Convergence of functions of self-adjoint operators and applications

The main result (roughly) is that if (H_i) converges weakly to H and if also f(H_i) converges weakly to f(H), for a single strictly convex continuous function f, then (H_i) must converge strongly to H. One application is that if f(pr(H)) = pr(f(H)), where pr denotes compression to a closed subspace M, then M must be invariant for H. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If h and f(h) are both quasimultipliers, then h must be a multiplier. Also (still roughly stated) if h and f(h) are both in pA_sa p, for a closed projection p, then h must be strongly q-continuous on p.


Introduction.
Although the main result of this paper involves only elementary operator theory, the original motivation was from a couple of technical operator algebraic questions. In [B1,Proposition 2.59(a)] it was shown that if f is a non-linear operator convex function on an interval I, if h is a self-adjoint quasimultiplier of a C * -algebra A such that σ(h) ⊂ I, where σ(h) is the spectrum, and if f (h) is also a quasimultiplier, then h is in fact a multiplier of A. And in [B2,Theorem 4.14(i)] it was shown (for f as above) that if p is a closed projection in A * * , the bidual, and if h and f (h) are both in pA sa p, then h (which is still assumed self-adjoint) must be strongly q-continuous on p (provided also that either p is compact or 0 ∈ I and f (0) = 0). (In general S sa = {x ∈ S: x * = x}.) In both cases ad hoc methods could be used to generalize the result to certain non-operator convex functions f , and the motivating questions were to find for which f the results are true.
It turns out that both results are true for an arbitrary strictly convex continuous function f . Also the best approach is to start with an elementary result about convergence in the weak operator topology. This result, roughly stated in the abstract, is Theorem 2.1 below. The application to compressions is Corollary 2.7, and the proof of Arveson's conjecture is Corollary 2.8. The operator algebraic results are Theorems 3.1 and 3.6. (Theorem 3.6 is not strictly speaking an application of Theorem 2.1, but its proof is modeled on that of 2.1.) Remark 2.6 and Proposition 2.10 discuss converses to Theorem 2.1, and Remark 2.6 also discusses extensions of the theorem and relations to the operator-theoretic version of the Kaplansky Typeset by A M S-T E X 1 density theorem ( [Kap,Thm. 2] for the forward direction and [Kad,Cor. 3.7] for the converse, see also [Ped,§2.3]).
There are other relationships involving the results of this paper, and the full story of these relationships does not seem clear. K. Davidson suggested that our results have the flavor of a Korovkin type theorem and also pointed out the relationship to [Arv]. The relationship of Arveson's results to Korovkin type theorems is discussed in [Arv,Remarks 1.4 and 1.8]. Also, Theorem 2.1, in the special case where the interval I occurring there is compact, could be deduced from either Corollary 2.7 or Theorem 3.1, and Corollary 2.7 can be deduced from Theorem 3.6.
We begin with some elementary results, none of which are probably new, but the only reference we know is that Corollary 1.2 can be deduced from [BL,Lemma 3.4]. Corollary 1.2 is used in §2 and Corollary 1.3 in §3. The set of bounded linear operators on a Hilbert space H is denoted by B(H).
Proof. P can be represented by an n × n matrix (P ij ), where P ij ∈ B(H j , H i ). If w = u 1 ⊕ . . . ⊕ u n and Q is the operator with matrix ((δ ij − t i t j )P ij ), then the conclusion is just the statement that (Qw, w) ≥ 0. Since the matrix (δ ij − t i t j ) is positive, this follows from the fact that the Hadamard product of positive matrices is positive.
Corollary 1.2. Let H = H 1 ⊕ . . . ⊕ H n be a direct sum of Hilbert spaces, P a positive operator in B(H), and let P ii be the compression of P to H i . Then P ≤ n 1 P ii .
Proof. Let v be a unit vector in H, and use the notation of the lemma. Then If A is a C * -algebra, its bidual A * * is a von Neumann algebra, called the enveloping von Neumann algebra of A. Bounded linear functionals on A are also regarded as weak * -continuous linear functionals on A * * . A state ϕ on A is said to be supported by a projection p in A * * if ϕ(1 − p) = 0 where 1 is the identity of A * * . For q, r ∈ A * * and ψ a bounded linear functional on A, qψr denotes the functional a → ψ(qar), which is again weak * -continuous on A * * . The set of positive elements of A is denoted by A + . Corollary 1.3. Let ϕ be a state on a C * -algebra A, and p = p 1 + . . . + p n , where p 1 , . . . , p n are mutually orthogonal projections in A * * . If ϕ is supported by p, then ϕ ≤ π(p i )H ϕ , and P the compression of π(a) to H for a in A + , noting that the state vector v is indeed in H. The conclusion is that ϕ(a) = (P v, v) ≤ n 1 (π(a)u i , u i ) = n 1 (1/ϕ(p i ))(p i ϕp i )(a).

