On a class of $C^*$-algebras determined uniquely by dual space

0.18 MB·English

by  Yurii Savchuk

ON A CLASS OF C*-ALGEBRAS DETERMINED UNIQUELY BY
4
DUAL SPACE.
0
0
2
SAVCHUKYURI

v
o
N

9 Mathematical Subject Classification: 55R10, 46L89, 46L99.
1
Abstract. EnvelopingC∗-algebrasforsomefinitelygenerated∗-algebrasare
]
considered. It is shown that all of the considered algebras are identically
A
defined by their dual spaces. The description in terms of matrix-functions is
O given.
. Keywords: algebraicbundles,finitelypresented C∗-algebras,dualspaces.
h
t
a
Introduction
m
[ One of the general approaches to description of C∗-algebras consists in realiza-
tion of them as ”functions” on dual space. All C∗-algebras having the irreducible
2
representationsindimensionlessorequaltosomenaturaln,usuallycanberealized
v
byn×n-matrix-functionswithsomeboundaryconditions. Thespecialcaseofsuch
1
2 C∗-algebras is given by homogeneous algebras - ones having all irreducible repre-
3 sentations in the same dimension n. The description of homogeneous C∗-algebras
1 intermsoffibrebundleswasobtainedin[4],[13](seeStatement3below). Theanal-
1 ogousdescriptionfor allC∗-algebrashaving irreducible representationsin different
4
dimensions less or equal to natural n was presented in [14].
0
/ ThehomogeneousC∗-algebraswhosespectraaretoriT2, T3 (see[2])andsphere
h Sk (see [1]) were studied by means of classification corresponding algebraic bun-
t
a dles. The concreteexample ofnon-homogeneousC∗-algebraisgivenbyC∗-algebra
m
generated by the free pair of projections (see [15]).
: Inthepresentpaperweconsiderwideclassofnon-homogeneousF -C∗-algebras
v 2n
and give a realization of algebras from this class in terms of matrix functions with
i
X
boundary conditions.
r
a
1. Preliminaries
Letusrecallsomepreparatoryfacts,definitionsandnotation,whichwillbeused
below.
1.1. EnvelopingC∗-algebras. InthispaperwestudyfinitelypresentedC∗-algebras.
ByfinitelypresentedC∗-algebrawemeananenvelopingC∗-algebraforfinitelypre-
sented ∗-algebra.
Definition. Let A be a ∗-algebra, having at least one representation. Then
a pair (A,ρ) of a C∗-algebra A and a homomorphism ρ : A −→ A is called an
enveloping pair for A if every irreducible representation π:A −→ B(H) factors
uniquelythroughtheA,i.e. thereexistspreciselyoneirreduciblerepresentationπ
1
of algebra A satisfying π ◦ρ=π. The algebra A is called an enveloping for A.
1
1
2 SAVCHUKYURI
1.2. F - algebras. Since all C∗-algebras considered below are examples of F -
2n 2n
C∗-algebras,werecallherethedefinitionandbasicpropertiesofalgebrassatisfying
the F polynomial identity. Let F denotes the following polynomial of degree n
2n n
in n non-commuting variables:
F (x ,x ,...,x )= (−1)p(σ)x ...x ,
n 1 2 n σ(1) σ(n)
σX∈Sn
where S is the symmetric group of degree n, p(σ) is the parity of a permuta-
n
tion σ ∈ S . We say that an algebra A is an algebra with F identity if for
n n
all x ,..., x ∈ A, we have F (x ,...x ) = 0. The Amitsur-Levitsky theorem
1 n n 1 n
says that the matrix algebra M (C) is an algebra with F identity. Moreover,
n 2n
C∗-algebra A has irreducible representations of dimension less or equal to n iff A
satisfies the F condition (see [10]). An important class of F -C∗-algebras is
2n 2n
formed by n-homogeneous algebras - ones having all irreducible representations of
dimension n. The simplest example of n-homogeneous C∗-algebra is 2×2-matrix
algebra over C(X), where X is a compact Hausdorff space - it is so-called trivial
algebra. Inthiscasethedualspace(spaceofprimitiveideals,see[3])iscanonically
homeomorphic to X. Simple example of non-homogeneous C∗-algebra is given by
”fixing” the homogeneous algebra above in a point x ∈X. Namely, consider
0
A(x )={f ∈C(X −→M (C))|f(x )∈M (C)⊕M (C)}.
