras are *-isomorphic to for some locally compact Hausdorff space . Moreover, is compact if and only if the -algebra has an identity element. This is the content of the Gelfand-Naimark theorem.
Furthermore, there is a correspondence between properties of the topological space and properties of the -algebra. For example: a compactification of the space corresponds to a unitization of the -algebra; the space is connected if and only if the -algebra has no non-trivial projections, among many other interesting correspondences.
For this reason, the theory of (noncommutative) -algebras is many times called noncommutative topology (click on the link for more information).
The second example is also more than just an example of -algebras. In fact, by the Gelfand-Naimark representation theorem, all -algebras are *-isomorphic to a norm closed *-subalgebra of , for some Hilbert space .
Note, however, that this does not provide a “classification” of -algebras since we do not know in general what are the closed *-subalgebras of . This is merely a (very-important) structural theorem. The classification problem for -algebras is still open.
6 Additional Examples
Compact operators in a Hilbert space form a closed ideal of . Moreover, this ideal is also closed for the involution of operators. Hence, the algebra of compact operators, , is a -algebra.
Let be a measure space. The space (http://planetmath.org/LpSpace) is an algebra under pointwise operations. We can define an involution again by complex conjugation and we consider the essential supremum norm . It can be readily verified that, under these operations and norm, is a -algebra.
The algebras are also particularly important since they are examples of von Neumann algebras, which are a specific kind of -algebras.
|Date of creation||2013-03-22 12:57:55|
|Last modified on||2013-03-22 12:57:55|
|Last modified by||asteroid (17536)|