Induced quadratic modules in $*$-algebras

0.3 MB·

by  Jaka Cimpric

INDUCED QUADRATIC MODULES IN -ALGEBRAS
∗
JAKA CIMPRICˇ, YURII SAVCHUK
2
1
0
Abstract. Positivity in -algebras can be defined either alge-
2 ∗
braically,byquadraticmodules,oranalytically,by -representations.

n ∗
By the induction procedure for -representations we can lift the
a ∗
analytical notion of positivity from a -subalgebra to the entire
J
∗
-algebra. The aim of this paper is to define and study the in-
1 ∗
1 duction procedure for quadratic modules. The main question is
when a given quadratic module on the -algebra is induced from
∗
] its intersection with the -subalgebra. This question is very hard
G ∗
even for the smallest quadratic module (i.e. the set of all sums of
A
hermitiansquares)andwillbeansweredonlyinveryspecialcases.
.
h
t
a
m
1. Introduction
[

2 Non-commutative real algebraic geometry studies real and complex
v
associative algebras with involution from a geometric perspective. Her-
4
7 mitian elements are considered as non-commutative real polynomi-
3 als and “well-behaved” (not necessarily bounded) -representations as
1 ∗
non-commutative real points, see [S2]. For every finite set S of hermit-
.
1
ian elements one tries to understand the set of all hermitian elements
0
2 that are positive (semi-)definite in every well-behaved -representation
∗
1 in which all elements from S are positive semi-definite. An algebraic
:
v description of this set is called a Positivstellensatz for S. They are
i
X well-understood in the commutative case, see [M1] for a survey, but in
r the non-commutative case they are very difficult to find. Most alge-
a
bras with involution for which Positivstellens¨atze are known (e.g. Weyl
algebras, quantum polynomials, matrix polynomials, central simple al-
gebras) have the property that their well-behaved -representations
∗
are induced from one-dimensional -representations of an appropriate
∗
commutative subalgebra. However, thequest forgeneral “Induced Pos-
itivstellensa¨tze” is still on.
This paper addresses only a small part of the big problem. We
define and study the induction procedure for various order-theoretic
structures (i.e. quadratic modules, preorderings, orderings) related to
Positivstellens¨atze.
Date: January 12, 2012.
1
2 JAKA CIMPRICˇ, YURIISAVCHUK
In Section 2 we discuss the relationship between -representations
∗
and quadratic modules whose special case is the relationship between
bounded -representation and archimedean preorderings. We recall the
∗
definition of a bimodule projection from a -algebra to its -subalgebra
∗ ∗
and the corresponding induction procedure for -representations.
∗
InSection3weextendtheinductionprocedurefrom -representations
∗
to quadratic modules. Let be a -subalgebra of an -algebra and
B ∗ ∗ A
let p: be a bimodule projection (i.e. a - bimodule map
A → B B B
which commutes with the involution and preserves identity). For every
quadratic module in we define the set
N B
Ind := a a = a∗ and p(x∗ax) for all x
N { ∈ A | ∈ N ∈ A}
which is a quadratic module in if and only if it contains 1. If is a
B N
preordering, then Ind need not be a preordering even if it contains
N
1. For every quadratic module in the set
M A
Res := p(m) m M
M { | ∈ }
is a quadratic module in and the following questions make sense:
B
(Q1) Is = IndRes ?
M M
(Q2) Is Res = ?
M M∩B
(Q3) Is generated by ?
M M∩B
These questions are surprisingly hard even for very special p and .
M
The bimodule projections that we are interested in come either from
group actions by -automorphisms (we discuss matrix -algebras, reg-
∗ ∗
ular functions on affine varieties, Galois extensions of fields) or group
gradings (we discuss group algebras, cyclic algebras, quantum poly-
nomials, Weyl algebra). Interesting quadratic modules include 2,
A
the set of all sums of hermitian squares, and , the set of all her-
+
A P
mitian elements that are positive semi-definite in all well-behaved -
∗
representations.
For example, let p be the “natural” bimodule projection from the
Weyl algebra = C a,a∗ aa∗ a∗a = 1 toitssubalgebra = C[a∗a].
