Best approximations, distance formulas and orthogonality in C*-algebras
aa r X i v : . [ m a t h . OA ] J a n BEST APPROXIMATIONS, DISTANCE FORMULAS ANDORTHOGONALITY IN C ∗ -ALGEBRAS PRIYANKA GROVER AND SUSHIL SINGLA
Abstract.
For a unital C ∗ -algebra A and a subspace B of A , a charac-terization for a best approximation to an element of A in B is obtained.As an application, a formula for the distance of an element of A from B has been obtained, when a best approximation of that element to B exists. Further, a characterization for Birkhoff-James orthogonality ofan element of a Hilbert C ∗ -module to a subspace is obtained. Introduction
Let A be a unital C ∗ -algebra over F (= R or C ) with the identity element1 A . The C ∗ -subalgebras of A are assumed to contain 1 A . For a ∈ A and B a subspace of A , dist( a, B ) denotes inf {k a − b k : b ∈ B} . An element b ∈ B is said to be a best approximation to a in B if k a − b k = dist( a, B ). It is awell known fact that b is a best approximation to a in B if and only if thereexists a functional ψ ∈ A ∗ such that ψ ( a − b ) = dist( a, B ) and ψ ( b ) = 0 forall b ∈ B (see [15, Theorem 1.1]).Let ( C ( X ) , k · k ∞ ) be the C ∗ -algebra of real or complex continuous functionson a compact Hausdorff space X , where k f k ∞ = sup x ∈ X | f ( x ) | . It wasproved in Theorem 1.3 of [15] that if f ∈ C ( X ) and B is a subspace of C ( X ), then g is a best approximation to f in B if and only if there existsa regular Borel probability measure µ on X such that the support of µ iscontained in the set { x ∈ X : | ( f − g )( x ) | = k f − g k ∞ } and R X ( f − g ) h dµ =0 for all h ∈ B . The condition that the support of µ is contained in the set { x ∈ X : | ( f − g )( x ) | = k f − g k ∞ } is equivalent to R X | f − g | dµ = k f − g k ∞ .A positive linear map from A to another C ∗ -algebra A is a linear map thatmaps positive elements of A to positive elements of A . For F = C , a state on A is a positive linear functional φ on A such that φ (1 A ) = 1. For F = R , anadditional requirement for φ to be a state is that φ ( a ∗ ) = φ ( a ) for all a ∈ A .Let S A denotes the set of states on A . Using Riesz Representation Theorem,the above characterization for best approximation in C ( X ) is equivalent tosaying that there exists φ ∈ S C ( X ) such that(1) φ ( | f − g | ) = k f − g k ∞ and φ (( f − g ) h ) = 0 for all h ∈ B . Mathematics Subject Classification.
Primary 46L05, 46L08, 41A50; Secondary46B20, 41A52, 47B47.
Key words and phrases.
Best approximation, conditional expectation, Birkhoff-Jamesorthogonality, cyclic representation, state, Hilbert C ∗ -module. For a ∈ A and B a subspace of A , a is said to be Birkhoff-James orthogonal to B (or B -minimal ) if k a k ≤ k a + b k for all b ∈ B . Note that this isequivalent to saying that 0 is a best approximation to a in B . It was provedin Theorem 2 of [16] that 0 is a best approximation to an element a of acomplex C ∗ -algebra A in C A if and only if there exists φ ∈ S A such that φ ( a ∗ a ) = k a k and φ ( a ) = 0. Theorem 6.1 in [13] shows that if B is a C ∗ -subalgebra containing 1 A of a complex C ∗ -algebra A and if 0 is a bestapproximation to a Hermitian element a of A in B , then there exists φ ∈ S A such that φ ( a ) = k a k and φ ( ab + b ∗ a ) = 0 for all b ∈ B . In Proposition 4.10of [4], it was proved that for any elements a and b of a complex C ∗ -algebra A , 0 is a best approximation to a in C b if and only if there exists φ ∈ S A such that φ ( a ∗ a ) = k a k and φ ( a ∗ b ) = 0. The main result of this articleshows the existence of such a state for any element a and for any subspace B of a C ∗ -algebra over F . Theorem 1.1.
