Dual C * algebra

The dual C * -algebras , also called C * -algebras of compact operators , are a special subclass of C * -algebras considered in mathematics . They are characterized by a particularly simple structure.

definition

If it is a subset of an algebra , then the left annulator of is called . Correspondingly, the right canceler is called by . In general, a Banach algebra is called dual if the following duality relationships exist: ${\displaystyle M\subset A}$ ${\displaystyle A}$ ${\displaystyle {\mbox{lan}}(M):=\{x\in A;\,xM=\{0\}\}}$ ${\displaystyle M}$ ${\displaystyle {\mbox{ran}}(M):=\{x\in A;\,Mx=\{0\}\}}$ ${\displaystyle M}$

• ${\displaystyle {\mbox{lan}}({\mbox{ran}})(I)\,=\,I}$for all completed link ideals , ${\displaystyle I\subset A}$
• ${\displaystyle {\mbox{ran}}({\mbox{lan}})(I)\,=\,I}$for all completed legal ideals . ${\displaystyle I\subset A}$

In C * algebras, each of the conditions follows from the other, since left and right ideals correspond one- to -one via involution .

Characterizations

A C * -algebra is called elementary if there is a Hilbert space such that it is isomorphic to the algebra of the compact operators on . The restricted product of a family of C * -algebras is the subalgebra of the Cartesian product of which consists of all tuples for which for each is finite. Together with the norm , this is again a C * algebra. With these conceptualizations the following applies: ${\displaystyle H}$ ${\displaystyle K(H)}$ ${\displaystyle H}$ ${\displaystyle (A_{i})_{i}}$ ${\displaystyle A_{i}}$ ${\displaystyle (x_{i})_{i}}$ ${\displaystyle \{i;\,\|x_{i}\|>\epsilon \}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \|(x_{i})_{i}\|:=\sup _{i}\|x_{i}\|}$

For a C * -algebra the following statements are equivalent: ${\displaystyle A}$

• ${\displaystyle A}$ is a dual C * algebra.
• The sum of the minimum left ideals is close to . ${\displaystyle A}$
• The sum of the minimum legal ideals is close to . ${\displaystyle A}$
• ${\displaystyle A}$ is isomorphic to a sub-C * -algebra of an elementary C * -algebra.
• ${\displaystyle A}$ is isomorphic to a restricted product of a family of elementary C * algebras.
• The Gelfand spectrum of any maximal commutative sub-C * algebra is discrete .
• For each , the left multiplication operator is a weakly compact operator . ${\displaystyle x\in A}$ ${\displaystyle L_{x}:\,A\rightarrow A,\,y\mapsto xy}$
• For each , the right multiplication operator is a weakly compact operator. ${\displaystyle x\in A}$ ${\displaystyle R_{x}:\,A\rightarrow A,\,y\mapsto yx}$

An operator is called weakly compact if the image of a restricted set in the weak topology has a compact closure .

Because of this characterization, dual C * -algebras are also called C * -algebras of compact operators .

Examples

• The matrix algebras are elementary and therefore dual, more generally all finite-dimensional C * algebras are dual. ${\displaystyle M_{n}(\mathbb {C} )}$ ${\displaystyle ={\mathbb {C} }^{n\times n}}$
• The sequence algebra of complex zero sequences is a restricted product of countably many copies of and is therefore dual. ${\displaystyle c_{0}}$ ${\displaystyle \mathbb {C} \cong M_{1}(\mathbb {C} )}$
• Is a Hilbert space and is a sub-C * -algebra of , then is dual. According to the above characterization, all dual C * algebras are obtained except for isomorphism. ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle K(H)}$ ${\displaystyle A}$
• The function algebra is not dual, because it is commutative and has no discrete Gelfand spectrum. For the same reason the sequence algebras and the convergent or bounded sequences are not dual. ${\displaystyle C([0,1])}$ ${\displaystyle c}$ ${\displaystyle \ell ^{\infty }}$

properties

• From the above characterizations it is easy to see that sub-C * -algebras of dual C * -algebras and restricted products of dual C * -algebras are again dual.

• Dual C * algebras are liminal .

• The representation theory of dual C * algebras is very simple. If the C * -algebra is available as a restricted product of elementary C * -algebras , then the irreducible representations are exactly the projections onto the components except for equivalence . ${\displaystyle K(H_{i})}$ ${\displaystyle K(H_{i})}$

swell

• W. Arveson : Invitation to C * -algebras , ISBN 0387901760
• J. Dixmier : Les C * -algèbres et leurs représentations , Gauthier-Villars, 1969