A h | − i B
We consider the Fock-Bargmann -representation as the only well-
∗
behaved -representation. For , the answers to (Q1) and (Q2) turn
+
∗ A
out to be positive, while the answer to (Q3) turns out to be negative.
For 2, (Q1) is still open while the answers to (Q2) and (Q3) are
A
trivially positive. For other -algebras we have similar situations.
P ∗
Finally, intheAppendixweextendourconstructionofIndandResof
quadraticmodulesfrombimoduleprojectionstorigged - bimodules.
A B
INDUCED QUADRATIC MODULES IN ∗-ALGEBRAS 3
2. Preliminaries
Inthissectionweexplaintherelationshipbetweenquadraticmodules
and -representations. We also recall the construction of an induced
∗
-representation via a bimodule projection.
∗
2.1. Quadratic modules. Let be a -algebra over a field F. This
A ∗
means that is an associative unital algebra over F and is equipped
A A
with an involution such that F∗ F (F is identified with F 1 ). We
A
∗ ⊆ ·
will denote by the set a a∗ = a of all self-adjoint elements
h
A { ∈ A | }
of . In the following we will also assume that the field F := F
h h
A ∩A
is equipped with an ordering such that ff∗ 0 for every f F.
≥ ≥ ∈
(For F = C this assumption implies that is the conjugation. )
C
∗|
A subset is called a quadratic module (q.m.) in if the
h
M ⊆ A A
following axioms are satisfied:
(QM1) + and λ for all λ F≥0,
M M ⊆ M M ⊆ M ∈
(QM2) c∗ c for every c ,
M ⊆ M ∈ A
(QM3) 1 .
A
∈ M
We say that is proper if 1 . For a subset S we denote
A h
M − 6∈ M ⊆ A
by QM (S) the q.m. generated by S, that is
A
m
QM (S) = λ a∗s a s S, a , λ F≥0 .
A { i i i i | i ∈ i ∈ A i ∈ }
i=1
X
The set 2 = QM ( 1 ) is the smallest quadratic module in . In
A
A { } A
all our examples 2 is proper. The largest q.m. in is the set of
P A A
all symmetric elements . It is never proper.
h
P A
Remark 2.1. Note that for F = Q, F = R and F = C the second
part of (QM1) follows from (QM2) and (QM3), so it can be omitted.
In particular, we can also omit λ in the definition of QM (S).
i A
A quadratic module in is archimedean if for every a
h
M A ∈ A
there is n N such that n1 a . It is a preordering if for every
A
∈ − ∈ M
x,y such that xy = yx we have that xy . Note that F≥0 is a
∈ M ∈ M
preordering. For F = Q, F = R and F = C, F≥0 is also archimedean.
2.2. -Representations. Let be an inner product space over F and
∗ V
let L( ) be the set of all linear operators on . A non-zero algebra
V V
homomorphism π : L( ) is called a -representation of on if
A → V ∗ A V
ρ(b)v,w = v,ρ(b∗)w and ρ(1 )v = v for all v,w . We will also
A
h i h i ∈ V
write (π) for . A -representation is called cyclic if it has a cyclic
D V ∗
vector, i.e. if there is v (π) such that π(a)v a = (π).
∈ D { | ∈ A} D
4 JAKA CIMPRICˇ, YURIISAVCHUK
Let π be a -representation of on . One easily checks that
∗ A V
QM (π) := a ρ(a)v,v 0 for all v
A h
{ ∈ A | h i ≥ ∈ V}
is a q.m. in . Note that QM (π) is archimedean if and only if π is
A
A
a bounded -representation (i.e. π(a) is bounded for every a ). If
∗ ∈ A
QM (π) is archimedean, it is also a preordering.
A
A non-zero linear functional ϕ on is called positive if ϕ( 2)
A A ≥
0. It is called hermitian if ϕ(a∗) = ϕ(a)∗ for every a . By the
∈ AP
GNS-construction (see [S1, Section 8.6]) for each positive hermitian
linearfunctionalϕthereexistsacyclic -representationπ withacyclic
ϕ
∗
vector ξ (π ) such that ϕ(a) = π (a)ξ ,ξ for every a A.
