Leibniz Seminorms and Best Approximation from C*-subalgebras

0.22 MB·English

by  Marc A. Rieffel

LEIBNIZ SEMINORMS AND BEST APPROXIMATION
FROM C*-SUBALGEBRAS
1
1 MARC A. RIEFFEL
0
2 In celebration of the successful completion by Richard V. Kadison of 85
circumnavigations of the sun
n
a
J
Abstract. We showthat if is a C*-subalgebraofa C*-algebra
5 B
suchthat containsaboundedapproximateidentityfor ,and
1
A B A
if L is the pull-back to of the quotient norm on / , then L is
A A B
] stronglyLeibniz. Inconnectionwiththissituationwestudycertain
A
aspects of best approximation of elements of a unital C*-algebra
O
by elements of a unital C*-subalgebra.
.
h
t
a
m
1. Introduction
[

4 From my attempts to understand C*-metrics, as defined in [18], I
v have recently been trying to discover the mechanisms that can lead
3
to (continuous) seminorms, L, defined on a C*-algebra , that are
3
A
7 Leibniz, that is, satisfy the Leibniz inequality
3
. L(AC) L(A) C + A L(C)
8
≤ k k k k
0
for all A,C . Also of much interest to me, because of their impor-
0
∈ A
1 tance in [17] and in potential non-commutative versions of the results
v: in [17], are seminorms on a unital C*-algebra that are strongly-Leibniz,
i that is, satisfy, in addition to the Leibniz inequality, the property that
X
if A and if A is invertible in , then
r
∈ A A
a
L(A−1) A−1 2L(A),
≤ k k
and also L(1 ) = 0, again as defined in [18]. This last inequality seems
A
to have received virtually no attention in the mathematics literature.
Actually, for infinite-dimensional C*-algebras, the C*-metrics as de-
fined in[18] are discontinuous and only densely defined. But they are
2000 Mathematics Subject Classification. Primary 46L87; Secondary 53C23,
58B34,81R15, 81R30.
Key words and phrases. C*-subalgebras, best approximation, distance formula,
Leibniz inequality, strongly-Leibniz, minimal elements.
The research reported here was supported in part by National Science Founda-
tion grant DMS-0753228.
1
2 MARCA.RIEFFEL
required to be lower semi-continuous with respect to the C*-norm, and
in all of the examples that I know of one proves that they are lower
semi-continuous by showing that they are the supremum of an infinite
family of continuous strongly-Leibniz seminorms. This provides ample
reason for studying continuous strongly-Leibniz seminorms.
My investigations have led me to consider the situation in which
B
is a C*-subalgebra of a C*-algebra and L is the quotient norm on
A
/ pulled back to . That is,
A B A
L(A) = inf A B : B
{k − k ∈ B}
for A , so that L(A) is the norm-distance from A to . One
∈ A B
motivation for studying such quotient seminorms is that they arise
naturally when considering matricial Lipschitz seminorms, as defined
in [25, 26, 27] (where the Leibniz aspect was not used) and as dis-
cussed in the final section of [18]. These involve a unital C*-algebra ,
D
and for each natural number m the corresponding matrix C*-algebra
= M ( ) = M (C) over . The Leibniz seminorm on
m m
A D ⊗ D D A
from a matricial Leibniz seminorm on will take value 0 on the C*-
subalgebra = M (C) 1 , and soDcan be viewed as a seminorm
m D
B ⊗
on / . Furthermore, in this situation is finite-dimensional, and
A B B
so elements of will always have a best approximation by an ele-
A
ment of (probably not unique). This is good motivation for studying
B
best approximations. Although there is a very large literature dealing
with best approximation of elements of a Banach space by elements
of a closed subspace [22], I have found almost no literature concerning
this topic for the case of C*-algebras and their C*-subalgebras. The
later sections of this paper are devoted to developing some basic results
about this for C*-algebras. We show that the situation for C*-algebras
has some nice features in comparison to the general case, and we are
able to tie together a few results scattered in the literature.
Somewhat to my surprise I have found that the seminorms from
quotients of C*-algebras are usually strongly-Leibniz. To be specific,
the most important theorem of this paper states:
Theorem. Let be a C*-algebra and let be a C*-subalgebra of .