Let a ∈ A . Let B be a subspace of A . Then b is a bestapproximation to a in B if and only if there exists φ ∈ S A such that (2) φ (( a − b ) ∗ ( a − b )) = k a − b k and φ ( a ∗ b ) = φ ( b ∗ b ) for all b ∈ B . For φ ∈ S A and a , a ∈ A , define h a | a i φ = φ ( a ∗ a ). This is a semi-innerproduct on A . Let k a k φ = h a | a i / φ . In this notation, the above theoremsays that b is a best approximation to a in B if and only if there exists φ ∈ S A such that k a − b k φ = k a − b k and h a − b | b i φ = 0 for all b ∈ B . We note that (2) is a Pythagoras theorem in the semi-inner product space( A , h·|·i φ ). Consider the triangle with vertices 0 , a, b in ( A , h·|·i φ ). If a / ∈ B ,then (2) gives that k a k φ = k b k φ + k a − b k and h a − b | b i φ = 0 for all b ∈ B .If k b k φ = 0, then we have k a k φ = k a − b k . This means that the length ofthe base and the length of the perpendicular are 0 and k a − b k , respectively.Suppose k b k φ = 0. Let θ a ,a φ = cos − h a | a i φ k a k φ k a k φ ! be the angle betweenthe vectors a and a in ( A , h·|·i φ ), when k a k φ , k a k φ = 0. Then we have k a − b k φ = k a − b k and θ a − b ,bφ = π/ b ∈ B . In particular, the abovetriangle becomes a right angled triangle and the length of the perpendicularis k a − b k .As a consequence, we obtain a distance formula of an element a ∈ A from asubspace B of A . Corollary 1.2.
Let a ∈ A . Let B be a subspace of A . If b is a bestapproximation to a in B , then (3)dist( a, B ) = max { φ ( a ∗ a ) − φ ( b ∗ b ) : φ ∈ S A and φ ( a ∗ b ) = φ ( b ∗ b ) for all b ∈ B} . A special case of the above corollary is the below result by Williams [16].He proved that for a ∈ A ,(4) dist( a, C A ) = max { φ ( a ∗ a ) − | φ ( a ) | : φ ∈ S A } . EST APPROXIMATIONS, DISTANCE FORMULAS AND ORTHOGONALITY IN C ∗ -ALGEBRAS3 See [14, Theorem 3.10] for a different proof of (4). For n × n complexmatrices, a different proof has also been given in [2, Theorem 9].As a direct consequence of Theorem 1.1, we get the following characteriza-tion of Birkhoff-James orthogonality to a subspace in a C ∗ -algebra. Corollary 1.3.
Let a ∈ A . Let B be a subspace of A . Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ S A such that φ ( a ∗ a ) = k a k and φ ( a ∗ b ) = 0 for all b ∈ B . Geometrically, this says that a is Birkhoff-James orthogonal to B if and onlyif there exists φ ∈ S A and a corresponding semi-inner product h·|·i φ on A such that k a k φ = k a k and a is perpendicular to B in ( A , h·|·i φ ).In Section 2, we give the proofs of Theorem 1.1 and Corollary 1.2. InSection 3, we give some other applications of Theorem 1.1. In Theorem3.1, we show that 0 is a best approximation to a in B if and only if 0 is abest approximation to a ∗ a in a ∗ B . In Theorem 3.4, it is shown that for anyelement a ∈ A and a subspace B of A , there exists a cyclic representation( H , π, ξ ) of A and a unit vector η ∈ H such that dist( a, B ) = h