The theory of scalar means was developed since the ancient Greek by the Pythagoreans until the last century by many famous mathematicians. See the development of this subject in a survey article . In Pythagorean school, various means are defined via the method of proportions (in fact, they are solutions of certain algebraic equations). The theory of matrix and operator means started from the presence of the notion of parallel sum as a tool for analyzing multi-port electrical networks in engineering; see . Three classical means, namely, arithmetic mean, harmonic mean and geometric mean for matrices and operators are then considered, e.g., in , , , , . These means play crucial roles in matrix and operator theory as tools for studying monotonicity and concavity of many interesting maps between algebras of operators; see the original idea in . Another important mean in mathematics, namely the power mean, is considered in . The parallel sum is characterized by certain properties in . The parallel sum and these means share some common properties. This leads naturally to the definitions of the so-called connection and mean in a seminal paper . This class of means cover many in-practice operator means. A major result of Kubo-Ando states that there are one-to-one correspondences between connections, operator monotone functions on the non-negative reals and finite Borel measures on the extended half-line. The mean theoretic approach has many applications in operator inequalities (see more information in Section 8), matrix and operator equations (see e.g. , ) and operator entropy. The concept of operator entropy plays an important role in mathematical physics. The relative operator entropy is defined in  for invertible positive operators by
In fact, this formula comes from the Kubo-Ando theory–is the connection corresponds to the operator monotone function . See more information in Chapter IV and its references.
In this chapter, we treat the theory of operator means by weakening the original definition of connection in such a way that the same theory is obtained. Moreover, there is a one-to-one correspondence between connections and finite Borel measures on the unit interval. Each connection can be regarded as a weighed series of weighed harmonic means. Hence, every mean in Kubo-Ando's sense corresponds to a probability Borel measure on the unit interval. Various characterizations of means are obtained; one of them is a usual property of scalar mean, namely, the betweenness property. We provide some new properties of abstract operator connections, involving operator monotonicity and concavity, which include specific operator means as special cases.
For benefits of readers, we provide the development of the theory of operator means. In Section 2, we setup basic notations and state some background about operator monotone functions which play important roles in the theory of operator means. In Section 3, we consider the parallel sum together with its physical interpretation in electrical circuits. The arithmetic mean, the geometric mean and the harmonic mean of positive operators are investigated and characterized in Section 4. The original definition of connection is improved in Section 5 in such a way that the same theory is obtained. In Section 6, several characterizations and examples of Kubo-Ando means are given. We provide some new properties of general operator connections, related to operator monotonicity and concavity, in Section 7. Many operator versions of classical inequalities are obtained via the mean-theoretic approach in Section 8.
Throughout, let be the von Neumann algebra of bounded linear operators acting on a Hilbert space . Let be the real vector space of self-adjoint operators on . Equip with a natural partial order as follows. For , we write if is a positive operator. The notation or means that is a positive operator. The case that and is invertible is denoted by or . Unless otherwise stated, every limit in is taken in the strong-operator topology. Write to indicate that converges strongly to . If is a sequence in , the expression means that is a decreasing sequence and . Similarly, tells us that is increasing and . We always reserve for positive operators. The set of non-negative real numbers is denoted by .
Remark 0.1 It is important to note that if is a decreasing sequence in such that , then if and only if for all . Note first that this sequence is convergent by the order completeness of . For the sufficiency, if , then
and hence .
The spectrum of is defined by
Then is a nonempty compact Hausdorff space. Denote by the C-algebra of continuous functions from to
Moreover, is isometric. We call the unique isometric -homomorphism which sends to the continuous functional calculus of . We write for .
If , then .
If , then
If for and , then we define . Equivalently, is the unique positive square root of .
If for and , then we define . Equivalently, .
A continuous real-valued function on an interval is called an operator monotone function if one of the following equivalent conditions holds:
for all Hermitian matrices of all orders whose spectrums are contained in ;
for all Hermitian operators whose spectrums are contained in and for an infinite dimensional Hilbert space ;
for all Hermitian operators whose spectrums are contained in and for all Hilbert spaces .
This concept is introduced in ; see also , , , . Every operator monotone function is always continuously differentiable and monotone increasing. Here are examples of operator monotone functions:
on , for and ,
on , for ,
on , where and .