A B A
Assume that contains a bounded approximate identity for . Let L
B A
be defined as above. Then L is Leibniz, that is,
L(AC) L(A) C + A L(C)
≤ k k k k
for all A,C . If is unital and if 1 , then L is strongly-
A
∈ A A ∈ B
Leibniz, that is, L is Leibniz, L(1) = 0, and if A is invertible in
A
then
L(A−1) A−1 2L(A).
≤ k k
BEST APPROXIMATION FROM C*-SUBALGEBRAS 3
This theorem is a fairly simple consequence of a C*-algebra version
of the Arveson distance formula [2] that has been widely used in the
study of nest algebras. This C*-algebra version and the proof of the
above theorem are given in Section 3.
One can, of course, ask what happens for Banach algebras and
A
their closed subalgebras . In Section 2 we discuss a property, that we
B
callthe“same-normapproximationproperty”, thatguaranteesthatthe
seminorm on pulled back from the quotient norm on / is Leibniz.
A A B
We then show that if is a C*-algebra and is a C*-subalgebra of
A B A
that iscentral in , then hasthe same-norm approximation property
A B
in .
A
But a main application of our later discussion of best approxima-
tion is to give an example of a finite-dimensional C*-algebra and a
(non-central) unital C*-subalgebra that does not have the same-norm
approximation property. The nature of the construction strongly sug-
gests that the same-norm approximation property will usually fail un-
less one is dealing with central subalgebras.
It is a pleasure to thankMan-Duen Choi, Erik Christensen, and Dick
Kadison for very helpful conversations concerning the subject of this
paper.
2. The Leibniz inequality and Banach algebras
In this section we make some elementary observations in the setting
of Banach algebras. These observations relate the Leibniz inequality
for quotient seminorms with a property of approximations by elements
of a subalgebra. We then apply these observations to the setting of
C*-algebras.
Our experience with Hilbert spaces leads us to expect that if B is a
best approximation to A then B A . But this easily fails for the
k k ≤ k k
seminorms coming from best approximation in C*-algebras, as we will
see later. In general, the most that one can say is that
B 2 A
k k ≤ k k
because B must be at least as close to A as is 0. Easy examples show
that 2 is the best constant that holds here in general. But the stronger
inequality is desirable for our purposes, as we will show in this section,
and so we make the following definition:
Definition 2.1. Let be a normed vector space and let be a closed
A B
subspace of . Let M be the quotient norm on / , pulled back to
A A B
. That is,
A
M(A) = inf A B : B .
{k − k ∈ B}
4 MARCA.RIEFFEL
Given A , we say that A is same-norm approximatable in if for
∈ A B
every ε > 0thereis aB with B A and A B < M(A)+ε.
∈ B k k ≤ k k k − k
If every element of is same-norm approximatable in then we say
A B
that has the same-norm approximation property in .
B A
Proposition 2.2. Let be a normed algebra, let be a closed sub-
A B
algebra of , and let M be the quotient norm on / , pulled back to
A A B
. Let A . If A is same-norm approximatable in , then for every
A ∈ A B
C the Leibniz inequality
∈ A
M(AC) M(A) C + A M(C)
≤ k k k k
holds for all C , and similarly for M(CA).
∈ A
Proof. Given ε > 0, choose D such that C D < M(C) + ε,
∈ B k − k
and choose B such that B A and A B < M(A) + ε.
∈ B k k ≤ k k k − k
Then
M(AC) AC BD A B C + B C D
≤ k − k ≤ k − kk k k kk − k
(M(A)+ε) C + A (M(C)+ε).
≤ k k k k
Since ε is arbitrary, we obtain the desired inequality. A similar calcu-
lation works for M(CA). (cid:3)
We remark that the above proof is somewhat parallel to the proof of
proposition 5.2 of [18].
In the setting of normed algebras it would be interesting to have con-
ditions that also ensure that M is strongly-Leibniz, as defined earlier.
For this to make sense we need that is unital, (There may well be a
A
suitable generalization of the notion of strongly-Leibniz to non-unital
algebras, but I have not explored this possibility.)
