a r
X
i
v
:08
8
.
2
3
7
v
1
[
you tubem
a
t h
.
F
A ]
14
A u
g
复合句
2
好听的经典英文歌
8
WEAK*CONTINUOUS STATES ON BANACH ALGEBRAS BOJAN MAGAJNA Abstract.We prove that if a unital Banach algebra A is the dual of a Banach space A ♯,then the t of weak*continuous states is weak*den in the t of all states on A .Further,weak*continuous states linearly span A ♯.1.Introduction An important tool in functional analysis is Goldstine’s theorem [5,3.27],which says that for a dual Banach space A the unit ball of its predual A ♯is weak*den in the unit ball B A ♯of the dual A ♯of A .Given a norm one element x ∈A ,we may consider the t of ‘states’S x (A )={ρ∈B A ♯:ρ(x )=1}and the subt of ‘normal states’S x n (A )=S x ∩A ♯.In general S x n (A )need not be weak*den in S x (A ),for S x n (A )may even be empty if x does not achieve its norm as a functional on A ♯.In this note we show that if A is a unital Banach algebra and x =1is the unit of A (with 1 =1),then the t of normal states S n (A ):=S 1n (A )is weak*den in the t S (A ):=S 1(A )of all states.Using this,we also show that S n (A )spans the predual of A .Of cour,all this is well known for von Neumann algebras.That S (A )spans A ♯for any unital Banach algebra A was proved by Moore [7]([10]and [1]contain simpler proofs).Our method is bad on a consideration of dissipative elements.Recall that an element a ∈A is dissipative if its numerical range W (a ):={ρ(a ):ρ∈S (A )}is c
ontained in the left half-plane Re z ≤0.To show that the t D A of all such elements is weak*clod,if A is a dual space,we will need a suitable metric char-acterization of dissipative elements (Lemma 2.1below).A similar,but not the same,characterization was obrved in [2]for C ∗-algebras;however,the argument from [2]does not apply to Banach algebras.For each a ∈D A the element 1−a is invertible since its numerical range (hence also its spectrum)is contained in the half-plane Re z ≥1.We will only need the estimate
(1.1) (1−a )−1 ≤1(a ∈D A ),
which is known from the Hille–Yosida theorem on generators of operator mi-groups.In our prent context it can easily be derived by applying the well known estimate e ta
≤1,t ≥0(e [4,p.55]or [11,A13(4)])to the integral reprenta-
tion (1−a )−1= ∞0e −t (1−a )dt ,which can be verified directly.
2BOJAN MAGAJNA
2.Dissipative elements and normal states
Lemma 2.1.An element a of a unital Banach algebra A is dissipative if and only if
(2.1) 1+ta ≤1+t 2 a 2for all t ≥0.
新加坡留学条件In particular,if a ≤1,then a ∈D A if and only if 1+ta ≤1+t 2for all t ≥0.Proof.If a satisfies (2.1),then for every state ρ∈S (A )and t >0we have that |1+tρ(a )|2=|ρ(1+ta )|2
≤ 1+ta 2≤(1+t 2 a 2)2.This implies that 2Re ρ(a )≤t (2 a 2−|ρ(a )|2)+t 3 a 4,hence (letting t →0)Re ρ(a )≤0.
For a proof of the rever direction,note that by (1.1)each a ∈D A satisfies
1+a = (1−a )−1(1−a 2) ≤ 1−a 2 ≤1+ a 2.
But,since ta is also dissipative if t ≥0,we may replace a by ta in the last inequality,which yields (2.1).The last ntence of the lemma follows now easily.
In C ∗-algebras the estimate (2.1)can be improved to 1+ta 2
≤1+t 2 a 2(a
∈D A ,t ≥0),a conquence of the C ∗-identity [2].This sharper estimate holds also in some other natu
ral examples of Banach algebras,but the author does not know if it holds in general.Since this topic is not esntial for our purpos here,we will postpone further discussion on it to the end of the paper.
Theorem 2.2.If a unital Banach algebra A is a dual Banach space,then D A is a weak*clod subt of A .Moreover,S n (A )is weak*den in S (A ).
Proof.The proof is the same as for operator spaces [2].Since it is very short,we will sketch it here for completeness.Since D A is convex and tD A
⊆D A if t ≥0,
to prove that D A is weak*clod,it suffices to show that the interction of D A with the clod unit ball of A is weak*clod ([5,4.44]).But this follows immediately from Lemma 2.1.
