AF-C * algebra

AF-C * -algebras , or AF-algebras for short , form a class of C * -algebras considered in the mathematical sub-area of functional analysis , which can be built up from finite-dimensional C * -algebras, AF stands for approximately finite (almost finite). These C * -algebras can be related to certain groups using K-theory and thus fully described.

definition

An AF-algebra is a C * -algebra for which there is a sequence of finite-dimensional C * -algebras such that ${\displaystyle A}$ ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$

• ${\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \ldots \subset A}$,
• ${\displaystyle \bigcup _{n\in \mathbb {N} }A_{n}}$lies close in . ${\displaystyle A}$

Examples

• Finite-dimensional C * -algebras are AF-algebras.
• The C * -algebra of compact operators on the Hilbert space is an AF-algebra. If the canonical basis of is , then be the subalgebra of the linear operators that map the linear envelope of and vanish on the orthogonal complement of it. They are obviously isomorphic to matrix algebra and meet the above definition. ${\displaystyle K(\ell ^{2})}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle (e_{n})_{n}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle A_{n}}$ ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ ${\displaystyle A_{n}}$ ${\displaystyle M_{n}}$
• Be the Cantor set . Then the commutative C * -algebra of continuous functions is an AF-algebra. Let be the subalgebra of the constant functions. Then they form a sequence of -dimensional C * -algebras and meet the above definition. ${\displaystyle X}$ ${\displaystyle A=C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle A_{n}}$ ${\displaystyle ({\tfrac {m}{3^{n}}},{\tfrac {m+1}{3^{n}}})\cap X}$ ${\displaystyle A_{n}}$ ${\displaystyle 2^{n}}$
• The C * -algebra of the continuous functions on is not an AF-algebra, because the zero-algebra and the algebra of the constant functions are the only finite-dimensional subalgebras. ${\displaystyle C([0,1])}$ ${\displaystyle [0,1]}$
• UHF algebras are AF algebras.

properties

• AF algebras are separable .
• AF algebras are nuclear .
• A separable C * -algebra is an AF-algebra if and only if there is a finite-dimensional sub-C * -algebra for a finite number of elements and each , so that for all . ${\displaystyle A}$ ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle B}$ ${\displaystyle \operatorname {dist} (a_{j},B)<\varepsilon }$ ${\displaystyle j=1,\ldots ,n}$
• The AF algebras are precisely the countable inductive limits of finite dimensional C * algebras in the category of C * algebras.
• Completed , two-sided ideals and quotients of AF algebras are again AF algebras. Sub-C * algebras of AF algebras are generally not AF algebras, for example the irrational rotation algebras are contained in AF algebras.
• Countable inductive limits of AF algebras are again AF algebras.
• Tensor products of AF algebras are again AF algebras.
• Is an AF algebra and goes by adjunction of the element made out, it is also an AF-algebra. ${\displaystyle A}$ ${\displaystyle A_{1}}$ ${\displaystyle A}$ ${\displaystyle A_{1}}$
• A commutative C * -algebra , X compact Hausdorff space , is an AF-algebra if and only if it is totally disconnected (see example above = Cantor set). ${\displaystyle C(X)}$ ${\displaystyle X}$ ${\displaystyle X}$
• Every separable, commutative C * -algebra is isomorphic to the center of an AF-algebra. This theorem provides slightly more examples of sub-algebras that are not AF algebras. A separable, commutative C * -algebra with not totally disjointed is not an AF-algebra, but occurs as the center of one. ${\displaystyle C(X)}$ ${\displaystyle X}$

K ₀ group of an AF algebra

Dimension group

The -function assigns a scaled, ordered, Abelian group to every C * -algebra (more generally to every ring ) . The set of isomorphism classes of the finitely generated projective modules is more precise . The direct sum makes this set a commutative semigroup. is defined as the Grothendieck group of and is the image of in . Finally, one can show that each projection via a projective defined module; is the image of the set of projections from in and is called the scale. Instead of group, one also says dimension group . ${\displaystyle K_{0}}$ ${\displaystyle A}$ ${\displaystyle (K_{0}(A),K_{0}(A)^{+},\Sigma (A))}$ ${\displaystyle V(A)}$ ${\displaystyle A}$ ${\displaystyle K_{0}(A)}$ ${\displaystyle (V(A),\oplus )}$ ${\displaystyle K_{0}(A)^{+}}$ ${\displaystyle V(A)}$ ${\displaystyle K_{0}(A)}$ ${\displaystyle p\in A}$ ${\displaystyle Ap}$ ${\displaystyle A}$ ${\displaystyle \Sigma (A)}$ ${\displaystyle A}$ ${\displaystyle K_{0}(A)}$ ${\displaystyle K_{0}}$