For C*-algebras the main situation in which the same-norm approx-
imation property seems to arise is the following.
Proposition 2.3. Let be a C*-algebra and let be a C*-subalgebra
A B
of such that is central in . Then has the same-norm approx-
A B A B
imation property in .
A
More specifically, let A , let F , and let G be the radial
∈ A ∈ B
retraction of F to the ball about 0 of radius A in defined by the
k k B
continuous-function calculus for normal elements. Then A G
k − k ≤
A F , and, of course, G A .
k − k k k ≤ k k
Proof. The assertion in the second paragraph implies the assertion in
the first paragraph because it shows that whatever approximation we
have, we canalways get fromitanother that isasclose but also satisfies
the same-norm condition.
BEST APPROXIMATION FROM C*-SUBALGEBRAS 5
If is not unital, then when we adjoin an identity element to the
A A
subalgebra is still central, and distances are not changed. So we will
assume nowBthat is unital. Let ˜denote with the identity element
A B B
of adjoined (if it is not already in ).
A B
To prove the assertion of the second paragraph, notice that when
we view ˜ as the algebra of continuous functions on a compact space,
we see thBat G = HF where H = K−1 and K = max 1, F / A as
{ | | k k}
functions. Notice that K 1 will be in , as will then H 1, so that K
− B −
and H are both multipliers of . To prove the assertion of the second
B
paragraph, it suffices to show that as operators
∗ ∗
(A F) (A F) (A G) (A G),
− − ≥ − −
that is, upon simplification, that
F 2(1 H2) (1 H)(F∗A+A∗F).
| | − ≥ −
Let (π, ) be an irreducible representation of , so that π(F) is a
H A
scalar multiple, say, λ, of I , since is central. Then π(H) = µI
H H
B
where µ = 1 if λ A while µ = A / λ otherwise. Then for the
| | ≤ k k k k | |
above inequality we need to know that
λ 2(1 µ 2) (1 µ)(λ¯π(A)+λπ(A∗)).
| | −| | ≥ −
Now λ¯π(A)+λπ(A∗) has norm 2 λ A and is Hermitian, and thus
≤ | |k k
it suffices to know that
λ 2(1 µ 2) (1 µ)2 λ A .
| | −| | ≥ − | |k k
Ifµ = 1thenbothsidesare0, whileif λ > A sothatµ = A /λ < 1
| | k k k k
then a simple calculation shows that this inequality holds. Since all of
this works for any irreducible representation, we obtain the desired
(cid:3)
inequality.
Corollary 2.4. Let be a C*-algebra and let be a C*-subalgebra of
A B
such that is central in , and let M be the quotient norm on /
A B A A B
pulled back to A. Then M satisfies the Leibniz inequality.
This corollary is related to the main result of [10], which, however,
is more in the spirit of the next section.
We remark that if is unital and if 1 / , then M(1 ) = 0, so
A A
A ∈ B 6
that M can not be strongly-Leibniz.
Corollary 2.5. Let be a unital C*-algebra, let = C1 , and let
A
A B
M be the corresponding seminorm on . Then M satisfies the Leibniz
A
inequality.
6 MARCA.RIEFFEL
ThislastcorollaryshowsthatwedoobtainaLeibnizseminorminthe
m = 1 case of the situation in which = M ( ) and = M (C1 )
m m D
A D B
that was mentioned in introduction. But it does not show that the
seminorm is strongly-Leibniz. (We will see in the next section that it
is indeed strongly-Leibniz.) This corollary is also strongly related to
an often-cited result of Stampfli [23], and to, for example, equation 1.6
of [3] and section 3.1 of [5].
We remark that if = M (C) and if is the C*-subalgebra of all
2
A B
diagonalmatricesin , then hasthesame-normapproximationprop-
A B
erty in . This is seen easily by direct calculation. Thus a proper C*-
A
subalgebra does not need to be central in order to have the same-norm
approximation property. But it would be interesting to know if there
areanyotherC*-algebrasthathaveapropernon-centralC*-subalgebra
that has the same-norm approximation property. This may be related
to the fact shown in [16] that M (C) is the only C*-algebra that has a
2
proper Chebyshev C*-subalgebra of dimension strictly greater than 1.