η | π ( a ) ξ i and h η | π ( b ) ξ i = 0 for all b ∈ B . In Theorem 3.5, a characterization forBirkhoff-James orthogonality of an element of a Hilbert C ∗ -module to asubspace is given. It is proved that an element e of a Hilbert C ∗ -module E over A is Birkhoff-James orthogonal to a subspace B of E if and only ifthere exists φ ∈ S A such that φ ( h e, e i ) = k e k and φ ( h e, b i ) = 0 for all b ∈ B .In [14], it was desired to have the generalization of distance formula (4) interms of conditional expectations from A to B . In Section 4, we make someremarks on our progress towards obtaining this. Corollary 1.2, Corollary1.3, Theorem 3.5 and Equation (15) are mentioned in the survey article [9].We provide the complete details here.2. Proofs
Few notations are in order. Let H be a Hilbert space over F . The innerproduct is assumed to be conjugate linear in the first coordinate and linearin the second coordinate. Let B ( H ) be the C ∗ -algebra of bounded F -linearoperators on H . The symbol I denotes the identity in B ( H ). The triple( H , π, ξ ) denotes a cyclic representation of A where k ξ k = 1, π : A → B ( H )is a ∗ -algebra map satisfying π (1 A ) = I and closure of { π ( a ) ξ : a ∈ A} is H . Proof of Theorem 1.1 If φ is a state such that (2) holds, then for every b ∈ B , k a − b k = φ (( a − b ) ∗ ( a − b )) ≤ φ (( a − b ) ∗ ( a − b )) + φ ( b ∗ b )= φ (( a − b − b ) ∗ ( a − b − b )) ≤ k a − b − b k . So b is a best approximation to a in B . For the other side, first let us assumethat A is a complex C ∗ -algebra. By the Hahn-Banach theorem, there exists ψ ∈ A ∗ such that k ψ k = 1, ψ ( a − b ) = dist( a, B ) = k a − b k and ψ ( b ) = 0 GROVER AND SINGLA for all b ∈ B . By Lemma 3.3 of [13], there exists a cyclic representation( H , π, ξ ) of A and a unit vector η ∈ H such that(5) ψ ( c ) = h η | π ( c ) ξ i for all c ∈ A . Now ψ ( a − b ) = h η | π ( a − b ) ξ i = k a − b k . So by using the condition forequality in Cauchy-Schwarz inequality, we obtain k a − b k η = π ( a − b ) ξ .Equation (5) gives ψ ( c ) = 1 k a − b k h π ( a − b ) ξ | π ( c ) ξ i for all c ∈ A . Therefore(6) h π ( a − b ) ξ | π ( a − b ) ξ i = k a − b k and(7) h π ( a − b ) ξ | π ( b ) ξ i = 0 for all b ∈ B . Define φ ∈ A ∗ as φ ( c ) = h ξ | π ( c ) ξ i . Then φ ∈ S A and by (6) and (7), weobtain (2).Next, let A be a real C ∗ -algebra. Let A c be the complexification of ( A , k · k )with the unique norm k · k c such that ( A c , k · k c ) is a C ∗ -algebra and thenatural embedding of A into A c is an isometry [7, Corollary 15.4]. Fromthe above case, there exists ψ ∈ S A c such that ψ (( a − b ) ∗ ( a − b )) = k a − b k and ψ ( a ∗ b ) = ψ ( b ∗ b ) for all b ∈ B . Let φ = Re ψ | A . Then φ ∈ S A , φ (( a − b ) ∗ ( a − b )) = k a − b k and φ ( a ∗ b ) = φ ( b ∗ b ) for all b ∈ B . (cid:3) Another proof of Theorem 1.1, in the case when A is a complex C ∗ -algebra,can be given as follows. The importance of this