Postliminal C * algebra

Postliminal C * -algebras form a class of C * -algebras considered in mathematics . Alternative names that will be motivated below are GCR algebra or Type IC * algebra . It is a generalization of the class of liminal C * algebras .

definition

A C * -algebra is called postliminal if for every true, closed, two-sided ideal the quotient algebra contains a different liminal ideal. ${\displaystyle A}$ ${\displaystyle I\subset A}$ ${\displaystyle A/I}$ ${\displaystyle \{0\}}$

The concept of post-liminal C * -algebra is thus traced back to that of liminal C * -algebra and apparently represents a generalization. This is also clear from the first of the following characterizations.

Characterizations

Images of irreducible representations

If an irreducible representation of the C * -algebra is on the Hilbert space , then by definition it contains a different liminal ideal. One can show that an irreducible representation of this ideal defines. Since is liminal, the image coincides with the algebra of compact operators and it follows . The picture of every irreducible representation therefore includes the compact operators and the converse is true: ${\displaystyle \pi }$ ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle A/{\mbox{ker}}(\pi )}$ ${\displaystyle \{0\}}$ ${\displaystyle {\tilde {\pi }}(a+{\mbox{ker}}(\pi )):=\pi (a)}$ ${\displaystyle {\tilde {\pi }}}$ ${\displaystyle I}$ ${\displaystyle {\tilde {\pi }}(I)}$ ${\displaystyle K(H)}$ ${\displaystyle \pi (A)\supset K(H)}$

• A C * -algebra is postliminal if and only if for every irreducible representation of . ${\displaystyle A}$ ${\displaystyle \pi (A)\supset K(H)}$ ${\displaystyle \pi :A\rightarrow L(H)}$ ${\displaystyle A}$

For liminal C * -algebras one has an almost identical characterization, the inclusion is simply replaced by an equality (see article liminal C * -algebra ). Since liminal C * algebras are also called CCR algebras (CCR = completely continuous representations) because of this relationship to the compact operators, postliminal C * algebras are also called GCR algebras (GCR = generalized completely continuous representations) for the same reason .

Composition series

A composition series of a C * -algebra is a family of closed, two-sided ideals , where ${\displaystyle A}$ ${\displaystyle (I_{\beta })_{0\leq \beta \leq \alpha }}$ ${\displaystyle I_{\beta }\subset A}$

1. ${\displaystyle \alpha }$is an ordinal number (it runs through all ordinal numbers up to and including.) ${\displaystyle \beta }$ ${\displaystyle \alpha }$
2. ${\displaystyle I_{0}\,=\,\{0\}}$and . ${\displaystyle I_{\alpha }\,=\,A}$
3. for true ${\displaystyle 0\leq \beta \leq \gamma \leq \alpha }$ ${\displaystyle I_{\beta }\subset I_{\gamma }}$
4. If a limit number is the conclusion of . ${\displaystyle \gamma \in [0,\alpha ]}$ ${\displaystyle I_{\gamma }}$ ${\displaystyle \bigcup _{0\leq \beta <\gamma }I_{\beta }}$

With this concept formation one can prove the following characterization:

• A C * -algebra is postliminal if and only if there is a composition series of such that all quotients are liminal. ${\displaystyle A}$ ${\displaystyle (I_{\beta })_{0\leq \beta \leq \alpha }}$ ${\displaystyle A}$ ${\displaystyle I_{\beta +1}/I_{\beta }}$

In the case of separable C * algebras, the term ordinal number can be avoided here, because if the ideals of the composition series are all different (which is not a further restriction), then is maximal and the composition series can be indexed with natural numbers. ${\displaystyle \alpha }$ ${\displaystyle \omega =\mathbb {N} }$

Type I.