Corollary 2.6. Let be a C*-algebra, let be a C*-subalgebra of
A B
, and let M be the quotient norm on / pulled back to A. Let
A A B
A . If there is a best approximation B to A which is normal
∈ A ∈ B
and commutes with A, then A is same-norm approximatable by (so
B
Proposition 2.2 is applicable).
Proof. By Fuglede’s theorem (theorem 4.76 of [8]) A also commutes
with B∗. Thus the C*-algebra C∗(B) generated by B is a central C*-
subalgebra ofthe C*-algebra C∗(A,B) generated by AandB. Thus we
can apply Proposition 2.3. Since C∗(B) contains a best approximation
to A in , namely B, the desired conclusion follows. (cid:3)
B
Corollary 2.7. Let be a C*-algebra, and for a natural number m
let = M ( ) andDlet be the C*-subalgebra = M (C) 1 as
m m D
A D B B ⊗
mentioned in the introduction. Then any element A of that is in the
A
commutant of in (i.e. is in I ) is same-norm approximatable
m
B A ⊗D
in (so that Proposition 2.2 is applicable).
B
Proof. It is easily seen that if A = I D for some D , and if
m
λ C is such that λ1 is a best approx⊗imation to D in C∈1D, chosen
D D
∈
via Corollary 2.5 so that λ D , then B = λI 1 is a best
m D
approximation to A in a|nd| ≤Bk k A . ⊗ (cid:3)
B k k ≤ k k
It would be interesting to have examples of Banach algebras that are
not C*-algebras but that have subalgebras that satisfy the same-norm
approximation property.
BEST APPROXIMATION FROM C*-SUBALGEBRAS 7
3. Many quotient norms of C*-algebras are
strongly Leibniz
Let be a C*-algebra and let be a C*-subalgebra of . As before
A B A
we let L denote the pull-back to of the corresponding quotient norm
A
on / , so that
A B
L(A) = inf A B : B
{k − k ∈ B}
for all A . For the purposes of my general investigation of Leibniz
∈ A
seminorms on C*-algebras, the most important theorem of this paper
is the following:
Theorem 3.1. Let be a C*-algebra and let be a C*-subalgebra of
A B
. Assume that contains a bounded approximate identity for . Let
A B A
L be defined as above. Then L is Leibniz, that is,
L(AC) L(A) C + A L(C)
≤ k k k k
for all A,C . If is unital and if 1 , then L is strongly-
A
∈ A A ∈ B
Leibniz, that is, L is Leibniz, L(1) = 0, and if A is invertible in
A
then
L(A−1) A−1 2L(A).
≤ k k
Inproposition1.2iiiof[18]itisseenthatanysupremumofa(possibly
infinite) family of Leibniz seminorms will again be Leibniz, with a
similar statement for strongly-Leibniz seminorms. In proposition 2.1 of
[18] it is seen that for any derivation d from into a normed bimodule,
A
if we set L (A) = d(A) then L is a Leibniz seminorm on , and
d d
k k A
if is unital and the bimodule is non-degenerate, then L is strongly-
A
Leibniz. Combining these two facts gives the last part of the next
theorem. Furthermore, Theorem 3.1 is an immediate consequence of
this next theorem.
Theorem 3.2. Let be a C*-algebra and let be a C*-subalgebra
A B
of . Assume that contains a bounded approximate identity for .
A B A
Let L be defined as above. For any A there is a non-degenerate
∈ A
-representation, ( ,π), of , and a Hermitian unitary operator U
∗ H A ∈
( ), such that [U,π(B)] = 0 for all B , and
L H ∈ B
L(A) = (1/2) [U,π(A)] .
k k
Furthermore, L is the supremum of all the seminorms L defined
(H,π,U)
by L (C) = (1/2) [U,π(C)] for C , as ( ,π,U) ranges over
(H,π,U)
k k ∈ A H
all non-degenerate -representations, ( ,π), of and all Hermitian
∗ H A
unitary elements U ( ) such that [U,π(B)] = 0 for all B .