Simple examples are or . ${\displaystyle K_{0}(M_{n})\cong (\mathbb {Z} ,\mathbb {N} _{0},[0,n])}$ ${\displaystyle K_{0}(K(\ell ^{2}))\cong (\mathbb {Z} ,\mathbb {N} _{0},\mathbb {N} _{0})}$

It can also be shown that a * homomorphism between C * algebras induces a homomorphism between the associated dimension groups, this is positive, i.e. maps the positive semigroups into one another, and scales, i.e. it maps the scales into one another. That is clear, because it induces a semigroup homomorphism , and for the scalability note that of course projections from are mapped onto those from . Overall, defines a functor from the category of C * -algebras to the category of scaled, ordered, Abelian groups. ${\displaystyle \varphi \colon A\rightarrow B}$ ${\displaystyle \varphi }$ ${\displaystyle V(A)\rightarrow V(B)}$ ${\displaystyle \varphi }$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle K_{0}}$

Elliott's theorem

• Elliott's theorem : Two AF algebras are isomorphic if and only if the corresponding dimension groups are isomorphic as scaled, ordered, Abelian groups. Every group isomorphism between two dimensional groups is induced by an * isomorphism of the associated AF algebras.

You can also put it succinctly like this:

• For AF algebras, the assigned dimension group is a complete isomorphism invariant.

Isomorphic invariant means that the dimensional groups of isomorphic AF algebras are isomorphic. This is clear because of the functional properties described above and even applies to all C * algebras. Completeness of the isomorphic invariant means that non-isomorphic AF algebras can be distinguished by their dimension groups, that is, that the associated dimension groups are not isomorphic either. That's the difficult part of Elliot's theorem.

Effros-Handelman-Shen's theorem

Since an AF algebra is determined by its dimension group, except for isomorphism, the question naturally arises as to which groups can appear as dimension groups of AF algebras. This question is fully answered by the

• Effros-Handelman-Shen's theorem : The countable, imperforate, scaled, ordered, Abelian groups with Riesz's interpolation property are precisely the dimensional groups of AF algebras.

For the terms of order theory appearing here, consult the article on ordered Abelian groups .

meaning

With the above theorems of Elliot and Effros-Handelman-Shen, the study of AF algebras can be reduced to the study of countable, imperforate, scaled, ordered, Abelian groups with Riesz's interpolation property.

So one can show that the closed, two-sided ideals of an AF algebra correspond in a one-to-one way to the order ideals of the dimension group, i.e. those subgroups with , where , and the property that follows from . ${\displaystyle A}$ ${\displaystyle H\subset K_{0}(A)}$ ${\displaystyle H=H^{+}-H^{+}}$ ${\displaystyle H^{+}:=H\cap K_{0}(A)^{+}}$ ${\displaystyle x\in H}$ ${\displaystyle 0\leq x\leq h\in H}$

So one can construct simple AF algebras, i.e. those without real two-sided ideals different from one another, by finding countable, imperforate, scaled, ordered groups with Riesz's interpolation property that have no real order ideals. The dense subgroups of , which include with as a scale, are examples (UHF algebras), but also groups such as with and the scale defined by the unit of order . ${\displaystyle \{0\}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle [0,1]}$ ${\displaystyle G=\mathbb {Q} ^{n}}$ ${\displaystyle G^{+}=\{0\}\cup \mathbb {Q} _{>0}^{n}}$ ${\displaystyle (1,\ldots ,1)}$

Bratteli diagrams

Another important tool for studying AF algebras are Bratteli diagrams , certain infinite, directed graphs that represent the structure of the algebra.

Individual evidence

1. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-8218-0599-1 , Example III.2.5
2. ↑ Ola Bratteli : The Center of Approximately Finite-Dimensional C * -Algebras , Journal of Functional Analysis 21 (1976), pages 195-202
3. ^ B. Blackadar: K-Theory for Operator-Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , chapters 5, 6
4. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-8218-0599-1 , Theorem IV.4.3
5. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-8218-0599-1 , IV.7.3
6. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-8218-0599-1 , IV.5.1
7. ^ B. Blackadar: K-Theory for Operator-Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , 7.5, 7.6

swell

• B. Blackadar: K-Theory for Operator-Algebras , Springer Verlag (1986), ISBN 3-540-96391-X
• KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-8218-0599-1
• EG Effros, DE Handelman, CL Shen: Dimension groups and their affine transformations , Amer. J. Math. (1980) Vol. 102, pp. 385-402.
• KR Goodearl: Notes on real and complex C * -algebras , Shiva Publishing Limited (1982), ISBN 0-906812-16-X