0 2 0 1 1
ThedualspaceP(A(x ))ofalgebraA(x )isT ∪T ,whereT isthespaceofirre-
0 0 2 1 2
ducible2-dimensionalrepresentations,homeomorphictoX\{x }andT ={x′,x′′}
0 1 0 0
is the space of 1-dimensional representations. Here the bases of neighborhoods of
thepointsx′ andx′′arethesameasforthex inX,i.e. wehavethespacesatisfying
0 0 0
T but notT separabilityaxiom(suchspacessometimesarecalledquasicompact).
0 1
The description above can be easily generalized, for example taking n≥2:
A(x )={f ∈C(X −→M (C))|f(x )∈M (C)⊕...⊕M (C)},
0 n 0 n1 nk
or ”fixing” a homogeneous C∗-algebra in a few points. Studying some examples
below, we will describe dual spaces in terms of algebras
A (m(1),...,m(1);m(2),...,m(2);m(3),...,m(3))=
n 1 k1 1 k2 1 k3
={f ∈C(S2 −→M (C))|f(x )∈M (C)⊕...⊕M (C), x ∈S2, i=1,2,3},
n i m(i) m(i) i
1 ki
where x ,x ,x ∈ S2 are fixed points (it can be checked that the different triples
1 2 3
x ,x ,x give isomorphic algebras).
1 2 3
Finitely presented C∗-algebras are mostly non-homogeneous. Below we present
some important examples of such algebras.
Example 1. Mainexamples of the paper concern∗-algebras(C∗-algebras)associ-
atedwith extendedDynkindiagramsD ,E ,E ,E , see[16], fordetails. Belowwe
4 6 7 8
recall how these algebras are defined by generators and basic relations:
f f f f
C∗(D )=C∗hp ,p ,p ,p |p +p +p +p =2e, p∗ =p =p2,i=1,4i,
4 1 2 3 4 1 2 3 4 i i i
C∗(E )=C∗hp ,p ,q ,q ,r ,r |p +2p +q +2q +r +2r =3e,
f6 1 2 1 2 1 2 1 2 1 2 1 2
p∗ =p =p2,q∗ =q =q2,r∗ =r =r2,i=1,2,p p =q q =r r =0,j 6=ki,
i fi i i i i i i i j k j k j k
C∗(E )=C∗hp ,p ,p ,q ,q ,q ,r|p +2p +3p +q +2q +3q +2r=4e,
7 1 2 3 1 2 3 1 2 3 1 2 3
p∗ =p =p2,q∗ =q =q2,i=1,3,r∗ =r =r2,p p =q q =0,j 6=ki,
i fi i i i i j k j k
C∗(E )=C∗hp ,p ,q ,q ,q ,q ,q ,r|2p +4p +q +2q +3q +4q +5q +3r=6e,
8 1 2 1 2 3 4 5 1 2 1 2 3 4 5
p∗ =p =p2,q∗ =q =q2,r∗ =r =r2,i=1,2,j =1,5,p p =q q =0,l6=mi.
i fi i j j j l m l m
ON A CLASS OF C*-ALGEBRAS DETERMINED UNIQUELY BY DUAL SPACE. 3
Irreducible representations of the algebras above were classified in the series of
works (see [10], [7], [11], [8]) and the description of dual spaces follows easily from
the classification (see for example [12]). The dual spaces of C∗-algebras presented
above are the following:
• for C∗(D ) the dual space is the same as for A (1,1; 1,1; 1,1)
4 2
• for C∗(E ) the dual space is the same as for A (2,1; 1,1,1; 1,1,1)
f6 3
• for C∗(E ) the dual space is the same as for A (2,2; 2,1,1; 1,1,1,1)
f7 4
• forC∗(E )thedualspaceisthesameasforA (3,3; 2,2,1,1; 1,1,1,1,1,1).
f8 6
Example 2. Gfeneralizing the relations defined by D one can consider the ∗-
4
algebra:
f
P =Chp ,p ,p ,p |α(p +p )+β(p +p )=Ii,
α,β 1 2 3 4 1 2 3 4
where α,β > 0,α+β = 1. The case P corresponds to D . When α 6= β, the
1,1 4
dual space for C∗(P ) is the same as fo2r2algebra:
α,β f
{f ∈C(S2 −→M (C))|f(x ), f(x )∈M (C)⊕M (C), x , x ∈S2},
2 1 2 1 1 1 2
it can be checked that the different pairs (x ,x ) give isomorphic algebras, see [5]
1 2
for more details.