∈ L H ∈ B
Since each L is Leibniz, so is L. If is unital and if 1 ,
(H,π,U) A
A ∈ B
then L is strongly-Leibniz.
8 MARCA.RIEFFEL
Theorem3.2isinturnaC*-algebraicvariationonArveson’s distance
formula for nest algebras [2], as used in the case of von Neumann alge-
bras [6]. I thank Erik Christensen for elucidating for me this formula
for the case of von Neumann algebras.
Ihave notseenagoodwaytoweaken theassumptionthat contains
B
a bounded approximate identity for .
A
Proof of theorem 3.2. Notice that if C then for any ( ,π,U) as
∈ A H
abovewe have [U,π(C)] = [U,π(C B)] 2 C B forall B ,
k k k − k ≤ k − k ∈ B
and thus
L (C) L(C)
(H,π,U)
≤
for all C . The fact that L is the supremum of the seminorms
∈ A
L thus follows from the first statement in the theorem. The fact
(H,π,U)
that L is strongly-Leibniz if is unital and if 1 then follows
A
A ∈ B
from the comments made before the statement of the theorem. Thus
it remains to prove the first statement of the theorem.
So let A be given. By scaling, we see that it suffices to treat
∈ A
the case in which L(A) = 1, and so for ease of bookkeeping we assume
this. We now use thefirst basic toolof linear approximation theory. By
applying the Hahn-Banach theorem to the image of A in / , we see
A B
that there is a linear functional, ψ, on , such that ψ = 1, ψ(A) = 1,
A k k
and ψ(B) = 0 for all B . Our proof is then based on the following
∈ B
key lemma, which is very closely related to the polar decomposition
of linear functionals in the predual of a von Neumann algebra. See
notably corollary 7.3.3 of [15]. This lemma can also be obtained by
examining the universal representation of , as suggested in the proof
A
of lemma 1 of [16].
Lemma 3.3. Let be a C*-algebra, and let ψ be a linear functional on
A
such that ψ = 1. Then there is a cyclic -representation, ( ,π,ξ),
A k k ∗ H
of , and a vector η , such that ξ = 1 = η and
A ∈ H k k k k
ψ(C) = π(C)ξ,η
h i
for all C .
∈ A
Before giving the proof of this lemma, we show how to use it to
complete the proof of Theorem 3.2. As in the first fragment of the
proof of Theorem 3.2 given above, let ψ be the linear functional on
with ψ = 1, ψ(A) = L(A) = 1, and ψ(B) = 0 for all B .
A k k ∈ B
According to Lemma 3.3 there is a non-degenerate -representation
∗
( ,π), of , and vectors ξ,η , such that ξ = 1 = η and
H A ∈ H k k k k
ψ(C) = π(C)ξ,η
h i
BEST APPROXIMATION FROM C*-SUBALGEBRAS 9
for all C . Let P denote the orthogonal projection of onto the
∈ A H
closure of π( )ξ. Since the closure of π( )ξ is clearly invariant, we
B B B
have [P,π(B)] = 0 for all B . Notice that for all B we have
∈ B ∈ B
0 = π(B)ξ,η , and thus η π( )ξ, so that P(η) = 0. Also, because
h i ⊥ B
contains a bounded approximate identity for , we see that ξ is in
B A
π( )ξ, so that P(ξ) = ξ. For notational simplicity, let us now write C
B
for π(C), etc. Then
[C,P]ξ,η = CPξ,η PCξ,η = Cξ,η = ψ(C)
h i h i−h i h i
for all C . In particular, [A,P] ψ(A) = L(A). This does
∈ A k k ≥ | |
not quite fit our needs, as we only have the general estimate that for
C and B
∈ A ∈ B
[C,P] = [C B,P] 2 C B ,
k k k − k ≤ k − k
so that [C,P] 2L(C), and we do not want the factor of 2 here.
k k ≤
(But notice the importance of having P commute with all the B’s.) To
rectify this, let U = 2P I, so that U is a Hermitian unitary, and in
−
particular, U = 1. Then we will have
k k
ψ(C) = (1/2) [C,U]ξ,η
h i
for all C , so that in particular (1/2) [U,A] L(A). But for any
∈ A k k ≥
C and B we still have, much as above,
∈ A ∈ B
[C,U] = [C B,U] 2 C B ,
k k k − k ≤ k − k
so that (1/2) [U,C] L(C). This gives the desired result. (cid:3)
k k ≤
For a related result in the finite-dimensional case see remark 3.1 of
[3].