The main result and basic applications.
Theorem 2.1. Let f be a strictly convex continuous function on an interval I, let H be a Hilbert space, and let (H i ) be a net in B(H) sa such that σ(H i ) ⊂ I, ∀i. If (H i ) converges weakly to a bounded operator H with σ(H) ⊂ I, and if also (f (H i )) converges weakly to f (H), then (ϕ(H i )) converges strongly to ϕ(H) for every bounded continuous function ϕ on I. In particular if the net (H i ) is bounded, then (H i ) converges strongly to H.
Note. The applications described in the introduction depend only on the special case where the interval I is compact.
Beginning of the proof. Since it is sufficient to prove that every subnet of (H i ) has a further subnet for which the conclusion is true, we may replace the given net with a subnet. For a Borel set E, let P i (E) = χ E (H i ) and P (E) = χ E (H), the spectral projections. Choose the subnet so that for each interval E, the net (P i (E)) converges weakly to a positive contraction Q(E). Note that Q(·) is finitely additive. We are going to prove a close relationship between Q(·) and P (·) via several lemmas. The first lemma is needed to deal with the endpoints, and it also illustrates one of the main ideas of the proof.
Proof. Let g be the function obtained by subtracting a linear function from f so that g(a) = g(x) = 0. Note that (g(H i )) converges weakly to g(H). Let −γ = g(c) be the minimum value of g. Because of strict convexity, γ > 0 and a < c < x. Also g is strictly increasing to the right of x and in particular β = g(y) > 0. For

Hence
( Now the one-sided derivatives g ′ (z±) exist and are finite, ∀z ∈ (a, b). By strict The next lemma is a standard real analysis fact stated in our notation. We need x 0 > a because of the possibility that f ′ (a+) = −∞.
Proof. There are only finitely many points y in ( This is proved in the same way as the uniform continuity of continuous functions on J. So we can obtain the desired partition as a suitable refinement of x 0 < y 1 < . . . < y n < x. For each j = 1, . . . , n let g j be the function obtained by subtracting a linear function from f so that g j (x j − 1) = g j (x j ) = 0. Let −γ j = g(c j ) be the minimum value of g j , and let β j = g j (y). As in the proof of (1) above, we see that Since ǫ is arbitrary, we conclude that P ([a, x])Q([y, ∞))P ([a, x]) = 0, and since Lemma 2.5. If a < y < b, then , y)). Using left-right symmetry, we can also prove Q((−∞, y]) ≤ P ((−∞, y]) and Q((y, ∞)) ≥ P ((y, ∞)). If b is the right endpoint of I, then End of the proof. It is enough to show that (ϕ((H i )) converges weakly to ϕ(H) for each bounded continuous function ϕ on I, since then we also get the same conclusion for |ϕ| 2 , and the facts that (ϕ(Hi)) converges weakly to ϕ(H) and (ϕ(H i ) 2 ) converges weakly to |ϕ(H)| 2 imply that (ϕ(H i )) converges strongly to ϕ(H). Then fix ϕ and a unit vector v in H. Let ε > 0 and choose δ such that has only countably many discontinuities in (a, b), and at each continuity point, Now let s be any cluster point of the net ((ϕ(H i )v, v)). Note that the bounded nets (P i ((−∞, a))) and (P i ((b, ∞))) converge strongly to 0, since they are positive and converge weakly to 0. Therefore ( , and hence (ϕ(H i )) converges weakly to ϕ(H).
Remarks 2.6. (i) If the net is not bounded, then (H i ) may not converge strongly to H.
(ii) The Kaplansky density theorem can be deduced from Theorem 2.1. If (H i ) is a net in B(H) sa which converges strongly to H, then (H 2 i ) converges weakly to H 2 .
Thus the hypotheses of 2.1 are met with I = R and f (x) = x 2 , and the conclusion that (ϕ(H i )) converges strongly to ϕ(H) for ϕ bounded and continuous is sufficient for the Kaplansky density theorem.