ϕ ϕ ϕ ϕ ϕ
∈ D h i ∈
Every -algebra over C will be considered with the finest locally
∗ A
convex topology (defined by the family of all seminorms on ). Closed
A
quadratic modules in play an important role in the real algebraic
A
geometry, see [M1, S2].
Proposition 2.2. A quadratic module in a complex -algebra is
M ∗ A
closed if and only if = QM (π) for some -representation π of .
A
M ∗ A
Proof. Let π be a -representation of on an inner-product space .
∗ A V
For each v denote by the set a π(a)v,v 0 . The
v h
∈ V A { ∈ A | h i ≥ }
set is closed, thus QM (π) = is closed.
v A v∈V v
A ∩ A
Let be a closed q.m. and let a . By the separation
h h
M ⊆ A ∈ A \M
theorem for closed convex sets there exists a R-linear functional ϕ on
a
such that ϕ ( ) 0 and ϕ (a) < 0. We denote the corresponding
h a a
A M ≥
C-linear functional on by ϕ . Let π be the GNS representation of
a a
ϕ and let π := π . TAhen = QM (π) by construction. (cid:3)
a a a A
⊕ M
2.3. Inclusion between quadratic modules. This subsection set-
tles an important basic question of non-commutative real algebraic
geometry. It will only be used in Example 5.4.
Let ρ and π be -representations of a complex -algebra . We
∗ ∗ A
say that ρ is weakly contained in π if every functional ω (a) :=
ρ,ψ
π (a)ξ ,ξ can be weakly approximated by functionals m ω .
h ϕ ϕ ϕi i=1 π,ξi
Thatis,forarbitraryε > 0anda ,...,a thereexistξ ,ξ ,...,ξ
1 n 1 2 m
∈ A P ∈
(π) such that
D
m
(1) ω (a ) ω (a ) < ε, for all j = 1,...,n.
| ρ,ψ j − π,ξi j |
i=1
X
Proposition 2.3. A -representation ρ is weakly contained in π if and
∗
only if QM (π) QM (ρ).
A A
⊆
INDUCED QUADRATIC MODULES IN ∗-ALGEBRAS 5
Proof. Let ρ be weakly contained in π and let a QM (π). Then
A
∈
ω (a) 0 for all ξ (π) and by (1) we have ω (a) 0 for all
π,ξ ρ,ψ
≥ ∈ D ≥
ψ (ρ). Hence a QM (ρ), which proves the ”if” direction.
A
∈ D ∈
To prove the ”only if” direction, we use some basics from the theory
oflocallyconvexspaces. Let (π)betheconvexhullof ω ξ = 1 .
π,ξ
P { | k k }
Then for ϕ (π) we have ϕ(1) = 1 and
∈ P
QM (π) = a ϕ(a) 0, ϕ (π) .
A h
{ ∈ A | ≥ ∀ ∈ P }
Consider as an R-vector space and let ′ be the space of all linear
Ah Ah
functionals on , so that , ′ is a dual pair. Consider (π) as a
Ah hAh Ahi P
subset of ′ and let
Ah
(π)◦ = a ϕ(a) 1, ϕ (π)
h
P { ∈ A | ≤ ∀ ∈ P }
be its polar set in . Fix an arbitrary ω , ψ (ρ) and assume
h ρ,ψ
A ∈ D
without loss of generality that ψ = 1. Then for a we have
h
k k ∈ A
a (π)◦ ϕ (π) ϕ(a) 1 ϕ (π) ϕ(1 a) 0
∈ P ⇒ ∀ ∈ P ≤ ⇒ ∀ ∈ P − ≥ ⇒
1 a QM (π) 1 a QM (ρ) ω (a) 1,
A A ρ,ψ
⇒ − ∈ ⇒ − ∈ ⇒ ≤
that is ω is contained in ( (π)◦)◦. By the bipolar theorem ω is
ρ,ψ ρ,ψ
weakly approximated by the ePlements of (π). (cid:3)
P
2.4. Bimodule projections. Let be -algebras over a field F.
B ⊆ A ∗
A mapping p: is said to be a bimodule projection if it satisfies
A → B
the following properties:
(CE1) p(a +a ) = p(a )+p(a ) for every a ,a ,
1 2 1 2 1 2
∈ A
(CE2) p(b ab ) = b p(a)b for every b ,b and a ,
1 2 1 2 1 2
∈ B ∈ A
(CE3) p(a∗) = p(a)∗,
(CE4) p(1 ) = 1 .