A representation of a C * algebra is of type I, if the image from the generated Von Neumann algebra from the type I , the is, when the Bikommutant a Type I Von Neumann algebra. ${\displaystyle \pi :A\rightarrow L(H)}$ ${\displaystyle A}$ ${\displaystyle \pi (A)}$ ${\displaystyle \pi (A)''\subset L(H)}$

• A C * -algebra is postliminal if and only if every representation is of type I. ${\displaystyle A}$

This is why postliminal C * algebras are also called type IC * algebras. This designation can give rise to confusion, because a type I Von Neumann algebra, which is also a C * algebra, is generally not a type IC * algebra, as the example with an infinite-dimensional Hilbert space shows. ${\displaystyle A=L(H)}$ ${\displaystyle H}$

spectrum

If an equivalence class of irreducible representations of , i.e. an element of the spectrum , then the ideal depends only on the equivalence class and not on the concrete representation . Since the kernels of irreducible representations are by definition the primitive ideals, the kernel formation,, is a mapping from the spectrum into the space of primitive ideals. According to its construction , this is surjective , but generally not injective . ${\displaystyle [\pi ]}$ ${\displaystyle A}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle {\mbox{ker}}(\pi )}$ ${\displaystyle [\pi ]}$ ${\displaystyle \pi }$ ${\displaystyle [\pi ]\mapsto \ker(\pi )}$ ${\displaystyle {\hat {A}}\to {\mbox{Prim}}(A)}$

• If a postliminal C * -algebra, then the kernel map is injective. If it is separable, the reverse is true. ${\displaystyle A}$ ${\displaystyle {\hat {A}}\to {\mbox{Prim}}(A)}$ ${\displaystyle A}$

It is unclear whether this inversion also applies to non-separable C * algebras.

Examples

• Liminal C * algebras are postliminal.

• Let it be the C * algebra generated by the shift operator , the so-called Toeplitz algebra (after Otto Toeplitz ). Since the orthogonal projection onto the subspace generated by the basis vectors and is therefore a compact operator, one can show that . Furthermore , where the circular line is, it is generated by the remainder of the class , and this has the circular line as a spectrum . You even have an exact sequence ${\displaystyle T}$ ${\displaystyle s\in L(\ell ^{2})}$ ${\displaystyle 1-s^{n}s^{*n}}$ ${\displaystyle e_{1},\ldots e_{n}\in \ell ^{2}}$ ${\displaystyle K(H)\subset T}$ ${\displaystyle T/K(H)\cong C(S^{1})}$ ${\displaystyle S^{1}\subset \mathbb {C} }$ ${\displaystyle T/K(H)}$ ${\displaystyle s+K(H)}$

${\displaystyle \{0\}\rightarrow K(H)\rightarrow T\rightarrow C(S^{1})\rightarrow \{0\}}$

.

In any case, is given by a series of compositions , and the quotients and are liminal. Hence T is postliminal, but not liminal, because it is an irreducible representation that contains the non-compact operator in the image. ${\displaystyle I_{0}:=\{0\},I_{1}:=K(H),I_{2}:=T}$ ${\displaystyle T}$ ${\displaystyle T/K(H)\cong C(S^{1})}$ ${\displaystyle K(H)/\{0\}\cong K(H)}$ ${\displaystyle {\mbox{id}}_{T}:T\rightarrow L(\ell ^{2})}$ ${\displaystyle s}$

• ${\displaystyle L(\ell ^{2})}$is an example of a C * algebra that is not postliminal. The Calkin algebra is another example of a non-postliminal C * algebra.

properties

• A sub-C * -algebra of a postliminal C * -algebra is again postliminal.
• If there is a closed, two-sided ideal in postliminal C * -algebra , then is also postliminal. ${\displaystyle I\subset A}$ ${\displaystyle A}$ ${\displaystyle A/I}$
• If a closed, two-sided ideal in C * -algebra and are and are postliminal, then is also postliminal. ${\displaystyle I\subset A}$ ${\displaystyle A}$ ${\displaystyle I}$ ${\displaystyle A/I}$ ${\displaystyle A}$
• Postliminal C * algebras are nuclear .
• If postliminal, then has a composition series such that all quotients are C * -algebras with a continuous trace . This intensifies the characterization given above by means of series of compositions. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle (I_{\beta })_{0\leq \beta \leq \alpha }}$ ${\displaystyle I_{\beta +1}/I_{\beta }}$
• A postliminal C * -algebra is liminal if and only if every point in is closed with respect to the Zariski topology , that is, if the spectrum is a T ₁ -space . ${\displaystyle A}$ ${\displaystyle {\hat {A}}\cong {\mbox{Prim}}(A)}$ ${\displaystyle {\hat {A}}}$

swell

• W. Arveson : Invitation to C * -algebras , Springer-Verlag (1976), ISBN 0387901760
• J. Dixmier : Les C * -algèbres et leurs représentations , Gauthier-Villars, 1969