(iii) We sketch an argument for how to fill the gap between what was proved in the theorem and what was disproved in (i). Assume the hypotheses of the theorem and let g be a function, obtained by subtracting a linear function from f , such that g(a) > 0 and g ′ (a−) < 0 if a is not the left endpoint of I and g(b) > 0 and g ′ (b+) > 0 if b is not the right endpoint of I. (If a 0 < b 0 , the most natural choice is to take g(a 0 ) = g(b 0 ) = 0; and if a 0 = b 0 and is in the interior of I, the most natural choice is to make the x-axis a line of support for g at a 0 .) The arguments at the end of the proof, with g in place of ϕ, show that(g(H i )P i ([a, b])) converges weakly to g(H). Since also (g(H i )) converges weakly to g(H), we conclude that (g(H i )(P i (−∞, a)+P i (b, ∞)) converges weakly to 0. So if ϕ is a continuous function on I such that |ϕ|/g(x) is bounded on I\[a, b], the arguments in the proof show that (ϕ(H i )) converges weakly to ϕ(H). In particular if x 2 /g(x) is bounded on I\[a, b], the weak convergence of (H 2 i ) to H 2 implies that (H i ) converges strongly to H. These conditions can be restated, without mentioning g, a, or b, as follows: For ϕ(H i ) to converge weakly to ϕ(H), it is sufficient that ϕ(x)/max(1, |x|, f (x)) be bounded on I. And for (H i ) to converge strongly to H, it is sufficient that if I is unbounded on the right, then f (x) > 0 for x sufficiently large and x 2 = O(f (x)) as x → ∞; and if I is unbounded on the left, then f (x) > 0 for x sufficiently small and The argument in [Kad] can be adapted to show that the above results are sharp in the following sense: If for all nets and a particular f and ϕ the hypotheses of the theorem imply that (ϕ(H i )) converges weakly to ϕ(H), then the first condition above must be satisfied; and if for all nets and a particular f the hypotheses imply that (H i ) converges strongly to H, then the second condition must be satisfied.
(iv) It is possible to weaken the hypotheses of the theorem: Instead of assuming that (f (H i )) converges weakly to f (H), assume only that (f (H)v, v) ≥ lim sup(f (H i )v, v) for each v in H. The same proof works. We do not know whether this strengthening of the theorem is valuable, but note that it is irrelevant if f is operator convex and I is compact, as in the result cited from [B1]. In this case it is automatic that Then (V n HV * n + x 0 (1 − V n V * n )) converges weakly to pr(H) then (f (V n HV * n + x 0 (1 − V n V * n )) converges weakly to pr(f (H)) + f (x 0 )1 M ⊥ = f (pr(H) ⊕ x 0 1 M ⊥ ). So the theorem implies that (V n HV * n + x 0 (1 − V n V * n )) converges strongly to pr(H) ⊕ x 0 1 M ⊥ . If u is a vector in M and Hu = u ′ ⊕ w, this implies that S n w → 0 and hence w = 0.
Corollary 2.7 leads to the elimination of the finite dimensionality hypothesis in a theorem of Arveson. The special case where f is operator convex had previously been proved by Petz.
Finally, we prove a converse to Theorem 2.1. [a, b] which is neither strictly convex nor strictly concave, then there are an operator H and a sequence (H n ) in B(ℓ 2 ) sa with σ(H), σ(H n ) ⊂ [a, b], such that (H n ) converges weakly to H and (f (H n )) converges weakly to f (H), but (H n ) does not converge strongly to H.

Proposition 2.10. If f is a continuous real-valued function on a compact interval
Proof. By the proof of [Arv,Proposition 9.2] there are x, y ∈ [a, b] and t ∈ (0, 1) such that x < y and f (tx + (1 − t)y) = tf (x) + (1 − t)f (y). Let H 0 be the 2 × 2 matrix Then σ(H 0 ) = {x, y} ⊂ [a, b], and f (pr(H 0 )) = prf (H 0 ) but M is not invariant for H 0 , where M is the one-dimensional subspace of C 2 consisting of vectors of the form * 0 . The proof of Corollary 2.7 produces the required counterexample.

Operator algebraic applications.