Denote by A ♯+the t of all nonnegative multiples of states on A and by (D A )◦the t of all ρ∈A ♯such that Re ρ(a )≤0for all a ∈D A .Clearly A ♯+
⊆(D A )◦.To prove that A ♯+=(D A )◦,let ρ∈(D A )◦.Since it 1∈D A for all t ∈R and −1∈D A ,it follows that ρ(1)≥0.Since a − a 1∈D A for each a ∈A ,we have that Re ρ(a )≤ a ρ(1).Replacing in this inequality a
generational
by zx for all z ∈C with |z |=1,it follows that |ρ(a )|≤ a ρ(1),hence ρ∈A ♯+.Now put A +♯=A ♯
∩A ♯+
and (D A )◦=(D A )◦∩A ♯.Then A +
♯=(D A )◦.Since
D A is weak*clod,a bipolar type argument shows that D A =((D A )◦)◦and that (D A )◦is weak*den in (D A )◦.This means that A +♯is weak*den in A ♯+.Now it follows easily that S n (A )is weak*den in S (A ). Corollary 2.3.Let A be as in Theorem 2.2.For every clod convex subt C of C the t A C ={a ∈A :W (a )⊆C }is weak*clod in A .
Proof.Since C is the interction of half-planes containing it,this follows from the fact that A {Re z ≤0}=D A is weak*clod. Theorem 2.4.If A is as in Theorem 2.2,then S n (A )spans A ♯.Each ω∈A ♯with ω <(e √
WEAK*CONTINUOUS STATES ON BANACH ALGEBRAS 3
amount
Proof.Put S =S (A )and S n =S n (A ).For a subt V of A define the polar V ⋄by V ⋄={ρ∈A ♯:|Re ρ(a )|
≤1∀a ∈V }.In the same way define also polars of subts of A ♯and ‘prepolars’V ⋄of subts of A or A ♯.Let U =S ⋄.Then U is the t of all elements a ∈A with the numerical range contained in the strip |Re z |≤1,hence U is weak*clod by Corollary 2.3.Since S n is weak*den in S n by Theorem 2.2and U =S ⋄,it follows that U =S ⋄n ,hence by the bipolar theorem U ⋄=
co(−S n
∪S n ))V ⋄is the closure of the t S 0:=co(S n ∪(−S n )∪(iS n )∪(−iS n )).On the other hand,by the definition of U ,V is just the t of all a ∈A with the numerical range W (a )contained in the square [−1,1]×[−i,i ].Since for every a ∈A the inequality a ≤ew (a )holds,where w (a )is the numerical radius of a (e [4]or [8,2.6.4]),it
follows that V is contained in the clod ball (√2e .Conquently 2e )−1B A
♯.Let T ={tω:ω∈S n ,t ∈[0,1]}.Since S n is norm clod and bounded and the
interval [0,1]is compact,it is not hard to verify that T is norm clod.Further,since S n
⊆T is convex and tT ⊆T for all t ∈[0,1],it follows from the definition of S 0that S 0⊆T 0:=T −T +iT −iT .Therefore we conclude from the previous paragraph that B A ♯⊆
√T 0.Thus,given ω∈A ♯and δ∈(0,1),there exists ω0∈ ω √2eT 0such
that ω−ω0−ω1 ≤δ2.Continuing,we find a quence of
functionals ωn ∈δn √2e ( ω ρ0+∞ n =1
δn ρn ),
where ρn
∈T 0.By the definition of T 0we have that ρn =ρn,1
−ρn,2+i (ρn,3−ρn,4),where ρn,j ∈T =co({0}∪S n ).Put γ= ω +
∞n =1δn = ω +δ(1−δ)−1.Since
T is clod and convex,ψj :=γ−1( ω ρ0,j + ∞n =1δn ρn,j )∈T for each j .From
(2.2)we have now
(2.3)ω=(√
2)−1,we may choo δso
small that γ≤(e √
4BOJAN MAGAJNA
Proof.If(2.4)holds,then for eachρ∈S(A)
(2.5)|ρ(h)+it|2=|ρ(h+it1)|2≤ h 2+t2(t∈R),
which implies(by letting t→∞)thatρ(h)∈R,hence h is hermitian.Converly, if h is hermitian then(2.5)holds.But,by a result of Sinclair[9]the norm of an element of the form a=h+λ1,where h is hermitian andλ∈C,is equal to the spectral radius r(a),hence also to the numerical radius w(a)(since in general r(a)≤w(a)≤ a ).Thus,taking in(2.5)the supremum over all statesρ∈S(A), we get(2.4).
