Difference between revisions of "AF-algebra"
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− | It follows from (an analogue of) Wedderburn's theorem (cf. [[Wedderburn–Artin theorem|Wedderburn–Artin theorem]]), that every finite-dimensional$C^*$-algebra is isomorphic to the direct sum of full matrix algebras over the field of complex numbers. Property 2 says that each AF-algebra is the [[Inductive limit|inductive limit]] of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras, where the connecting mappings $A_j\rightarrow A_{j+1}$ are $*$-preserving homomorphisms. If two such sequences $A_1\rightarrow A_2\rightarrow\dots$ and $B_1\rightarrow B_2\rightarrow\dots$ define isomorphic AF-algebras, then already the algebraic inductive limits of the two sequences are isomorphic (as algebras over $\mathbb C$). | + | It follows from (an analogue of) Wedderburn's theorem (cf. [[Wedderburn–Artin theorem|Wedderburn–Artin theorem]]), that every finite-dimensional$C^*$-algebra is isomorphic to the direct sum of full matrix algebras over the field of complex numbers. Property 2 says that each AF-algebra is the [[Inductive limit|inductive limit]] of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras, where the connecting mappings $A_j\rightarrow A_{j+1}$ are ${}^*$-preserving homomorphisms. If two such sequences $A_1\rightarrow A_2\rightarrow\dots$ and $B_1\rightarrow B_2\rightarrow\dots$ define isomorphic AF-algebras, then already the algebraic inductive limits of the two sequences are isomorphic (as algebras over $\mathbb C$). |
All essential information of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras with connecting mappings can be expressed in a so-called Bratteli diagram. The Bratteli diagram is a graph, divided into rows, whose vertices in the $j$th row correspond to the direct summands of $A_j$ isomorphic to a full matrix algebra, and where the edges between the $j$th and the $(j+1)$st row describe the connecting mapping $A_j\rightarrow A_{j+1}$. By the facts mentioned above, the construction and also the classification of AF-algebras can be reduced to a purely combinatorial problem phrased in terms of Bratteli diagrams. (See [[#References|[a2]]].) | All essential information of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras with connecting mappings can be expressed in a so-called Bratteli diagram. The Bratteli diagram is a graph, divided into rows, whose vertices in the $j$th row correspond to the direct summands of $A_j$ isomorphic to a full matrix algebra, and where the edges between the $j$th and the $(j+1)$st row describe the connecting mapping $A_j\rightarrow A_{j+1}$. By the facts mentioned above, the construction and also the classification of AF-algebras can be reduced to a purely combinatorial problem phrased in terms of Bratteli diagrams. (See [[#References|[a2]]].) |
Revision as of 19:19, 19 April 2014
Approximately Finite-dimensional algebra.
AF-algebras form a class of $C^*$-algebras that, on the one hand, admits an elementary construction, yet, on the other hand, exhibits a rich structure and provide examples of exotic phenomena. A (separable) $C^*$-algebra $A$ is said to be an AF-algebra if one of the following two (not obviously) equivalent conditions is satisfied (see [a1], [a2] or [a6]):
- for every finite subset $\{a_1,\dots,a_n\}$ of $A$ and for every $\epsilon>0$ there exists a finite-dimensional sub-$C^*$-algebra $B$ of $A$ and a subset $\{b_1,\dots,b_n\}$ of $B$ with $\|a_j-b_j\|<\epsilon$ for all $j=1,\dots,n$;
- there exists an increasing sequence $A_1\subseteq A_2\subseteq\dots$ of finite-dimensional sub-$C^*$-algebras of $A$ such that the union $\bigcup_{j=1}^\infty A_j$ is norm-dense in $A$.
Bratteli diagrams.
It follows from (an analogue of) Wedderburn's theorem (cf. Wedderburn–Artin theorem), that every finite-dimensional$C^*$-algebra is isomorphic to the direct sum of full matrix algebras over the field of complex numbers. Property 2 says that each AF-algebra is the inductive limit of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras, where the connecting mappings $A_j\rightarrow A_{j+1}$ are ${}^*$-preserving homomorphisms. If two such sequences $A_1\rightarrow A_2\rightarrow\dots$ and $B_1\rightarrow B_2\rightarrow\dots$ define isomorphic AF-algebras, then already the algebraic inductive limits of the two sequences are isomorphic (as algebras over $\mathbb C$).
All essential information of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras with connecting mappings can be expressed in a so-called Bratteli diagram. The Bratteli diagram is a graph, divided into rows, whose vertices in the $j$th row correspond to the direct summands of $A_j$ isomorphic to a full matrix algebra, and where the edges between the $j$th and the $(j+1)$st row describe the connecting mapping $A_j\rightarrow A_{j+1}$. By the facts mentioned above, the construction and also the classification of AF-algebras can be reduced to a purely combinatorial problem phrased in terms of Bratteli diagrams. (See [a2].)
UHF-algebras.
AF-algebras that are inductive limits of single full matrix algebras with unit-preserving connecting mappings are called UHF-algebras (uniformly hyper-finite algebras) or Glimm algebras. A UHF-algebra is therefore an inductive limit of a sequence , where, necessarily, each
divides
. Setting
and
for
, this UHF-algebra can alternatively be described as the infinite tensor product
. (See [a1].)