A B
A bimodule projection is a conditional expectation if it satisfies
(CE5) p( 2) 2 (or equivalently p( 2) = 2 .)
A ⊆ A A A ∩B
We will not need properties (CE4) and (CE5) until subsection 3.3.
P P P P
Example 2.4. Every hermitian linear functional φ: A F satisfies
→
(CE1)-(CE3) and 1 φ is a bimodule projection. If also φ is positive
φ(1)
and F≥0 = F2, then 1 φ is a conditional expectation.
φ(1)
Many morePconditional expectations will be given in Sections 4 and 5.
Example 2.5. Let g be the Heisenberg Lie algebra, i.e. the real Lie
algebra with generators a,b,c, relations [a,b] = c, [a,c] = [b,c] = 0
and involution a∗ = a, b∗ = b, c∗ = c. Let h be its subalgebra
− − −
generated by a,c. We claim that there exists no bimodule projection
from = (g)to = (h). Assume tothecontrary, thatp : is
A U B U A → B
6 JAKA CIMPRICˇ, YURIISAVCHUK
a bimodule projection. Then c = p(c) = p(ab ba) = ap(b) p(b)a = 0,
− −
by commutativity of . A contradiction.
B
2.5. Induced -representations. Let be -algebras over a
∗ B ⊆ A ∗
field F and let p : be a bimodule projection. Further, let ρ
A → B
be a -representation of on an inner-product space . Let us briefly
∗ B V
recall the construction of the induced representation Indρ, see [SS].
Let denote the quotient space of by the linear hull
B
A ⊗ V A ⊗ V
of ab v a ρ(b)v, a , b , v . Note that is
B
{ ⊗ − ⊗ ∈ A ∈ B ∈ V} A ⊗ V
an induced -module in the sense of [H]. Representation ρ is called
A
inducible if the sesquilinear form , on defined by
0 B
h· ·i A⊗ V
x v , y w := ρ(p(y∗x ))v ,w , x ,y , v ,w
h i⊗ i j⊗ ji0 h j i i ji i j ∈ A i j ∈ V
i j i,j
X X X
is positive semi-definite. The kernel of the form , is invariant
ρ 0
N h· ·i
under the action of on . For a v X denote by
B B
A A ⊗ V ⊗ ∈ A ⊗
[a v] X/ the image under the quotient mapping. Thus,
B ρ
⊗ ∈ A ⊗ N
X/ is an inner-product space and there is a well-defined -
B ρ
A ⊗ N ∗
representation π = Indρ of on the unitary space X/ which
B ρ
A A⊗ N
acts by
π(a)( [x v ]) := [ax v ].
i i i i
⊗ ⊗
i i
X X
Example 2.6. Let φ be a hermitian linear functional from to F.
A
The left regular representation λ of F is inducible if and only if φ is
positive. In this case Indλ = π , i.e. the -representation associated
φ
∗
to φ by the GNS construction. Note also that QM (λ) = F≥0 and
F
QM (π ) = a ϕ(x∗ax) 0 for all x .
A ϕ h
{ ∈ A | ≥ ∈ A}
3. Induced quadratic modules
3.1. Definition. Let be a -algebras over a field F. Motivated
B ⊆ A ∗
by Example 2.6 we define for every quadratic module
N ⊆ B
Ind := a p(x∗ax) for all x .
h
N { ∈ A | ∈ N ∈ A}
The set Ind is called induction of from to via p. We will also
N N B A
write Ind or Indp for Ind .