The next result is called the Löwner-Heinz's inequality .
Theorem 0.3 For and , if , then . That is the map is an operator monotone function on for any .
A key result about operator monotone functions is that there is a one-to-one correspondence between nonnegative operator monotone functions on and finite Borel measures on via integral representations. We give a variation of this result in the next proposition.
Proposition 0.4 A continuous function is operator monotone if and only if there exists a finite Borel measure on such that
Here, the weighed harmonic mean is defined for by
and extended to by continuity. Moreover, the measure is unique. Hence, there is a one-to-one correspondence between operator monotone functions on the non-negative reals and finite Borel measures on the unit interval.
Recall that a continuous function is operator monotone if and only if there exists a unique finite Borel measure on such that
Consider the Borel measurable function Then, for each ,
Now, set . Since is bijective, there is a one-to-one corresponsence between the finite Borel measures on of the form and the finite Borel measures on of the form . The map is clearly well-defined and bijective.
3. Parallel sum: A notion from electrical networks
In connections with electrical engineering, Anderson and Duffin  defined the parallel sum of two positive definite matrices and by
The impedance of an electrical network can be represented by a positive (semi)definite matrix. If and are impedance matrices of multi-port networks, then the parallel sum indicates the total impedance of two electrical networks connected in parallel. This notion plays a crucial role for analyzing multi-port electrical networks because many physical interpretations of electrical circuits can be viewed in a form involving parallel sums. This is a starting point of the study of matrix and operator means. This notion can be extended to invertible positive operators by the same formula.
Lemma 0.5 Let for all .
If , then . If , then .
If and , then .
If and , then .
If and , then exists and does not depend on the choices of .
(1) Assume . Then is increasing and, for each ,
(2) Follow from (1).
(3) Let be such that and where . Then and . So, is an increasing sequence in such that
i.e. . By (1), we thus have
(4) Let be such that and . Then, by (2), is a decreasing sequence of positive operators. The order completeness of guaruntees the existence of the strong limit of . Let and be another sequences such that and . Note that for each , we have and . Then
Note that as , and . We have that as ,
Hence, and By symmetry, .
We define the parallel sum of to be
where the limit is taken in the strong-operator topology.
Lemma 0.6 For each ,
First, assume that are invertible. Then for all ,
With , we have
Hence, we have the claim for . For , consider and where .
Remark 0.7 This lemma has a physical interpretation, called the Maxwell's minimum power principle. Recall that a positive operator represents the impedance of a electrical network while the power dissipation of network with impedance and current is the inner product . Consider two electrical networks connected in parallel. For a given current input , the current will divide , where and are currents of each network, in such a way that the power dissipation is minimum.
Theorem 0.8 The parallel sum satisfies
transformer inequality: for every .
continuity from above: if and , then .
(2) For each such that , by the previous lemma,
Again, the previous lemma assures .
(3) Let and be decreasing sequences in such that and . Then is decreasing and for all . We have that, by the joint monotonicity of parallel sum, for all
Since and , by Lemma we have .
Remark 0.9 The positive operator represents the impedance of a network connected to a transformer. The transformer inequality means that the impedance of parallel connection with transformer first is greater than that with transformer last.
Proposition 0.10 The set of positive operators on is a partially ordered commutative semigroup with respect to the parallel sum.
For , we have and . The continuity from above in Theorem ▭ implies that and for all . The monotonicity of the parallel sum means that the positive operators form a partially ordered semigroup.
Theorem 0.11 For , we have the series-parallel inequality
In other words, the parallel sum is concave.
For each such that , we have by the previous lemma that
Applying the previous lemma yields .
Remark 0.12 The ordinary sum of operators represents the total impedance of two networks with series connection while the parallel sum indicates the total impedance of two networks with parallel connection. So, the series-parallel inequality means that the impedance of a series-parallel connection is greater than that of a parallel-series connection.
4. Classical means: arithmetic, harmonic and geometric means
Some desired properties of any object that is called a “mean” on should have are given here.
positive homogeneity: for ;
transformer inequality: for ;
congruence invariance: for invertible ;
concavity: for ;
continuity from above: if and , then ;
betweenness: if , then ;
fixed point property: .