Example 3. We canconsidermuch morecomplicatedrelations using the algebras
P from previous example. Construct the next algebra:
α,β
∗ ∗ ∗
P =C hp ,p ,p ,p ,α,β|α(p +p )+β(p +p )=I,α=α ,β =β
ε 1 2 3 4 1 2 3 4
[α,p ]=[β,p ]=[α,β]=0, i=1,4, α,β ≥εI, α+β =Ii,
i i
where 0 < ε < 1. Using the previous example one can describe the dual space for
2
the P . We will see later that all algebras P , ε∈(0,1/2) are isomorphic.
ε ε
Example 4. The group algebra for G = Z ∗Z gives an example of F -algebra
2 2 4
corresponding to infinite discrete group.
C∗(Z ∗Z )=C∗hp ,p |p2 =p =p∗, k =1,2i:=P .
2 2 1 2 k k k 2
1.3. Fibre bundles. Allthenecessaryinformationaboutfibrebundlesreadercan
find in [9].
From the general theory of fibre bundles it is known that locally-trivial bundle
is defined by its 7-tuple
(E,B,F,G,p,{V } ,{φ } ), where E is a bundle space, B is a base, F is a
i i∈I i i∈I
fibre, G is a structure group, p : E → B is a projection, V is an open covering of
i
base B and φ : V ×F → p−1(V ) are coordinate homeomorphisms. Note that in
i i i
Vasil’ev’spaper[14]coordinateandskewproductsareconsideredinsteadoflocally
trivial bundles. Coordinate product is a 7-tuple (E,B,F,G,p,{V } ,{φ } ),
i i∈I ji i,j∈I
where φji =φ−j1◦φi|Vi∩Vj :(Vi∩Vj)×F →(Vi∩Vj)×F are sewing maps. Every
φ is uniquely determined by continuous map (which we also call sewing map)
ji
φ :V ∩V →G by formulae: φ (x,f)=(x,φ (x)◦f), ∀x∈V ∩V .
ji i j ji ji i j
In this paper we consider only algebraic bundles, i.e. bundles with F ≃ M (C)
n
andG⊆U(n),whereU(n)isagroupofinnerautomorphismsofM (C),i.e. group
n
of unitary matrices U(n) factorized by it’s center. Group U(n) is supposed to act
on the M (C) in the following way. Let gˆ ∈ U(n) and g ∈ U(n) is corresponding
n
matrix, then gˆ(m) = g∗mg, m ∈ M (C). If G is a subgroup of U(n), we denote
n
4 SAVCHUKYURI
by G the quotient of G by it’s center. Sometimes we suppose sewing maps φ to
ji
havevaluesin U(n), insteadof U(n),one has to takea compositionofφ with the
ji
canonical map of U(n) onto U(n).
The following statements on triviality will be used below.
Statement 1. (see [6]) Every locally-trivial bundle over n-dimensional disk Dn is
isomorphic to trivial.
Statement 2. (see [1]) Every algebraic bundle over some compact subset of C is
isomorphic to trivial.
In this article by B(X,F,G) we denote some bundle with base X, fibre F and
group G. The algebra of sections of bundle B is denoted by Γ(B).
1.4. Structure of F -C∗-algebras. Next statement (Fell-Tomiyama-Takesaki
2n
theorem) gives a description of n-homogeneous C∗-algebras in terms of algebraic
bundles.
Statement 3. ([13],[14]) For every n-homogeneous C∗-algebra A there exists a
bundle B(P(A),M (C),U(n)), such that A ≃ Γ(B), where P(A) is space of all
n
pairly non-equivalent irreducible representations.
Analogousresultfornon-homogeneousF -C∗-algebraswasobtainedinVasil’ev’s
2n
work [14]. The following statement is the straightforward corollary from the main
result of [14].
Statement 4. Let A be F -C∗-algebra, having finite number of irreducible repre-
2N
sentations in dimensions 1,2,...,N −1. Then there exist:
(1) finite sets X1,...,XN−1, compact space X and open dense in X subspace
X, such that X\X is finite,
(2) a formal decomposition for all x∈X\X :
r
x= d x ,
k k
Xk=1
where d are natural numbers, x ∈ X and d n = N, such that two
k k nk k k
points from X\X coincide iff they have the samPe decomposition,
(3) bundle B = B(X,M (C),U(N)) having n U(d )×...×n U(d ) as the
N 1 1 r r
structure group at the point x ∈ X\X, x = d x +...+d x (i.e. there
1 1 r r
exists a neighborhood O ⊆ X, such that the structure group of bundle B
x
reduces to n U(d )×...×n U(d ) at the subspace O ∩X),
1 1 r r x
such that algebra A can be realized as an algebra of N-tuples a = (a(1),...,a(N)),
where a(i) : X −→ M (C), i = 1,N −1 are some functions and a(N) ∈ Γ(B) is a
i i
section such that:
0
1d1 ⊗a(n1)(x1)
lim a(N)(x)= ... .