Proof of Lemma 3.3. For the reader’s convenience we now give a proof
ofLemma3.3thatusesasitsbiggesttooljusttheJordandecomposition
of Hermitian linear functionals into differences of two positive linear
functionals,
Let and ψ be as in the statement of Lemma 3.3. If is not
A A
unital, adjoin an identity element in the usual way so as to obtain a
C*-algebra. By the technical step in the proof of the Hahn-Banach
theorem we can extend ψ to the resulting C*-algebra, with no increase
in its norm. Until the end of the proof we will now assume that is
A
unital.
Form the algebra M ( ) of 2 2 matrices with entries in , with
2
A × A
its unique C*-algebra structure. Define on M ( ) a linear functional,
2
A
ψ , by
2
A C
ψ = (1/2)(ψ(B) +ψ(C∗)).
2(cid:18)(cid:18)B D(cid:19)(cid:19)
10 MARCA.RIEFFEL
Then ψ is a Hermitian linear functional, that is, ψ (T∗) = ψ(T) for
2 2
any T M ( ). Note also that ψ = 1.
2 2
Letψ∈ = (1A/2)(φ+ φ−)bethekJorkdandecompositionofψ (theorem
2 2
4.3.6 and remark 4.3−.12 of [13]), so that φ+ and φ− are positive linear
functionals on M ( ) such that ψ+ + φ− = 2 ψ = 2. Note
2 2
A k k k k k k
that ψ (I ) = 0, where I is the identity element of M ( ). It follows
2 2 2 2
that φ+(I ) = φ−(I ), so that φ+ = φ− = 1. ThusAboth φ+ and
2 2
φ− are states on M ( ). k k k k
2
Let ( +,π+,ξ+) aAnd ( −,π−,ξ−) be the GNS representations for
φ+ andHφ−. Set H
= + − and π = π+ π−,
2 2
H H ⊕H ⊕
and set
ξ = (ξ+ ξ−)/√2 and η = (ξ+ ξ−)/√2.
2 2
⊕ ⊕−
Note that ξ = 1 = η . Then for any T M ( ) we have
2 2 2
k k k k ∈ A
ψ (T) = (1/2)(φ+(T) φ−(T))
2
−
= (1/2)( π+(T)ξ+,ξ+ π−(T)ξ−,ξ− )
h i−h i
= π (T)ξ ,η .
2 2 2
h i
Now for any A we have
∈ A
0 0 0 1 A 0
= .
(cid:18)A 0(cid:19) (cid:18)1 0(cid:19)(cid:18)0 0(cid:19)
Thus
0 0 A 0 0 1
ψ(A) = ψ = π ξ , π η .
2(cid:18)(cid:18)A 0(cid:19)(cid:19) h 2(cid:18)(cid:18)0 0(cid:19)(cid:19) 2 2(cid:18)(cid:18)1 0(cid:19)(cid:19) 2i
Let P = π ((1 0)), and then set = P . Notice that π ((A 0))
2 0 0 H H2 2 0 0
carries into itself for every A . We obtain in this way a unital
H ∈ A
*-representation, π, of on . Set ξ = Pξ and η = Pπ ((0 1))η .
A H 2 2 1 0 2
From the equation displayed just above we see that
ψ(A) = π(A)ξ,η
h i
for all A . From the definitions given above it is easily seen that
∈ A
ξ 1 and η 1. One can restrict π to the cyclic -subspace of
k k ≤ k k ≤ A H
generatedbyξ. Thenfromthefactthat ψ = 1itfollowsthat ξ = 1
k k k k
and η = 1, as desired. If an identity element had been adjoined to
k k
the original , one can restrict π further to the cyclic -subspace of
generatedAby ξ. A (cid:3)
H