If A is a C * -algebra, let M (A) = {t ∈ A * * : tA + At ⊂ A}, the algebra of multipliers of A, and let QM (A) = {t ∈ A * * : AtA ⊂ A}, the set of quasimultipliers of A. Also S(A) denotes the state space of A and Q(A) denotes {ϕ ∈ A * : ϕ ≥ 0 and ϕ ≤ 1}, the quasi-state space of A. Proof. We will show that h 2 ∈ QM (A). The result then follows from [AP,Proposition 4.4]. By [AP,Theorem 4.1], it is sufficient to show ϕ i (h 2 ) → ϕ(h 2 ) whenever ϕ i and ϕ are in S(A) and ϕ i → ϕ weak * . We may assume that the GNS representations for ϕ and the ϕ i 's are all in Hilbert spaces of the same dimension. If not, replace A by A ⊗ K(H), where K(H) is the algebra of compact operators on a Hilbert space H of sufficiently large dimension, so that A * * is replaced by the W * -tensor product A * * ⊗ B(H). Then replace h by h ⊗ 1, ϕ by ϕ ⊗ ψ, and ϕ i by ϕ i ⊗ ψ, for a pure state ψ on K(H). Now, reverting to the original notation, and passing to a subnet, which is permissible, we can realize all the GNS representations by maps π i , π: A → B(H) with the same unit vector v as the state vector such that π i (a) → π(a) strongly, ∀a ∈ A, (cf. [Dix,Section 3.5]). Letπ i andπ denote the canonical extensions to A * * . If t ∈ QM (A) and a, b ∈ A, the facts that π i (a) * π i (t)π i (b) → π(a) * π (t)π(b) strongly, π i (a) → π(a) strongly, and π i (b) → π(b) strongly imply that (π i (t)π(b)u, π(a)w) → (π(t)π(b)u, π(a)w), ∀u, w ∈ H. Since π is non-degenerate, this impliesπ i (t) → π(t) weakly. So Theorem 2.1 now applies with H i =π i (h). Thusπ i (h) →π(h) strongly and henceπ i (h 2 ) =π i (h) 2 →π(h) 2π (h 2 ) weakly (also strongly). Therefore Remark. As in Remark 2.6(iv), we can weaken the hypotheses: Instead of assuming f (h) is in QM (A), assume only that f (h) is weakly upper semicontinuous. This strengthening of the theorem is irrelevant if f is operator convex, since then, by [B1,Proposition 2.34  Proof. It is easy to see that A * * can be identified with the algebra of bounded collections t = {T n : 1 ≤ n ≤ ∞, T n ∈ B(ℓ 2 )}. Then t is in QM (A) if and only if t n → t ∞ weakly and t ∈ M (A) if and only if t n → t strongly. So the result follows from Proposition 2.10.
For a projection p in A * * , F (p) denotes {ϕ ∈ Q(A): ϕ(1 − p) = 0}, the face of Q(A) supported by p. Then p is called closed [Ak1] if F (p) is weak * closed and compact ([Ak2]) if F (p) ∩ S(A) is weak * closed. If p is closed and h ∈ pA * * sa p, then h is called strongly lower semicontinuous on p ( [B2]) if the map ϕ → ϕ(h) is lower semicontinuous (lsc) on F (p), and h is called strongly q-continuous on p ( [B1], cf. also [APT]) if χ F (h) is closed whenever F is a closed subset of R and compact if also 0 ∈ F . Here χ F (h) denotes the spectral projection computed in pA * * p. If A is unital the qualifier "strongly" is unnecessary and every closed projection is compact. It was shown in [B1,Theorem 3.43] that h is strongly q-continuous on p if and only if h = pa for some a in A sa such that ap = pa.
Lemma 3.3. Let A be a unital C * -algebra, p a closed projection in A * * , and f a strictly convex continuous function on a compact interval [a, b]. If h ∈ pA sa p such that σ(h) ⊂ [a, b] and f (h) is (strongly) lower semicontinuous on p, then h is (strongly) q-continuous on p.
The proof follows the next two lemmas, in which p(E) denotes χ E (h), the spectral projection computed within pA * * p and the notation of Lemma 3.3 is assumed.
As in the proof of Lemma 2.2, γ/β ≤ x−a y−x .
Proof. Let η be as in the proof of Lemma 2.4, let ǫ > 0, and choose x 0 , . . . , x n as in the proof of 2.4. By Corollary 1.3, for each i we can write ϕ i ≤ n 0 ψ ij , where ψ io is a state of A supported by p([a, x 0 ]) and ψ ij , j ≥ 1, is a state of A supported by p([x j−1 , x j ]). Passing to a subnet, we may assume each net (ψ ij ) converges weak * to a state ϕ j . Clearly ϕ ≤ n 0 ϕ j . By Lemma 3.4, As in the proof of Lemma 3.4, for j ≥ 1, ϕ j (p([y, b])) ≤ γ j /β j , in the notation of Lemma 2.4, and as in the proof of 2.4, . So since ǫ is arbitrary, ϕ(p([y, b])) = 0.
