Bratteli diagram

Bratteli diagrams , named after Ola Bratteli , are special graphs used in the mathematical branch of functional analysis . They are used to study the structure of AF-C * algebras (AF algebras for short).

definition

The Bratteli diagrams are derived from the definition of the AF algebras; the latter are the completions of ascending sequences of finite-dimensional C * algebras . Each finite C * -algebra is isometrically isomorphic to a finite direct sum of complete matrix algebra over , that is, for every algebra applies ${\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \ldots \subset A}$ ${\displaystyle M_{n}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A_{j}}$

${\displaystyle A_{j}\cong M_{n_{j,1}}\oplus M_{n_{j,2}}\oplus \ldots \oplus M_{n_{j,m_{j}}}}$.

Except for the order, the numbers are clearly defined. These numbers form the points of the column-wise Bratteli diagram; The -th column contains exactly the numbers , i.e. the number is in the -th position . ${\displaystyle n_{j,1},\ldots ,n_{j,m_{j}}}$ ${\displaystyle j}$ ${\displaystyle n_{j,1},\ldots ,n_{j,m_{j}}}$ ${\displaystyle (i,j)}$ ${\displaystyle n_{i,j}}$

Between the points of the -th and -th column arrows are drawn according to the following criteria: The embeddings are as injective * -Homomorphismen until unitary equivalence already determined by at which multiplicity of th summand of the -th summands of being imaged. ${\displaystyle j}$ ${\displaystyle (j+1)}$ ${\displaystyle A_{j}\subset A_{j+1}}$ ${\displaystyle i}$ ${\displaystyle M_{n_{j,i}}}$ ${\displaystyle A_{j}}$ ${\displaystyle k}$ ${\displaystyle M_{n_{j+1,k}}}$ ${\displaystyle A_{j+1}}$

Example: embedding

${\displaystyle M_{2}\rightarrow M_{7}\oplus M_{3}\oplus M_{2},{\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}}\mapsto {\begin{pmatrix}x_{11}&x_{12}&0&0&0&0&0\\x_{21}&x_{22}&0&0&0&0&0\\0&0&x_{11}&x_{12}&0&0&0\\0&0&x_{21}&x_{22}&0&0&0\\0&0&0&0&x_{11}&x_{12}&0\\0&0&0&0&x_{21}&x_{22}&0\\0&0&0&0&0&0&0\end{pmatrix}}\oplus {\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\\\end{pmatrix}}\oplus {\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}}}$

has the multiples 3, 0 and 1. Now draw -arrows from the -th node to the -th node if the -th summand of is mapped with the multiplicity in the -th summand of . The numbers depend on and are subject to the restriction that arises from the fact that the summands in the -th column must be large enough to be able to accommodate the corresponding matrices of the -th column with the multiplicities. According to Bratteli’s theorem, every AF algebra, except for isomorphism, can be constructed by a sequence of finite direct sums full of matrix algebras with the specified special embeddings. ${\displaystyle p}$ ${\displaystyle (i,j)}$ ${\displaystyle (k,j+1)}$ ${\displaystyle i}$ ${\displaystyle A_{j}}$ ${\displaystyle p}$ ${\displaystyle k}$ ${\displaystyle A_{j+1}}$ ${\displaystyle p=p(i,j,k)}$ ${\displaystyle i,j,k}$ ${\displaystyle \textstyle \sum _{i=1}^{m_{j}}p(i,j,k)n_{j,i}\leq n_{j+1,k}}$ ${\displaystyle (j+1)}$ ${\displaystyle j}$

Examples

Compact operators

As is known, the ascending inclusions , with each matrix from being mapped onto the matrix from expanded by a zero row and a zero column , define an AF algebra that is isomorphic to the C * algebra of the compact operators on the Hilbert space . The corresponding Bratteli diagram has the form above ${\displaystyle M_{1}\subset M_{2}\subset M_{3}\subset \ldots }$ ${\displaystyle M_{j}}$ ${\displaystyle M_{j+1}}$ ${\displaystyle \ell ^{2}}$

${\displaystyle 1\rightarrow 2\rightarrow 3\rightarrow 4\rightarrow 5\rightarrow \ldots }$