approach is that it indicatesthat proving the theorem when B is a one dimensional subspace is sufficient.Since b is a best approximation to a in B , 0 is a best approximation to a − b in B . So without loss of generality, we assume b = 0. For b ∈ B , we have k a k ≤ k a + λb k for all λ ∈ C . By Proposition 4.1 of [4], there exists φ b ∈ S A such that φ b ( a ∗ a ) = k a k and φ b ( a ∗ b ) = 0. Let N = { αa ∗ a + β A + a ∗ b : α, β ∈ C , b ∈ B} , the subspace generated by a ∗ a , 1 A and a ∗ B . Define ψ : N −→ C as ψ ( αa ∗ a + β A + a ∗ b ) = α k a k + β for all α, β ∈ C and b ∈ B . To see that ψ is well defined, note that for any b ∈ B we have φ b ( αa ∗ a + β A + a ∗ b ) = α k a k + β . Since k φ b k = 1, we get(8) | α k a k + β | ≤ k αa ∗ a + β A + a ∗ b k . Thus αa ∗ a + β A + a ∗ b = 0 implies α k a k + β = 0 . Clearly ψ is a linear mapand equation (8) shows that k ψ k ≤
1. Since ψ (1 A ) = 1, we have k ψ k = 1.By the Hahn-Banach theorem, there exists a linear functional φ : A → C such that k φ k = 1 and φ | N = ψ . Since k φ k = 1 = φ (1 A ), using TheoremII.6.2.5(ii) of [5], we get that φ ∈ S A . By definition, φ satisfies the requiredconditions. Proof of Corollary 1.2
Let φ ∈ S A be such that φ ( a ∗ b ) = φ ( b ∗ b ) for all b ∈ B . In particular we have φ ( a ∗ b ) = φ ( b ∗ b ). So φ (( a − b ) ∗ ( a − b )) = φ ( a ∗ a ) − φ ( b ∗ b ) . Since φ (( a − b ) ∗ ( a − b )) ≤ k a − b k = dist( a, B ) , wehave φ ( a ∗ a ) − φ ( b ∗ b ) ≤ dist( a, B ) . EST APPROXIMATIONS, DISTANCE FORMULAS AND ORTHOGONALITY IN C ∗ -ALGEBRAS5 This givessup { φ ( a ∗ a ) − φ ( b ∗ b ) : φ ∈ S A , φ ( a ∗ b ) = φ ( b ∗ b ) for all b ∈ B} ≤ dist( a, B ) . By Theorem 1.1, there exists φ ∈ S A such thatdist( a, B ) = φ ( a ∗ a ) − φ ( b ∗ b ) and φ ( a ∗ b ) = φ ( b ∗ b ) for all b ∈ B . This completes the proof. (cid:3) Applications
An interesting fact arises out of Corollary 1.3, which is worth noting sepa-rately.
Theorem 3.1.
Let a ∈ A . Let B be a subspace of A . Then a is Birkhoff-James orthogonal to B if and only if a ∗ a is Birkhoff-James orthogonal to a ∗ B .Proof. First let a is Birkhoff-James orthogonal to B . Then by Corollary 1.3,there exists φ ∈ S A such that φ ( a ∗ a ) = k a k and φ ( a ∗ b ) = 0 for all b ∈ B .So for b ∈ B , φ ( a ∗ a + a ∗ b ) = k a k . Since k φ k = 1, we get k a ∗ a k = k a k ≤k a ∗ a + a ∗ b k . Conversely, suppose a ∗ a is Birkhoff-James orthogonal to a ∗ B ,that is, k a ∗ a k ≤ k a ∗ a + a ∗ b k for every b ∈ B . This implies k a k ≤ k a ∗ kk a + b k and thus k a k ≤ k a + b k for all b ∈ B . (cid:3) We now show that Theorem 1 of [8] can also be proved using Corollary1.3. We first prove the following lemma, which is of independent interest.The proof of the lemma is along the same lines as a portion of the proof ofTheorem 1 of [3]. For u, v ∈ H , u ¯ ⊗ v will denote the finite rank operator ofrank one on H defined as u ¯ ⊗ v ( w ) = h v | w i u for all w ∈ H . Lemma 3.2.