B↑A
N N N
Proposition 3.1. Let be a q.m. Then
N ⊆ B
(i) the set Ind is a quadratic module if and only if
N
(2) p(x∗x),x ,
{ ∈ A} ⊆ N
(ii) if (2) is satisfied, then Ind is proper if and only if is proper,
N N
(iii) if (2) is satisfied, then Ind is the largest q.m. contained in
N
p−1( ).
N
INDUCED QUADRATIC MODULES IN ∗-ALGEBRAS 7
Proof. (i): OneeasilychecksthatInd isasubsetof and(QM1),(QM2)
h
N A
are satisfied. Then p(x∗x) for all x if and only if 1 Ind ,
A
∈ N ∈ A ∈ N
i.e. (QM3) is satisfied.
(ii): Clearly, if = then Ind = is also not proper. Assume
h h
N B N A
now Ind = , that is 1 Ind . Then 1 = p(1∗ ( 1 )1 )
N Ah − A ∈ N − B A − A A ∈
, i.e. is not proper.
N(iii): NFollows from the definition of Ind . (cid:3)
N
In the view of Proposition 3.1 we say that a q.m. is inducible
N ⊆ B
if and only if (2) is satisfied. Equivalently, is inducible if and only
N
if Ind is a q.m.
N
In general, induction does not respect preorderings:
Example 3.2. Let = R[x,y] and = QM ( x,y ). By [M1],
A
A M { }
Example 4.1.5 and Theorem 4.1.2, is closed. A short computation
M
shows that xy . Then there exists a linear functional ϕ : R
6∈ M A →
such that ϕ( ) 0 and ϕ(xy) < 0. For p = 1 φ we have that x,y
M ≥ φ(1) ∈
IndpR≥0 and xy / IndpR≥0. Therefore IndpR≥0 is not a preordering.
∈
If 2 is archimedean then induction respects closed preorderings.
A
In view of subsection 2.2, this will follow from Proposition 3.3.
P
3.2. Relation to induced -representations. Let be a closed
∗ N
q.m. in a real or complex -algebra . By Proposition 2.2 there exists
∗ B
a -representation ρ of such that = QM (ρ). By the proof of
B
∗ B N
Proposition 2.2, we may assume that ρ is a direct sum of cyclic -
∗
representations. Proposition 3.3 will imply that Ind is also closed.
N
Proposition 3.3. If ρ is a direct sum of cyclic -representations then
∗
QM (ρ) is inducible if and only if ρ is inducible. Moreover if QM (ρ)
B B
is inducible, then QM (Indρ) = Ind(QM (ρ)).
A B
Proof. Let = be a decomposition of into orthogonal sum
i i
V ⊕ V V
of cyclic components and v be the corresponding cyclic vectors.
i i
∈ V
AssumethatQM (ρ)isinducible, thatis p(a∗a) a QM (ρ)
B B
{ | ∈ A} ⊆
or, equivalently, ρ(p(a∗a))w,w 0 for all a , w . We show
h i ≥ ∈ A ∈ V
that ρ is inducible. For let w and a . Then w = ρ(b )v
i ∈ V i ∈ A i ij ij j
and we get
P
a w , a w = ρ(p(a∗a ))w ,w =
h i ⊗ i i ⊗ ii0 h j i i ji
i i i,j
X X X
(3) = ρ(p(a∗a ))ρ(b )v ,ρ(b )v =
h j i im m jk ki
i,j k,m
XX
= ρ(p(b∗ x∗x b ))v ,v = ρ(p(z∗z ))v ,v 0,
h jk j i ik k ki h k k k ki ≥
k i,j k
XX X
8 JAKA CIMPRICˇ, YURIISAVCHUK
where z = x b . Thus ρ is inducible. The inverse statement is
k i i ik
trivial.