Proof of Lemma 3.3. Let a < x < b, and let the ϕ i 's be states of A supported by p([a, x]) such that ϕ i → ϕ weak * . Choose a monotone sequence (y n ) such that y n ∈ (x, b) and y n → x. Since p([y n , b]) → p((x, b)) in the weak* topology of A * * and since ϕ(p[y n , b]) = 0, ∀n, then ϕ(p(x, b]) = 0. In other words ϕ is supported by p([a, x]), whence p([a, x]) is closed. A symmetrical proof shows that p([x, b]) is closed. Akemann showed in [Ak1] that the set of closed projections is closed under arbitrary lattice meets and also that q 1 ∨ q 2 is closed if q 1 and q 2 are and if the angle between q 1 and q 2 is positive. The latter applies in particular if q 1 q 2 = q 2 q 1 . So we can conclude that p(E) is closed for all closed sets E.
Theorem 3.6. Let A be a C * -algebra, p a closed projection in A * * , and f a strictly convex continuous function on a compact interval [a, b]. Assume also that either p is compact or 0 ∈ [a, b] and f (0) = 0. If h ∈ pA sa p such that σ(h) ⊂ [a, b] and f (h) is strongly lower semicontinuous on p, then h is strongly q-continuous on p. Here σ(h) and f (h) are computed within pA * * p.
Proof. We apply Lemma 3.3 toÃ, the result of adjoining an identity to A, identi-fyingÃ * * with A * * ⊕ C. In the compact case, we use p ⊕ 0 in place of p and h ⊕ 0 in place of h. Since f (h ⊕ 0) = f (h) ⊕ 0, the conclusion is immediate.
In the non-compact case, we usep = p ⊕ 1 in place of p and h ⊕ 0 in place of h. It is still true that f (h ⊕ 0) = f (h) ⊕ 0, and we conclude that h ⊕ 0 is (strongly) q-continuous onp. If E is a closed set not containing 0, then χ E (h⊕0) = χ E (h)⊕0; and the fact that this is closed inÃ * * implies that χ E (h) is compact in A * * . If E is a closed set that contains 0, then χ E (h ⊕ 0) = χ E (h) ⊕ 1; and the fact that this is closed inÃ * * implies that χ E (h) is closed in A * * . Therefore h is strongly q-continuous on p.
Remarks 3.7. (i) (cf. Remark 2.6(iv) and the remark following Theorem 3.1) It follows from the theorem that f (h) is in pA sa p. We could have proved a slightly weaker theorem by assuming f (h) in pA sa p instead of f (h) strongly lsc, and we do not know whether the extra strength of the actual theorem is worthwhile. In [B2], this issue did not arise because f was assumed operator convex. Theorem 4.3 of [B2] implies in this case that f (h) is automatically strongly usc on p.
(ii) Remark 4.15 of [B2] gives some discussion, which will not be repeated here, of the hypothesis that 0 ∈ [a, b] and f (0) = 0. We mention only that if p is not compact, it is impossible to have f (0) < 0, given the other hypotheses.
Proposition 3.8. Let f be a continuous real-valued function on a compact interval [a, b]. If f is neither strictly convex nor strictly concave, then there are a unital C *algebra A, a closed projection p in A * * , and an h in pA sa p such that σ(h) ⊂ [a, b], f (h) ∈ pAp and h is not q-continuous on p.
Proof. Let A be the algebra of convergent sequences in M 2 , the algebra of 2 × 2 matrices. Then A * * can be identified with the set of bounded collections t = {T n : 1 ≤ n ≤ ∞, T n ∈ M 2 }. Let the projection p be given by p n = 1 0 0 0 for n < ∞ and p ∞ = 1 0 0 1 . Then p is closed. As in the proof of proposition 2.10, there is a self-adjoint matrix H = α β β γ such that σ(H) = {x, y} ⊂ [a, b], x < α < y, and f (H) = f (α) * * * . Let h be given by H n = α 0 0 0 for n < ∞ and H ∞ = H. Then h and f (h) are in pA sa p. If E is the closed set {α}, then χ E (h) is not closed.