Let A ∈ B ( H ) . Let T be a positive trace class operator with k T k = 1 and tr( AT ) = k A k . Then there is an at most countable indexset J , a set of positive numbers { s j : j ∈ J } and an orthonormal set { u j : j ∈ J } ⊆ Ker ( T ) ⊥ such that (i) P j ∈J s j = 1 , (ii) Au j = k A k u j for each j ∈ J , (iii) T = P j ∈J s j u j ¯ ⊗ u j .Proof. Using Corollary 5.4 of [6, Ch. II], there exists a sequence of realnumbers s , s . . . with orthonormal basis { u , u , . . . } of Ker( T ) ⊥ such that T = ∞ P i =1 s i u i ¯ ⊗ u i . Since T is positive, s i are non-negative. And k T k = 1implies ∞ P i =1 s i = 1. Now AT = ∞ P i =1 s i Au i ¯ ⊗ u i . Let J = { i ∈ N : s i = 0 } . Then GROVER AND SINGLA P j ∈J s j = 1 and AT = P j ∈J s j Au j ¯ ⊗ u j . So tr( AT ) = P j ∈J s j tr( Au j ¯ ⊗ u j ) = P j ∈J s j h u j | Au j i .Now k A k = tr( AT ) = X j ∈J s j h u j | Au j i = (cid:12)(cid:12)(cid:12)(cid:12) X j ∈J s j h u j | Au j i (cid:12)(cid:12)(cid:12)(cid:12) ≤ X j ∈J s j (cid:12)(cid:12)(cid:12)(cid:12) h u j | Au j i (cid:12)(cid:12)(cid:12)(cid:12) ≤ X j ∈J s j k Au j k≤ X j ∈J s j k A k = k A k . So X j ∈J s j (cid:12)(cid:12)(cid:12)(cid:12) h u j | Au j i (cid:12)(cid:12)(cid:12)(cid:12) = X j ∈J s j k Au j k = k A k . Therefore0 = X j ∈J s j (cid:18) k A k − (cid:12)(cid:12)(cid:12)(cid:12) h u j | Au j i (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = X j ∈J s j (cid:18) k Au j k − (cid:12)(cid:12)(cid:12)(cid:12) h u j | Au j i (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . Since s j > j ∈ J , we get(9) k A k = (cid:12)(cid:12)(cid:12)(cid:12) h u j | Au j i (cid:12)(cid:12)(cid:12)(cid:12) = k Au j k for all j ∈ J . By the condition of equality in Cauchy-Schwarz inequality, for every j ∈ J there exists α j ∈ C such that α j Au j = u j . And using (9), we get Au j = k A k u j . This completes the proof. (cid:3) Let M n ( F ) be the C ∗ -algebra of n × n matrices with entries in F . A densitymatrix A ∈ M n ( F ) is a positive element in M n ( F ) with tr( A ) = 1. A differentproof of Theorem 1 in [8] follows. Theorem 3.3. [8, Theorem 1]
Let A ∈ M n ( F ) . Let m ( A ) be the multiplicityof the maximum singular value k A k of A . Let B be a subspace of M n ( F ) .Then A is Birkhoff-James orthogonal to B if and only if there exists a densitymatrix T ∈ M n ( F ) of rank at most m ( A ) such that A ∗ AT = k A k T and tr( B ∗ AT ) = 0 for all B ∈ B .Proof. By Corollary 1.3, there exists a density matrix T such that tr( A ∗ AT ) = k A k and tr( B ∗ AT ) = 0 for all B ∈ B . Using Lemma 3.2, there ex-ists s , . . . , s m and a set of orthonormal vectors { u , . . . , u m } such that m P j =1 s j = 1, A ∗ Au j = k A k u j for every j = 1 , . . . , m and T = m P j =1 s j u j ¯ ⊗ u j .Clearly rank T ≤ m ≤ m ( A ) and A ∗ AT = k A k T . (cid:3) It is worth noting that from the proof of Theorem 3.3, we get that A ∗ AT = k A k T is equivalent to tr( A ∗ AT ) = k A k , where A, T ∈ M n ( F ) and T is adensity matrix. This supplements Remark 1 of [8].Next we note that the idea of the proof of Theorem 1.1 also proves thefollowing generalization of Corollary 2.8 in [1]. EST APPROXIMATIONS, DISTANCE FORMULAS AND ORTHOGONALITY IN C ∗ -ALGEBRAS7 Theorem 3.4.