Compact operators with one element

Adjoint you to the example above, the compact operators an identity, then comes to each one direct summand added and the embedding we see this: ${\displaystyle M_{j}}$ ${\displaystyle M_{1}\cong \mathbb {C} }$ ${\displaystyle \mathbb {C} \oplus M_{j}\rightarrow \mathbb {C} \oplus M_{j+1}}$

${\displaystyle x\oplus {\begin{pmatrix}x_{11}&\ldots &x_{1j}\\\vdots &\ddots &\vdots \\x_{j1}&\ldots &x_{jj}\\\end{pmatrix}}\mapsto x\oplus {\begin{pmatrix}x_{11}&\ldots &x_{1j}&0\\\vdots &\ddots &\vdots \vdots \\x_{j1}&\ldots &x_{jj}&0\\0&\ldots &0&x\end{pmatrix}}}$

This leads to the following Bratteli diagram:

${\displaystyle {\begin{array}{cccccccccc}1&\rightarrow &1&\rightarrow &1&\rightarrow &1&\rightarrow &1&\ldots \\&\searrow &&\searrow &&\searrow &&\searrow &&\\1&\rightarrow &2&\rightarrow &3&\rightarrow &4&\rightarrow &5&\ldots \\\end{array}}}$

Cantor set

Consider the C * -algebra of the continuous functions on the Cantor set . An increasing sequence of finite-dimensional partial algebras is obtained by embedding the finite-dimensional algebra of the functions that are constant on the intervals of length via restriction into the algebra of the functions that are constant on the intervals of length . This leads to the following Bratteli diagram: ${\displaystyle C(X)}$ ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A_{n}}$ ${\displaystyle {\tfrac {1}{3^{n}}}}$ ${\displaystyle A_{n+1}}$ ${\displaystyle {\tfrac {1}{3^{n+1}}}}$

On the left the first four columns of the Bratteli diagram. is the union of the corresponding intervals of length , is isomorphic to the algebra of the locally constant functions on ${\displaystyle X_{n}}$ ${\displaystyle 1/{3^{n}}}$ ${\displaystyle A_{n}}$ ${\displaystyle X_{n}}$

CAR algebra

The CAR algebra is obtained through inclusions

${\displaystyle M_{1}\subset M_{2}\subset M_{4}\subset M_{8}\subset \ldots \subset M_{2^{j}}\subset M_{2^{j+1}}\subset \ldots }$,

where the embedding is defined by. Here all the multiples are equal to 2 and you get the following Bratteli diagram: ${\displaystyle M_{2^{j}}\subset M_{2^{j+1}}}$ ${\displaystyle X\mapsto {\begin{pmatrix}X&0\\0&X\\\end{pmatrix}}}$

${\displaystyle 1\rightrightarrows 2\rightrightarrows 4\rightrightarrows 8\rightrightarrows \ldots \rightrightarrows 2^{j}\rightrightarrows 2^{j+1}\rightrightarrows \ldots }$

.

Applications

The Bratteli diagrams for an AF algebra are not clearly defined, because they depend on the concrete implementation as the completion of an ascending chain of finite-dimensional C * -algebras, and this is not unambiguous because, for example, one can omit an initial section or several on top of one another combine the following inclusions into one. However, apart from isomorphism, there is only one AF algebra for every Bratteli diagram and properties of this algebra can be read from such a diagram. It explains how to read information about the ideal structure and how to determine whether the AF algebra is liminal or postliminal .

Ideal structure

If there is a closed two-sided ideal in the AF-algebra given by, then it is also an AF-algebra and is an increasing sequence of finite-dimensional partial C * -algebras with a close union. It is chosen so large that . In this way, a subgraph of the Bratteli diagram of is assigned to each completed two-sided ideal . The empty subgraph corresponds to the zero ideal. ${\displaystyle I\subset A}$ ${\displaystyle A_{1}\subset A_{2}\subset \ldots }$ ${\displaystyle I}$ ${\displaystyle I\cap A_{p}\subset I\cap A_{p+1}\subset \ldots }$ ${\displaystyle I}$ ${\displaystyle p}$ ${\displaystyle I\cap A_{p}\not =\{0\}}$ ${\displaystyle D_{I}}$ ${\displaystyle D_{A}}$ ${\displaystyle A}$

