Liminal C * algebra

Liminal C * algebras are a class of C * algebras considered in mathematics . These are the "building blocks" from which the postliminal or type I C * algebras are built.

The liminal C * algebras are also called CCR algebras (CCR stands for completely continuous representations) by some authors , under this name they were introduced by Irving Kaplansky in 1951 . However, there is then a name conflict in the quantum field theory considered algebras (CCR is there for canonical commutation relations, that is canonical commutation ). We are following the name going back to Jacques Dixmier ( French : liminaire, English : liminal).

definition

A C * -algebra is called liminal if the images of irreducible representations consist of compact operators .

Examples

• Dual C * algebras are liminal.
• Commutative C * algebras are liminal, because every irreducible representation is one-dimensional. The commutative C * algebra of continuous functions is liminal, but not dual. ${\displaystyle C[0,1]}$ ${\displaystyle [0,1]\rightarrow \mathbb {C} }$
• If locally compact , then is liminal, because every irreducible representation has the form for one except for equivalence . ${\displaystyle X}$ ${\displaystyle C_{0}(X,M_{n}(\mathbb {C} )):=\{a:X\rightarrow M_{n}(\mathbb {C} );\,a{\mbox{ ist stetig und }}\lim _{x\to \infty }\|a(x)\|=0\}}$ ${\displaystyle \pi _{x}(a):=a(x)\in M_{n}(\mathbb {C} )=L(\mathbb {C} ^{n})}$ ${\displaystyle x\in X}$
• Let it be an infinitely dimensional Hilbert space. Then it is not liminal, because it is irreducible and has non-compact operators in the image. ${\displaystyle H}$ ${\displaystyle A=L(H)}$ ${\displaystyle {\mbox{id}}_{A}:A\rightarrow L(H)}$

The greatest liminal ideal

If there is a C * -algebra, then ${\displaystyle A}$

${\displaystyle I:=\{a\in A;\pi (a)\,{\mbox{ ist kompakt für jede irreduzible Darstellung von}}\,A\}}$

a closed , two-sided ideal that is liminal and contains every other liminal ideal, in short the greatest liminal ideal . Accordingly, a C * -algebra is liminal if and only if it coincides with its greatest liminal ideal. The quotient may well contain another liminal ideal; this observation leads to the important concept of postliminal C * algebra . ${\displaystyle A/I}$ ${\displaystyle \{0\}}$

properties

• Every sub-C * -algebra of a liminal C * -algebra is again liminal.
• If a liminal C * -algebra and a closed two-sided ideal is liminal again. ${\displaystyle A}$ ${\displaystyle I\subset A}$ ${\displaystyle A/I}$
• If a liminal C * -algebra and an irreducible representation, then we have . It is the algebra of compact operators on the definition only demanded the inclusion . ${\displaystyle A}$ ${\displaystyle \pi :A\rightarrow L(H)}$ ${\displaystyle \pi (A)=K(H)}$ ${\displaystyle K(H)}$ ${\displaystyle H}$ ${\displaystyle \pi (A)\subset K(H)}$

Antiliminal C * algebras

A C * algebra is said to be anti-criminal if the only liminal ideal is in the null ideal, that is, if the greatest liminal ideal is. The Calkin algebra is an example of an antiliminal C * algebra. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \{0\}}$

C * algebras with a continuous trace

For a C * -algebra, let the spectrum of , i.e. the set of all equivalence classes of irreducible representations of (see Hilbert space representation ). If and is positive , then a positive compact operator is on and the trace can be formed, whereby this number does not depend on but only on the equivalence class . Keep going ${\displaystyle A}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle A}$ ${\displaystyle [\pi ]}$ ${\displaystyle \pi }$ ${\displaystyle A}$ ${\displaystyle [\pi ]\in {\hat {A}}}$ ${\displaystyle a\in A}$ ${\displaystyle \pi (a)}$ ${\displaystyle H}$ ${\displaystyle Sp(\pi (a))\in [0,\infty ]}$ ${\displaystyle \pi }$ ${\displaystyle [\pi ]}$

${\displaystyle P:=\{a\in A;\,a\geq 0,\,[\pi ]\mapsto Sp(\pi (a)){\mbox{ ist eine stetige Funktion }}\,{\hat {A}}\rightarrow [0,\infty )\}}$.

Then the set of all for which holds is a two-sided ideal in . If this ideal is close to , it is said to be a C * -algebra with a continuous trace . The following sentence applies. ${\displaystyle a\in A}$ ${\displaystyle a^{*}a\in P}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$

• C * -algebras with a continuous trace are liminal, the spectrum of such a C * -algebra is a Hausdorff space .

The C * algebra mentioned above is an example of a C * algebra with a continuous trace. The sub-C * -algebra is not a C * -algebra with a continuous trace (for ), but as a sub- algebra liminal. ${\displaystyle C([0,1],M_{n}(\mathbb {C} ))}$ ${\displaystyle \{a\in C([0,1],M_{n}(\mathbb {C} ));\,a(0){\mbox{ ist eine Diagonalmatrix}}\}}$ ${\displaystyle n\geq 2}$

swell

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