Let a ∈ A . Let B be a subspace of A . Then there existsa cyclic representation ( H , π, ξ ) of A and a unit vector η ∈ H such that dist( a, B ) = h η | π ( a ) ξ i and h η | π ( b ) ξ i = 0 for all b ∈ B .Proof. By the Hahn-Banach theorem, there exists ψ ∈ A ∗ such that k ψ k = 1, ψ ( a ) = dist( a, B ) and ψ ( b ) = 0 for all b ∈ B . By Lemma 3.3 of [13], thereexists a cyclic representation ( H , π, ξ ) of A and a unit vector η ∈ H suchthat ψ ( c ) = h η | π ( c ) ξ i for all c ∈ A . (cid:3) It was shown in [3] that for any A ∈ M n ( C )dist( A, C A ) = max {|h y | Ax i| : x, y ∈ C n , k x k = k y k = 1 and x ⊥ y } . Using Theorem 3.4, we obtain a similar formula for dist( a, B ), in the generalcase of a unital C ∗ -algebra A and C A replaced with any subspace B . Wehavedist( a, B ) = max (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) h η | π ( a ) ξ i (cid:12)(cid:12)(cid:12)(cid:12) : ( H , π, ξ ) is a cyclic representation of A , η ∈ H , (10) k η k = 1 and h η | π ( b ) ξ i = 0 for all b ∈ B (cid:27) . Under the restriction that best approximation to a in B exists, the aboveformula was obtained in [9, Theorem 4.3]. Another formula for dist( a, B )when B is a C ∗ -subalgebra of A was proved in Theorem 3.2 of [13]. For moredistance formulas, see [3] and [8] for a discussion in M n ( C ), [1] and [12] for B ( H ) and [1] for general complex C ∗ -algebras and Hilbert C ∗ -modules overa complex C ∗ -algebra.A Hilbert C ∗ -module E over A is a right A -module with a function h· , ·i : E × E → A , known as A -valued semi-inner product, with the followingproperties for ξ, η, ζ ∈ E , a ∈ A , λ ∈ C :(1) h ξ, η + ζ i = h ξ, η + ζ i and h ξ, λη i = λ h ξ, η i ,(2) h ξ, ηa i = h ξ, η i a ,(3) h ξ, η i = h η, ξ i ∗ ,(4) h ξ, ξ i is a positive element of A .Let K be a Hilbert space. Let B ( H , K ) denotes the space of bounded F -linear operators from H to K . It is a Hilbert C ∗ -module over B ( H ) with h A, B i = A ∗ B for all A, B ∈ B ( H , K ). The below result extends Theorem2.7 of [1] and Theorem 4.4 of [4]. Theorem 3.5.
Let e ∈ E . Let B be a subspace of E . Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists φ ∈ S A such that φ ( h e, e i ) = k e k and φ ( h e, b i ) = 0 for all b ∈ B .Proof. We prove the theorem for the special case E = B ( H , K ) . The generalcase follows by Lemma 4.3 of [4]. The reverse direction is easy. Now let e beorthogonal to B . For any operator t ∈ B ( H , K ) we denote by ˜ t , the operator GROVER AND SINGLA on H ⊕ K given by ˜ t = (cid:20) t (cid:21) . We have e is Birkhoff-James orthogonalto B if and only if ˜ e is Birkhoff-James orthogonal to ˜ B = { ˜ b : b ∈ B} .Now using Corollary 1.3, we get that there exists ˜ φ ∈ S B ( H⊕K ) such that˜ φ (˜ e ∗ ˜ e ) = k ˜ e k and ˜ φ (˜ e ∗ ˜ b ) = 0 for all ˜ b ∈ ˜ B . Now φ defined as φ ( e ) = ˜ φ (˜ e ) isthe required state. (cid:3) Another approach to prove the above theorem has been briefly discussedafter Theorem 3.7 in [9]. We also remark that some related results withrestricted hypotheses for B ( H ) and B ( H , K ) have appeared recently in [11].The results in this article are stronger in these spaces.4. Remarks
Remark 4.1.