A subgraph of a Bratteli diagram is called directed if it contains all arrows starting from it with the associated target points for each point. ${\displaystyle S}$

A sub-graph is called hereditary (Engl. Hereditary ), if: Lying to be included in the lower graph for a point to all destination points emanating from his arrows in the lower graph, has already this point. The following sentence now applies:

• If an AF algebra with a Bratteli diagram , then the above assignment is a bijection from the set of closed two-sided ideals onto the set of directed, hereditary subgraphs of . ${\displaystyle A}$ ${\displaystyle D_{A}}$ ${\displaystyle I\mapsto D_{I}}$ ${\displaystyle D_{A}}$

A C * -algebra is called simple if it contains no closed two-sided ideals apart from the zero ideal and itself. From the above sentence one can easily derive the following corollary:

• An AF algebra with Bratteli diagram is just simply when it comes to the point of is a column, so you each point of this column can be made by way of arrows reach. ${\displaystyle A}$ ${\displaystyle D_{A}}$ ${\displaystyle x}$ ${\displaystyle D_{A}}$ ${\displaystyle x}$

In particular, the C * -algebras of the compact operators and the CAR-algebra are simple, because the associated Bratteli diagrams are linear chains. If you adjoint a unit element to the algebra of the compact operators, the resulting algebra is not easy, because on the Bratteli diagram you can easily see that no 1 in the upper row can be reached from any point in the lower row in a subsequent column. Obviously the lower line is a directed and hereditary subgraph, it corresponds to the ideal of compact operators.

Liminal and Post-Liminal AF Algebras

One can read from the Bratteli diagram of an AF algebra whether it is liminal or postliminal . To do this, one considers infinite paths in the Bratelli diagram, that is, sequences of points in the diagram so that at least one arrow leads from to for each . If and are two points, it is said to be the successor of with multiplicity if there are different paths from to . ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle n}$ ${\displaystyle x_{n}}$ ${\displaystyle x_{n+1}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle q}$ ${\displaystyle q}$ ${\displaystyle x}$ ${\displaystyle y}$

• Let be an AF algebra with a Bratteli diagram . is liminal if and only if there is an infinite path in natural numbers and to every infinite path such that is for all successors of with multiplicity . ${\displaystyle A}$ ${\displaystyle D_{A}}$ ${\displaystyle A}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle D_{A}}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle x_{n}}$ ${\displaystyle n\geq p}$ ${\displaystyle x_{1}}$ ${\displaystyle q}$

Accordingly, the above examples " compact operators " and " Cantor set " are liminal, because the Bratteli diagrams are trees with simple edges, which means that only the multiplicity 1 can occur anyway. The example " compact operators with one element " is not liminal, since there are more and more possible paths from 1 to for the path consisting of the 1 in the upper line and all points in the lower line , that is, the multiplicity cannot be from one certain place can be limited by a fixed one. ${\displaystyle 2,3,4,\ldots ,n,\ldots }$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle q}$

• Let be an AF algebra with a Bratteli diagram . is postliminal if and only if there is an infinite path into a natural number for each such that there is a successor of with multiplicity 1 for each . ${\displaystyle A}$ ${\displaystyle D_{A}}$ ${\displaystyle A}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle D_{A}}$ ${\displaystyle p}$ ${\displaystyle x_{n}}$ ${\displaystyle n>p}$ ${\displaystyle x_{n-1}}$

It is easy to see that the Bratteli diagram of the example " Compact operators with one element " has this property, so it is a post-minimal C * -algebra. The CAR-algebra does not have this property, because all occurring multiplicities between direct successors are equal to 2, the CAR-algebra is therefore not postliminal.

Individual evidence

1. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Chapter III
2. ↑ KR Goodearl: Notes on real and complex C * -algebras , Shiva Publishing Limited (1982), ISBN 0-906-81216-X , set 2/17
3. ^ AJ Lazar, DC Taylor: Approximately Finite Dimensional C * -Algebras and Bratteli Diagrams , Transactions of the American Mathematical Society, Volume 259 (1980), pp. 599-619, Theorem 3.8
4. ^ AJ Lazar, DC Taylor: Approximately Finite Dimensional C * -Algebras and Bratteli Diagrams , Transactions of the American Mathematical Society, Volume 259 (1980), pages 599-619, Theorem 3.13