abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, a unital map on a

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...

is a map

\phi

which preserves the identity element: :

\phi ( I ) = I.

This condition appears often in the context of

completely positive map In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition. Definition Let A and B be C*-algebras. A linear ...

s, especially when they represent

quantum operation In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed ...

s. If

\phi

is completely positive, it can always be represented as :

\phi ( \rho ) = \sum_i E_i \rho E_i^\dagger.

(The

E_i

are the

Kraus operator In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed ...

s associated with

\phi

). In this case, the unital condition can be expressed as :

\sum_i E_i E_i ^\dagger= I.

References

* C*-algebras {{Mathanalysis-stub