x→ rk=1dkxk  0 
P  1dr ⊗a(nr)(xr)
ON A CLASS OF C*-ALGEBRAS DETERMINED UNIQUELY BY DUAL SPACE. 5
1.5. Bundles over 2-sphere. Recall, that fibre bundles over the sphere Sk are
naturallyclassifiedby elements of the homotopic groupπk−1(G), whereG is struc-
ture group of bundle (see for example [9] or [6]).
Bundles over 2-dimensional sphere with fibre M (C) are classified in [1], in this
n
case π(U(n))≃Z . Let us give a sketch of this classification.
n
Consider an atlas on 2-sphere, containing 2 charts : upper and lower closed
half spheres (S2 = W and S2 = W correspondingly). As sewing map φ take
+ 1 − 2 1,2
continuousmapV′ :S1 −→U(n). ConstructabundleB′ withatlas{W , W }and
1 2
sewing map V′. Two bundles B′ and B′′ defined by sewing maps V′ and V′′ are
isomorphic iff ind V′−ind V′′ = nl, l ∈Z, where ind V = (2π)−1[argdetV(t)]
S1
is the winding number of the function V with respect to the circle. Define sewing
maps:
(1.1) V :S1 −→U(n),z 7→diag(zk,1,...,1), k =0,n−1.
k
Thenthe bundlesB , k =0,n−1,definedbyV arecanonicalrepresentatives
n,k k
of all isomorphism classes.
2. Enveloping C∗-algebras, an examples.
To give a description of C∗-algebras for examples from Section 1, we need the
following
Theorem 1. Let
Dl(m ,...,m )={f ∈C(Dl −→M (C))|
1 k m1+...+mk
f(0,...,0)∈M (C)⊕...⊕M (C)},
m1 mk
Every C∗-algebra Ahaving thedual spacesameas Dl(m ,...,m )is isomorphic
1 k
to Dl(m ,...,m ).
1 k
Proof. We give proof when l = 2, k = 2, m = m = 1, i.e. Dl(m ,...,m ) =
1 2 1 k
D2(1,1) and realize D2 as the unit disk in complex plane. For the general case
proof is analogous. Let us introduce some notaion:
D2(1,1)= f ∈C(D2 −→M (C))|f(0,0)∈M (C)⊕M (C) ,
± ± 2 1 1
(cid:8) (cid:9)
where D2 = z ∈D2|ℑm(z)≥0 and D2 = z ∈D2|ℑm(z)≤0 , and let A be
+ − +
C∗-algebra ha(cid:8)ving dual space sam(cid:9)e as for D2(cid:8). We have to prove(cid:9)that A ≃ D2.
+ + +
Thestatement4impliesthatthereexistsabundleB =B (D2\{0},M (C),U(2)),
+ + + 2
having U(1)×U(1) as the structure group at the point 0, such that A is isomor-
+
phic to the algebra of pairs:
(a′,a′′)|a′ =a′(x)∈Γ(B ),a′′ ∈M (C)⊕M (C),lima′(x)=a′′ .
+ 1 1
n x→0 o
We suppose B to be assigned by charts {V } , where
+ k k∈N
1 1
V = z ∈D2, ≤|z|≤ .
k (cid:26) + 2k 2k+1(cid:27)
Sewingmapsφ ofthebundleB are{µ } ,whereµ :V ∩V →U(2)
k,k+1 + k k∈N k k k+1
are some continuous functions.
The condition for B to have U(1)×U(1) as the structure group at the point 0
implies that there exists p ∈ N such that µ (z) ∈ U(1)×U(1),∀r > p. The fact
r
6 SAVCHUKYURI
thatU(2)andU(1)×U(1)areconnectedgroupsallowsus to constructcontinuous
maps
µ :V →U(2), k ≤p
k k
and
µ :V →U(1)×U(1),k >p
k k
such that
µ (z)= e, z ∈Vk−1∩Vk;
k (cid:26) µk(z), z ∈Vk∩Vk+1.