In order to study matrix or operator means in general, the first step is to consider three classical means in mathematics, namely, arithmetic, geometric and harmonic means.
The arithmetic mean of is defined by
Then the arithmetic mean satisfies the properties (A1)–(A9). In fact, the properties (A5) and (A6) can be replaced by a stronger condition:
for all .
Moreover, the arithmetic mean satisfies
affinity: for .
Define the harmonic mean of positive operators by
where and . Then the harmonic mean satisfies the properties (A1)–(A9).
The geometric mean of matrices is defined in  and studied in details in . A usage of congruence transformations for treating geometric means is given in . For a given invertible operator , define
Then each is a linear isomorphism with inverse and is called a congruence transformation. The set of congruence transformations is a group under multiplication. Each congruence transformation preserves positivity, invertibility and, hence, strictly positivity. In fact, maps and onto themselves. Note also that is order-preserving.
Define the geometric mean of by
Then for . This formula comes from two natural requirements: This definition should coincide with the usual geometric mean in : provided that . The second condition is that, for any invertible ,
The next theorem characterizes the geometric mean of and in term of the solution of a certain operator equation.
Theorem 0.13 For each , the Riccati equation has a unique positive solution, namely, .
The direct computation shows that . Suppose there is another positive solution . Then
The uniqueness of positive square roots implies that , i.e.,
Theorem 0.14 (Maximum property of geometric mean) For ,
where the maximum is taken with respect to the positive semidefinite ordering.
If , then
and i.e. by Theorem ▭.
Recall the fact that if
Lemma 0.15 Let for all .
If and , then .
If and , then .
If and , then exists and does not depend on the choices of .
(1) The extremal characterization allows us to prove only that . Indeed,
(2) Assume and . Then is a decreasing sequence of strictly positive operators which is bounded below by 0. The order completeness of implies that this sequence converges strongly to a positive operator. Since , the Löwner-Heinz's inequality assures that and hence for all . Note also that for all . Recall that the multiplication is jointly continuous in the strong-operator topology if the first variable is bounded in norm. So, converges strongly to . It follows that
Since is norm-bounded by by Löwner-Heinz's inequality, we conclude that
The proof of (3) is just the same as the case of harmonic mean.
We define the geometric mean of by
Then for any .
Theorem 0.16 The geometric mean enjoys the following properties
continuity from above: .
fixed point property: .
congruence invariance: for all invertible .
(2) Follows from Lemma ▭ and the definition of the geometric mean.
(3) The unique positive solution to the equation is .
(4) The unique positive solution to the equation is . But this equstion is equivalent to . So, .
(5) The equation has the same solution to the equation by taking inverse in both sides.
(6) We have
Then apply Theorem ▭.
The congruence invariance asserts that is an isomorphism on with respect to the operation of taking the geometric mean.
Lemma 0.17 For and , the operator
is positive if and only if is positive, i.e., .
Since preserves positivity, we obtain the desired result.
Theorem 0.18 The geometric mean of is the largest operator for which the operator
The Löwner-Heinz's inequality forces . Now, applying yields .
Remark 0.19 The arithmetric mean and the harmonic mean can be easily defined for multivariable positive operators. The case of geometric mean is not easy, even for the case of matrices. Many authors tried to defined geometric means for multivariable positive semidefinite matrices but there is no satisfactory definition until 2004 in .
5. Operator connections
We see that the arithmetic, harmonic and geometric means share the properties (A1)–(A9) in common. A mean in general should have algebraic, order and topological properties. Kubo and Ando  proposed the following definition:
Definition 0.20 A connection on is a binary operation on satisfying the following axioms for all :
joint continuity from above: if satisfy and , then .
The term “connection" comes from the study of electrical network connections.
Example 0.21 The following are examples of connections:
the left trivial mean and the right trivial mean
the parallel sum
arithmetic, geometric and harmonic means
the weighed arithmetic mean with weight which is defined for each by
the weighed harmonic mean with weight which is defined for each by and extended to the case by continuity.
From now on, assume . Consider the following property:
separate continuity from above: if satisfy and , then and .
The condition (M3') is clearly weaker than (M3). The next theorem asserts that we can improve the definition of Kubo-Ando by replacing (M3) with (M3') and still get the same theory. This theorem also provides an easier way for checking a binary opertion to be a connection.