P
Take a Ind(QM (ρ)). Then for all x and w holds
B
∈ ∈ A ∈ V
ρ(p(x∗ax))w,w 0. A calculation similar to (3) shows that
h i ≥
ax w , x w 0
i i i i 0
h ⊗ ⊗ i ≥
i i
X X
for arbitrary w and x . Thus Indρ(a) is positive or, equiva-
i i
lently a QM (∈InVdρ). The i∈nvAerse statement is trivial. (cid:3)
A
∈
3.3. Restriction. Let be -algebras and π a -representation
B ⊆ A ∗ ∗
of . The restriction of π to is defined by Resπ = π . It is easy to
B
A B |
see that QM (Resπ) = QM (π) . The most natural extension to
B A
∩B
quadratic modules would be that is considered as the restriction
M∩B
of a quadratic module on . However, we will not go this way.
M A
Let p : be a fixed bimodule projection. For a quadratic
A → B
module on we define
M A
Res := p(m) m .
M { | ∈ M}
The set Res is called restriction of onto via p. It can be easily
M M B
checked that Res is always a q.m. (property (CE4) is used here
M ⊆ B
for the first time), it is always inducible and it always contains .
M∩B
The advantage of this definition of Res is that we obtain a Galois
M
correspondence between quadratic modules on and inducible qua-
A
dratic modules on . The following is easy to verify:
B
Proposition 3.4. Let p: be a bimodule projection. For every
A → B
q.m. and every inducible q.m. we have that
M ⊆ A N ⊆ B
Res iff Ind .
M ⊆ N M ⊆ N
Here are some special cases:
(1) Ind(Res ) and Res(Ind ) .
M ⊇ M N ⊆ N
(2) Res iff Ind( ).
M ⊆ M M ⊆ M∩B
(3) Ind iff ResQM ( ) .
A
N ⊆ N N ⊆ N
By (1), Res(Ind(Res )) = Res and Ind(Res(Ind )) = Ind .
M M N N
We say that a q.m. is induced if = Ind for some
M ⊆ A M N
q.m. (or equivalently, if = IndRes ). We say that a
N ⊆ B M M
q.m. is restricted if = Res for some q.m. (or
N ⊆ B N M M ⊆ A
equivalently, if = Res(Ind )).
N N
By (2) we have that Res = iff Ind( ). It follows
M M∩B M ⊆ M∩B
that = Ind( ) iff Res = and = IndRes .
M M∩B M M∩B M M
By (3) we have that every inducible q.m. which satisfies
N ⊆ B
Ind is restricted (since QM ( ) ResQM ( )
A A
N ⊆ N N ∩B ⊆ N ⊆ N ⊆
INDUCED QUADRATIC MODULES IN ∗-ALGEBRAS 9
QM ( ) ). In many cases (see Propositions 4.1 and 5.1), we also
A
N ∩B
have the converse: every restricted q.m. satisfies Ind .
N ⊆ B N ⊆ N
We say that an inducible q.m. is perfect if
N ⊆ B
QM ( ) = Ind .
A
N N
In the situations from Propositions 4.1 and 5.1, every perfect q.m.
N
is restricted and Ind satisfies (Q1)-(Q3) by the discussion above.
N
3.4. Induction in stages. Induction in stages is a useful tool in the
theory of induced representations. The next proposition is a coun-
terpart of it for quadratic modules. Note that a composition of two
bimodule projections is always a bimodule projection.
Proposition 3.5. Let be -algebras, let p : , p :
1 2
C ⊂ B ⊂ A ∗ A → B
be bimodule projections and let be a q.m.
B → C K ⊆ C
Ind (Ind ) = Ind ( ).
B↑A C↑B C↑A
K K
Proof. Take x . Using properties (CE1)-(CE4) we get
h
∈ A
x Ind (Ind ) p (a∗xa) Ind , a
B↑A C↑B 1 C↑B
∈ K ⇐⇒ ∈ K ∀ ∈ A ⇐⇒
p (b∗p (a∗xa)b) = p (p (b∗a∗xab)) , a , b
2 1 2 1
⇐⇒ ∈ K ∀ ∈ A ∀ ∈ B ⇐⇒
p (p (a∗xa)) = p (p (1 a∗xa1 )) , a x Ind .
2 1 2 1 B B C↑A
⇐⇒ ∈ K ∀ ∈ A ⇐⇒ ∈ K
(cid:3)
This completes the proof.
Example 3.6. In this example we show that a composition of two
conditional expectations is in general not a conditional expectation.