Let us write sections of B explicitly:
+
′ ′
Γ(B )∋a ←→{a } ,
+ k k∈N
wherea′ :V →M (C)arecontinuousfunctionssatisfyingtheconditionofcompat-
k k 2
ibilitya′ (z)=µ (z)◦(a′(z)), z ∈V ∩V . Sowewriteelements(a′,a′′)∈A
k+1 k k k k+1 +
as ({a′k}k∈N,a′′).
Let us consider the map:
′ ′′ ′ ′′
A+ ∋({ak(z)}k∈N,a )7→({µk(z)◦ak(z)}k∈N,a ).
One can check that the map is the isomorphism of algebras A and D2(1,1).
+ +
Analogously the isomorphism A− ≃D−2(1,1) can be proved.
LetusconsiderbundleB(D2\{0},M (C),U(2))havingU(1)×U(1)asthestruc-
2
ture group at the point 0, such that:
A≃ (a′,a′′)|a′ =a′(x)∈Γ(B),a′′ ∈M (C)⊕M (C),lima′(x)=a′′ .
1 1
n x→0 o
The fact that A+ and A− are ”trivial” allows us to suppose B to have only two
charts: W =D2\{0}andW =D2\{0}andone sewingmapµ :[−1,1]\{0}→
1 + 2 − 1,2
U(2) such that ∃ε >0:∀z ∈(−ε ,0)∪(0,ε ) µ(z)∈U(1)×U(1). As before, by
0 0 0
connectivityofU(2)andU(1)×U(1)weconstructcontinuousmapµ:W →U(2)
1
possessing a values from U(1)×U(1) in O(ε ,0) = {z ∈ D2,0 < |z| < ε } such
0 + 0
that ∀z ∈W ∩W :µ(z)=µ (z).
1 2 1,2
As before we have an isomorphism given by µ:
A∋({a′(z),a′(z)},a′′)7→({µ(z)a′(z),a′(z)},a′′)∈D2(1,1).
1 2 1 2
The proof is completed. (cid:3)
Remark 1. One can regard the theorem above as a certain generalization of the
Statement 1. Indeed we have a C∗-algebra identically defined by it’s dual space.
As before, let
k
m =n, m ∈N,j =1,k
j j
Xj=1
and set
B (m ,...,m )={f ∈Γ(B )|f(x)∈M (C)⊕...⊕M (C)},
n,p 1 k n,p m1 mk
where x∈S2 is fixed point (we consider x to be covered only by one chart).
Next theorem is basic to study the structure of C∗-algebras corresponding to
Dynkin diagrams.
ON A CLASS OF C*-ALGEBRAS DETERMINED UNIQUELY BY DUAL SPACE. 7
Theorem 2. If l ≡l (mod(m ,...,m ) ), then
1 2 1 k
(2.1) B (m ,...,m )≃B (m ,...,m )
n,l1 1 k n,l2 1 k
(here (m ,...,m ) denotes the greatest common divider of m ,...,m )
1 k 1 k
Proof. Let us realize elements of algebras B (m ,...,m ) as sections of corre-
n,p 1 k
sponding bundles B (here p=l or p=l ):
n,p 1 2
B (m ,...,m )={(f ,f )|f ∈C(D2 →M (C)),
n,p 1 k 1 2 i n
f (z)=V∗(z)f (z)V (z), |z|=1, f (0)∈M (C)⊕...⊕M (C)},
1 p 2 p 1 m1 mk
where V are defined by (1.1).
p
Condition l ≡l (mod(m ,...,m ) ) implies that
1 2 1 k
∃c ,...,c ∈Z:c m +...+c m =l −l .
1 k 1 1 k k 1 2
LetH(t)=ei2πc1tEm1⊕...⊕ei2πcktEmk ∈U(n),whereEm denotesthe m×m-
identity matrix. Construct the map H :D2 −→U(n) by the rule :
z
z 7→H , z 6=0,
(cid:18)|z|(cid:19)
07→1 .
n
Nevertheless H is not continuous in 0, it is easy to check that H defines an
isomorphism :
∗ ′
B (m ,...,m )∋(f (z),f (z))7→(H (z)f (z)H(z),f )∈Γ(B ).
n,l1 1 k 1 2 1 2 n,l2
The bundle B′ is defined by the sewing map V′(z)=H∗(z)V (z). The equality
ind V′ =indnV,l2 gives an isomorphism (2.1). l2 l1 (cid:3)
l2 l2
Remark 2. The converse statement to the Theorem 2 takes place. We will not use
it below, so we don’t give proof.