The above argument also shows that a contraction is hermitian if and only if h+it1 2≤1+t2for all t≥0.This imlies that the interction A h∩B A is weak* clod,hence A h is weak*clod by the Krein–Smulian theorem.
The estimate
notyet
(2.6) 1+a 2≤1+ a 2,
which holds for all dissipative elements in C∗-algebras,holds in general Banach algebras at least for dissipative elements of a special form.For example,using the fact that for hermitian elements the norm is equal to the spectral radius,it can be shown that(2.6)holds for dissipative hermitian elements.
Proposition2.6.In any unital Banach algebra A each dissipative element of the form a=−tp,where p is an idempotent and t∈(0,∞),satisfies the estimate(2.6). Proof.By the well known criterion[4,p.55]−p is dissipative if and only if e−tp ≤1for all t≥0.From the Taylor ries expansion of e−tp we compute that e−tp= (1−s)q+s1,where q=1−p and s=e−t.So,−p is dissipative if and only if (1−s)q+s1 ≤1for all s∈[0,1],which is equivalent to q ≤1.Since the norm of any nonzero idempotent is at least1,we conclude that−p(hence also−sp,if s>0)is dissipative if and only if q =1
yui natsukiTo prove(2.6),where a=−tp is assumed dissipative(thus q =1),consider first the ca when t∈(0,1].Put s=1−t and note that
1−tp 2= s1+(1−s)q 2≤(s+(1−s) q )2=1≤1+t2 p 2.
On the other hand,if t>1,then
1−tp 2= q+(1−t)p 2≤(1+(t−1) p )2
=1+t2 p 2−2t p ( p −1)− p (2− p )≤1+t2 p 2, since1≤ p = 1−q ≤2.
Question.Do all dissipative elements in each unital Banach algebra satisfy(2.6)?
References
[1]L.Asimow and A.J.Ellis,On hermitian functionals on unital Banach algebras,Bull.London
Math.Soc.4(1972),333–336.
[2]D.P.Blecher and B.Magajna,Dual operator systems,arXiv:0807.4250v1[math.OA],July
2008.
[3]D.P.Blecher and M.Neal,Metric characterizations of isometries and of unital operator包装设计需求
spaces and systems,arXiv:0805.2166v2[math.OA],May2008.
[4]F.F.Bonsall and J.Duncan,Complete normed algebras,Springer-Verlag,Heidelberg,1973.
[5]M.Fabian et al.,Functional analysis and infinite-dimensional geometry,CMS Books in
Math.,Springer-Verlag,New York,2001.
[6]P.D.Lax,Functional analysis,Pure and Appl.Math.,Wiley-Interscience,New York,2002.
WEAK*CONTINUOUS STATES ON BANACH ALGEBRAS5 [7]R.T.Moore,Hermitian functionals on B-algebras and duality characterizations of C∗-
algebras,Trans.Amer.Math.Soc.162(1971),253–266.
[8]T.Palmer,Banach algebras and the general theory of∗-algebras,Vol.I,Algebras and Banach
algebras,Encyclopedia of Math.and its appl.49,Cambridge Univ.Press,Cambridge,1994.
[9]A.M.Sinclair,The norm of a hermitian element in a Banach algebra,Proc.Amer.Math.
Soc.28(1971),446–450.
[10]A.M.Sinclair,The states of a Banach algebra generate the dual,Proc.Edinburgh Math.
美国公立高中排名
Soc.17(1971),341–344.
[11]M.Takesaki,Theory of operator algebras III,Springer-Verlag,New-York,2001.
[12]I.Vidav,Eine metrische Kennzeichnung der lbstadjungierten Operatoren(German),Math.
Z.66(1956),121–128.
Department of Mathematics,University of Ljubljana,Jadranska21,Ljubljana1000, Slovenia
E-mail address:Bojan.Magajna@fmf.uni-lj.si