The UHF-algebra with is called the CAR-algebra; it is generated by a family of operators
, where
is some separable infinite-dimensional Hilbert space and
is linear and satisfies the canonical anti-commutation relations (cf. also Commutation and anti-commutation relationships, representation of):
![]() |
![]() |
(See [a7].)
-theory and classification.
By the -theory for
-algebras, one can associate a triple
to each
-algebra
.
is the countable Abelian group of formal differences of equivalence classes of projections in matrix algebras over
, and
and
are the subsets of those elements in
that are represented by projections in some matrix algebra over
, respectively, by projections in
itself. The
-group of an AF-algebra is always zero.
The classification theorem for AF-algebras says that two AF-algebras and
are
-isomorphic if and only if the triples
and
are isomorphic, i.e., if and only if there exists a group isomorphism
such that
and
. If this is the case, then there exists an isomorphism
such that
. Moreover, any homomorphism
such that
is induced by a
-homomorphism
, and if
are two
-homomorphisms, then
if and only if
and
are homotopic (through a continuous path of
-homomorphisms from
to
).
An ordered Abelian group is said to have the Riesz interpolation property if whenever
with
, there exists a
such that
.
is called unperforated if
, for some integer
and some
, implies that
. The Effros–Handelman–Shen theorem says that a countable ordered Abelian group
is the
-theory of some AF-algebra if and only if it has the Riesz interpolation property and is unperforated. (See [a3], [a5], [a8], and [a6].)
A conjecture belonging to the Elliott classification program asserts that a -algebra is an AF-algebra if it looks like an AF-algebra! More precisely, suppose that
is a separable, nuclear
-algebra which has stable rank one and real rank zero, and suppose that
and that
is unperforated (
must necessarily have the Riesz interpolation property when
is assumed to be of real rank zero). Does it follow that
is an AF-algebra? This conjecture has been confirmed in some specific non-trivial cases. (See [a9].)
Traces and ideals.
The -theory of an AF-algebra not only serves as a classifying invariant, it also explicitly reveals some of the structure of the algebra, for example its traces and its ideal structure. Recall that a (positive) trace on a
-algebra
is a (positive) linear mapping
satisfying the trace property:
for all
. An "ideal" means a closed two-sided ideal.
A state on an ordered Abelian group
is a group homomorphism
satisfying
. An order ideal
of
is a subgroup of
with the property that
generates
, and if
,
, and
, then
. A trace
on
induces a state on
by
![]() |
where ,
are projections in
(or in a matrix algebra over
); and given an ideal
in
, the image
of the induced mapping
(which happens to be injective, when
is an AF-algebra) is an order ideal of
. For AF-algebras, the mappings
and
are bijections. In particular, if
is simple as an ordered group, then
must be simple.
If a -algebra
has a unit, then the set of tracial states (i.e., positive traces that take the value
on the unit) is a Choquet simplex. Using the characterizations above, one can, for each metrizable Choquet simplex, find a simple unital AF-algebra whose trace simplex is affinely homeomorphic to the given Choquet simplex. Hence, for example, simple unital
-algebras can have more than one trace. (See [a3] and [a5].)
Embeddings into AF-algebras.
One particularly interesting, and still not fully investigated, application of AF-algebras is to find for a -algebra
an AF-algebra
and an embedding
which induces an interesting (say injective) mapping
. Since
is positive, the positive cone
of
must be contained in the pre-image of
. For example, the order structure of the
-group of the irrational rotation
-algebra
was determined by embedding
into an AF-algebra
with
(as an ordered group). As a corollary to this, it was proved that
if and only if
or
. (See [a4].)
Along another interesting avenue there have been produced embeddings of into appropriate AF-algebras inducing injective
-theory mappings. This suggests that the "cohomological dimension" of these AF-algebras should be at least
.
References
[a1] | J. Glimm, "On a certain class of operator algebras" Trans. Amer. Math. Soc. , 95 (1960) pp. 318–340 MR0112057 Zbl 0094.09701 |
[a2] | O. Bratteli, "Inductive limits of finite-dimensional ![]() |
[a3] | G.A. Elliott, "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras" J. Algebra , 38 (1976) pp. 29–44 MR0397420 Zbl 0323.46063 |
[a4] | M. Pimsner, D. Voiculescu, "Imbedding the irrational rotation algebras into AF-algebras" J. Operator Th. , 4 (1980) pp. 201–210 MR595412 |
[a5] | E. Effros, D. Handelman, C.-L. Shen, "Dimension groups and their affine representations" Amer. J. Math. , 102 (1980) pp. 385–407 MR0564479 Zbl 0457.46047 |
[a6] | E. Effros, "Dimensions and ![]() |
[a7] | O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , II , Springer (1981) MR0611508 Zbl 0463.46052 |
[a8] | B. Blackadar, "![]() |
[a9] | G.A. Elliott, "The classification problem for amenable ![]() |
AF-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=AF-algebra&oldid=31884