For a complex C ∗ -algebra A and a C ∗ -subalgebra B of A such that A ∈ B , a conditional expectation from A to B is a positive linearmap E of norm such that E (1 A ) = 1 A and E ( b ab ) = b E ( a ) b for all b , b ∈ B and a ∈ A . For any given conditional expectation E from A to B ,we can define a B -valued inner product on A given by h a | a i E = E ( a ∗ a ) (see [14] ). So h a − E ( a ) | a − E ( a ) i E = E (( a − E ( a )) ∗ ( a − E ( a )))= E ( a ∗ a ) − E ( E ( a ) ∗ a ) − E ( a ∗ E ( a )) + E ( E ( a ) ∗ E ( a ))= E ( a ∗ a ) − E ( a ) ∗ E ( a ) − E ( a ) ∗ E ( a ) + E ( a ) ∗ E ( a ) E (1 A )= E ( a ∗ a ) − E ( a ) ∗ E ( a ) . For φ ∈ S A , we have (11) φ ( h a − E ( a ) | a − E ( a ) i E ) = φ ( E ( a ∗ a )) − φ ( E ( a ) ∗ E ( a )) . Since a ∗ a ≤ k a k A and E (1 A ) = 1 A , we get φ ( E ( a ∗ a )) ≤ k a k . So (12) φ ( E ( a ∗ a )) − φ ( E ( a ) ∗ E ( a )) ≤ k a k . By (11) and (12) , we obtain (13) φ ( h a − E ( a ) | a − E ( a ) i E ) ≤ k a k . Now for b ∈ B , (14) h a − E ( a ) | a − E ( a ) i E = h a − b − E ( a − b ) | a − b − E ( a − b ) i E . By (13) and (14) , we obtain φ ( h a − E ( a ) | a − E ( a ) i E ) ≤ k a − b k for all b ∈ B , and so φ ( h a − E ( a ) | a − E ( a ) i E ) ≤ dist( a, B ) . Thus we obtain alower bound for dist( a, B ) as follows: (15)dist( a, B ) ≥ sup { φ ( E ( a ∗ a ) − E ( a ) ∗ E ( a )) : φ ∈ S A , E is a conditional expectation from A to B} , (where sup( ∅ ) = −∞ ). Remark 4.2.
In the case B = C A , equality holds in (15) . To see this,let h a, C A i be the subspace generated by a and A . Let λ A be a bestapproximation to a in C A . We define ˜ E : h a, C A i → C A as ˜ E ( a + λ A ) = EST APPROXIMATIONS, DISTANCE FORMULAS AND ORTHOGONALITY IN C ∗ -ALGEBRAS9 ( λ + λ )1 A . For any c ∈ A , the norm of the best approximation of c to C A is less than or equal to k c k . Since ( λ + λ )1 A is the best approximation to a + λ A , we get that k ˜ E k = 1 . By Hahn-Banach theorem, there exists anextension E of ˜ E which is of norm . By Corollary II.6.10.3 of [5] , E isa conditional expectation. By Theorem 1.1, there exists φ ∈ S A such that dist( a, B ) = φ ( a ∗ a ) − | λ | and φ ( a ) = λ = φ ( E ( a )) . Since φ ◦ E = φ , weget the required state for which equality in (15) holds. Remark 4.3.
It would be very interesting to find a counterexample to equal-ity in (15) when
B 6 = C A . Acknowledgments
We would like to thank Sneh Lata and Ved Prakash Gupta for many usefuldiscussions. We would also like to acknowledge several discussions withAmber Habib, which helped us to understand the geometric ideas behind thetheorems. The research of the first-named author is supported by INSPIREFaculty Award IFA14-MA-52 of DST, India, and by Early Career ResearchAward ECR/2018/001784 of SERB, India.
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Department of Mathematics, Shiv Nadar University, NH-91, Tehsil Dadri,Gautam Buddha Nagar, U.P. 201314, India.
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