Forlet = R[x], = R[x2], = R[x4]andletp : , p :
1 2
A B C A → B B → C
be bimodule projections defined by (p (f))(x) = 1(f(x)+f( x)) and
1 2 −
(p (g))(x2) = 1(f(x2) + f( x2)) respectively. It easy to verify that
2 2 −
p ,p are conditional expectations. On the other hand (p p )((x3
1 2 2 1
◦ −
x)2) = 2x4 / 2, i.e. (p p ) is not a conditional expectation.
2 1
− ∈ A ◦
4. APverages of actions of finite groups
Suppose that G is a finite group which acts by -automorphisms
∗
α , g G on a -algebra , and let := a α (a) =
g g
∈ ∗ A B { ∈ A |
a for all g G be the -subalgebra of stable elements. Let
∈ } ⊆ A ∗
p : be the average mapping given by
A → B
1
(4) p(a) = α (a), a .
g
G ∈ A
| | g∈G
X
It can be easily shown that p is a conditional expectation from onto
A
, see also [CKS, SS].
B
Proposition 4.1. Let p be defined by (4).
10 JAKA CIMPRICˇ, YURIISAVCHUK
(i) If is an inducible quadratic module then Ind is a
N ⊆ B N
G-invariant q.m. such that
Ind = Res(Ind )
N ∩B N
(ii) For every q.m. we have
M ⊆ A
Res Ind(Res ).
M ⊆ M
Proof. (i): Notethat Ind = p( Ind ) p(Ind ) = Res(Ind ).
B∩ N B∩ N ⊆ N N
Let a Ind . For arbitrary x we have
∈ N ∈ A
1 1
p(x∗p(a)x) = α (x∗)p(a)α (x) = α (x∗)α (a)α (x) =
G g g G 2 g h g
| | g∈G | | g,h∈G
X X
(5)
1 1
= α (α (x∗)aα (x)) = p(α (x∗)aα (x)) ,
h k k k k
G 2 G ∈ N
| | k∈Gh∈G | | k∈G
XX X
that is p(a) Ind . It proves Res(Ind ) Ind . To see, that
∈ N N ⊆ B ∩ N
Ind is invariant, take a Ind . Then for every x and g G we
N ∈ N ∈ A ∈
have p(x∗αg(a)x) = p(αg−1(x∗)aαg−1(x)) , hence αg(a) Ind .
∈ N ∈ N
(ii): Let a . For an arbitrary x calculation (5) yields
∈ M ∈ A
1
p(x∗p(a)x) = p(α (x∗)aα (x)) Res .
k k
G ∈ M
| | k∈G
X
Hence p(a) Ind(Res ), which implies the assertion. (cid:3)
∈ M
Assertion (i) suggests the following problem: Is every G-invariant
q.m. induced? Cf. 6.4 for a very special case of this. In view
M ⊆ A
of subsection 3.3, assertion (ii) implies that a q.m. is contained
N ⊆ B
in Ind iff it is restricted.
N
4.1. Matrix -algebras. Let be a -algebra, N a positive integer
∗ B ∗
and = M ( ) with involution [b ]∗ = [b∗ ]. Let G be semidirect
A N B ij ji
product(Z/2Z)N⋊ Z/NZwhereφ(1): (i ,i ,...,i ) (i ,i ,...,i ).
φ 1 2 N N 1 N−1
7→
The action of G on is defined by Ag = T∗AT where, for g =
A g g
(i ,...,i ,j),
1 N
( 1)i1 0 ... 0 0 0 1 0 ... 0 j
−
0 ( 1)i2 ... 0 0 0 0 1 ... 0
Tg =  ... −... ... ... ...  ... ... ... ... ... 
 0 0 ... ( 1)iN−1 0  0 0 0 ... 1 
 −  
 0 0 ... 0 ( 1)iN  1 0 0 ... 0 
 −  
  
If we identify I with , then the average map p(A) = 1 Ag
B⊗ N B |G| g∈G
coincides with the normalized trace tr(A) = 1(a +...+a ). Since
N 11 NNP