Approximate identity

An approximation of one is a concept from the mathematical theory of Banach algebras. Many important applications for Banach algebras do not have a unit element. An adjunction of the element would be an unnatural procedure generally. In such situations but can be present here to be discussed approximations of the One, then form a substitute for the missing one element.

After examples of Banach algebras without identity approximations of the one to be defined. Finally, the one specified for the examples mentioned approximations.

Examples of Banach algebras without identity

  • Let be a locally compact Hausdorff space. The C * - algebra of continuous functions which vanish at infinity, only has a unit element, if is compact. In this case, the constant function is one, the unit element. The C * - algebra has no unit element.
  • Let be a locally compact group. Then the convolution algebra has L1 ( G ) if a unit element if is discrete. In this case, for all, the unity element (the neutral element of the group). The examined as part of the Fourier transform algebra has no unit element.
  • The C * - algebra of compact operators, trace class and the Hilbert-Schmidt class on a Hilbert space if and only have a unit element, if the dimension of is finite. In this case, the identity map is the identity element. In the important cases for applications or are no identity elements.
  • The sequence spaces, are the component-wise multiplication Banach algebras without identity.

Definitions

A left - approximation of one ( or right - approximation of one) of a Banach algebra is a network with (respectively) for all.

A ( two-sided ) approximation of the One is a network that is left - and right - Approximation of the One at a time.

Characteristics of the network, such as countable or narrowness, are also attributed to the approximations of the one.

If an identity, then the one-element mesh is an approximation of one. Banach algebras with approximation of the one thus generalize Banach algebras with identity.

Limited approximations of One

Has a limited left - approximation of the one and a limited right - Approximation of the One, then by a simple calculation shows that a two-sided bounded approximation of unity.

A Banach space which is a left - module is called a Banach - Links module when there is one constant with all and. An important special case is the Banach algebras product as a module operation.

If a Banach - Links module, and has a limited approximation of the one with all, so you can each about factorize, ie there is one and one with, in formulas.

The special case deserves special mention: If a Banach algebra with limited approximation of the one so true, more precisely, each element of can be written as a product of two elements.

Examples

Zero multiplication

A Banach space is different from 0 at a Banach algebra, if one explains the product of two elements as 0. Such a Banach algebra can not contain approximation of one.

C *-algebras

  • Every C *-algebra has a bounded by 1 approximation of one.

With the help of the continuous functional calculus, one can show that with respect to the order ( see Positive Operator) on the set of self-adjoint elements of an upward quantity and therefore represents a network itself. This network is an approximation of one.

In many cases, but you can ( in the separable case even episodes) specify simpler networks. In the above example was

.

Then the result is an approximation of the one in.

Group algebras

  • This is a locally compact group, has a limited approximation of the one by one.

Be a Left - Hair measure on. Is a basis of neighborhoods of the neutral element of, then there exists for every a continuous function with compact, located in carrier, for all and. As is addressed as an environment base by inclusion, is a network, from which one can show that it is an approximation of the one for.

In the special case with the Lebesgue measure as a hair - measure you can take as a basis of neighborhoods of the sequence of quantities. Is obtained as follows

So is the result of an approximation of the one for. Can be found also infinitely differentiable functions, form an approximation of the one which plays a role in the theory of Fourier transformation and the theory distribution ( approximation of the delta function ).

Operator algebras

It should be the directed set of finite-dimensional subspaces of an infinite-dimensional Hilbert space, the orthogonal projection is on. Then, an approximation of the one for the C * - algebra of compact operators on even a limited approximation of the One, because orthogonal projections have the operator norm 1

This network is also an approximation of the one in the shadow classes, ie in particular in the trace class and the Hilbert-Schmidt class, but not limited, as is true for the trace norm applies to the Hilbert-Schmidt norm, generally applies to the norm of the shadow class. It can be shown that there is no limited to the one in the shadow approximations classes. For the Hilbert-Schmidt class that follows from the above theorem on Banach left modules, because.

Swell

  • FF Bonsall, J. Duncan: Complete Normed Algebras. Springer -Verlag 1973, ISBN 3,540,063,862th
  • J. Dixmier: Les C * - algebres et leurs représentations, Gauthier -Villars, 1969, ISBN 9782876470132. .
  • R.V. Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras. 1983, ISBN 0,123,933,013th
  • Gert K. Pedersen: C * - Algebras and Their automorphism groups. ISBN 0,125,494,505th
  • Functional Analysis
Mathematics C*-Algebra Hilbert–Schmidt operator Operatornorm Schatten class operator Richard Kadison
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