Remark 3. One can easily prove the following generalization of theorem 2: if we
”fix”sectionsofbundlesB andB inafewpointsinblockmatrices(withequal
n,l1 n,l2
dimensionofblocksforbothbundlesineverypoint)andsupposethatallblocksare
ofdimensions m ,...,m , thenl ≡l (mod(m ,...,m ))implies isomorphismof
1 k 1 2 1 k
corresponding algebras of sections.
Let us return to our examples of finitely generatedF -C∗-algebras(see prelim-
2n
inaries).
Proposition 1. For C∗-algebras associated with extended Dynkin diagrams one
has the next isomorphisms:
∗
C (D )≃A (1,1; 1,1; 1,1),
4 2
∗
C (Ef)≃A (2,1; 1,1,1; 1,1,1),
6 3
∗
C (Ef)≃A (2,2; 2,1,1; 1,1,1,1),
7 4
∗
C (Ef)≃A (3,3; 2,2,1,1; 1,1,1,1,1,1).
8 6
Proof. Let A denotes Cf∗(D ). We can consider this algebra to be realized by
4
functions on the S2 with values in matrices of corresponding dimension, not nec-
f
essary continuous yet. We can find a covering of S2 by closed subsets W ≃ D2
j
such that every W contains only representations in main dimension or contains
j
8 SAVCHUKYURI
one point with reducible representation of main dimension. Consider the ”restric-
tions” A(W ) of algebra A onto subsets W (i.e. factor-algebras A/I(W ), where
j j j
I(W ) = f ∈A,f| ≡0 ), then use Theorem 1 to realize every A(W ) as sec-
j Wj j
tions of so(cid:8)me algebraic bu(cid:9)ndle, i.e. everyone of A(Wj) coincides with one of the
B (m ,...,m ) (possibly with m =n, m =...=m =0).
n,l 1 k 1 2 k
DifferentA(W )are”sewed”bysomecontinuousmaps,whichweusetoconstruct
j
the bundle B such thatalgebraA is isomorphicto B (m ,...,m ). Theorem2
n,l n,l 1 k
shows that we can suppose l=0. (cid:3)
Proposition 2. Enveloping C∗-algebra for P is :
α,β
C∗(P )={f ∈C(S2 −→M (C))|f(x ), f(x )∈M (C)⊕M (C), x , x ∈S2}
α,β 2 1 2 1 1 1 2
Proof. See the proof of previous theorem. (cid:3)
The following example of enveloping C∗-algebra is considered in [15]. One can
get the result of [15] using our Theorem 1.
Proposition 3. Enveloping C∗-algebra P for free pair of projections (see 4) is :
2
P ={f ∈C([0,1]−→M (C))|f(0), f(1) are diagonal }.
2 2
Proof. As in previous case, we realize this algebra by sections of some bundle over
[0,1]. By statement 1 all bundles over [0,1] are trivial. (cid:3)
All the constructions above in the paper, including Statement 4, can be gener-
alized to prove the next proposition:
Proposition 4. Enveloping C∗-algebra for P (example 3) has the form:
ε
P ={f ∈C(S2×I −→M (C))|f(x ,1/2), f(x ,t), f(x ,t)∈M (C)⊕M (C),
e 2 1 2 3 1 1
t∈I, x , x , x ∈S2}.
1 2 3
Proof. Here we use the fact that the theory of algebraic bundles over S2×I is the
same as for S2. Then the proof of Theorem 2 should be transferred from S2 to
S2×I. (cid:3)
Remark 4. The definition of triviality for homogeneous C∗-algebras can be gener-
alized. One can define F -C∗-algebra satisfying conditions of the Statement 4 to
2N
be trivial iff it is isomorphic to the F -C∗-algebra with the trivial corresponding
2N
bundleB(X,M (C),U(N)). Thenallpropositionsabovestatethe trivialityofC∗-
N
algebrasunderconsideration. Exampleofnon-trivialnon-homogeneousC∗-algebra
one can constructby ”fixing”sections of bundle B in the 2×2-blockmatrices in
4,1
one point, it is also an example of algebra that is not defined by the dual space.
Acknowledgements.
AuthorexpresseshisgratitudetoProf. Yuri˘ıS.Samo˘ılenkofortheproblemstating
and permanent attention. I am also indebted to Dr. Daniil P. Proskurin for help
during the preparation of this paper.
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KievTarasShevchenkoNationalUniversity,FacultyofCybernetics,Volodymyrska,
64,01033,Kiev,Ukraine,
E-mail address: [email protected]