Theorem 0.22 If a binary operation on satisfies (M1), (M2) and (M3'), then satisfies (M3), that is, is a connection.
Denote by the set of operator monotone functions from to . If a binary operation has a property (A), we write . The following properties for a binary operation and a function play important roles:
: If a projection commutes with , then
: for any .
Proposition 0.23 The transformer inequality (M2) implies
Congruence invariance: For and , ;
Positive homogeneity: For and , .
For and , we have
and hence . The positive homogeneity comes from the congruence invariance by setting .
Lemma 0.24 Let be an increasing function. If satisfies the positive homogeneity, (M3') and (F), then is continuous.
To show that is right continuous at each , consider a sequence in such that . Then by (M3')
i.e. . To show that is left continuous at each , consider a sequence such that . Then and
Since is increasing, is decreasing. So, and are left continuous.
Lemma 0.25 Let be a binary operation on satisfying (M3') and (P). If is an increasing continuous function such that and satisfy (F), then for any .
First consider in the form where is an orthogonal family of projections with sum and for all . Since each commutes with , we have by the property (P) that
Now, consider . Then there is a sequence of strictly positive operators in the above form such that . Then and converges strongly to . Hence, .
Proof of Theorem ▭: Let . As in , the conditions (M1) and (M2) imply that satisfies (P) and there is a function subject to (F). If , then by (M1)
i.e. . The assumption (M3') is enough to guarantee that is continuous by Lemma ▭. Then Lemma ▭ results in for all . Now, (M1) and the fact that yield that is operator monotone. If there is another satisfying (F), then for each , i.e. . Thus, we establish a well-defined map such that and satisfy (F).
where the integral is taken in the sense of Bochner. Consider and set for each . Since and , we get
By Banach-Steinhaus' theorem, there is an such that for all . Hence,
So, is Bochner integrable. Since for all , . Thus, is a well-defined element in . The monotonicity (M1) and the transformer inequality (M2) come from passing the monotonicity and the transformer inequality of the weighed harmonic mean through the Bochner integral. To show (M3'), let and . Then for by the monotonicity and the separate continuity from above of the weighed harmonic mean. Let . Define a bounded linear map
For each , we obtain by (M3') that
where . That is . Therefore, there is a bijection between and . Every element in admits an integral representation (▭). Since the weighed harmonic mean possesses the joint continuity (M3), so is any element in .
The next theorem is a fundamental result of .
Theorem 0.26 There is a one-to-one correspondence between connections and operator monotone functions on the non-negative reals satisfying
There is a one-to-one correspondence between connections and finite Borel measures on satisfying
Moreover, the map is an affine order-isomorphism between connections and non-negative operator monotone functions on . Here, the order-isomorphism means that when for , for all if and only if .
Each connection on produces a unique scalar function on , denoted by the same notation, satisfying
Let . If , then . If , then .
Theorem 0.27 There is a one-to-one correspondence between connections and finite Borel measures on the unit interval. In fact, every connection takes the form
for some finite Borel measure on . Moreover, the map is affine and order-preserving. Here, the order-presering means that when (i=1,2), if for all Borel sets in , then for all .
The proof of the first part is contained in the proof of Theorem ▭. This map is affine because of the linearity of the map on the set of finite positive measures and the bijective correspondence between connections and Borel measures. It is straight forward to show that this map is order-preserving.
Remark 0.28 Let us consider operator connections from electrical circuit viewpoint. A general connection represents a formulation of making a new impedance from two given impedances. The integral representation (▭) shows that such a formulation can be described as a weighed series connection of (infinite) weighed harmonic means. From this point of view, the theory of operator connections can be regarded as a mathematical theory of electrical circuits.
Definition 0.29 Let be a connection. The operator monotone function in (▭) is called the representing function of . If is the measure corresponds to in Theorem ▭, the measure that takes a Borel set in to is called the representing measure of in the Kubo-Ando's theory. Here, is a homeomorphism .
Since every connection has an integral representation (▭), properties of weighed harmonic means reflect properties of a general connection. Hence, every connection satisfies the following properties for all and invertible :
transformer inequality: ;
congruence invariance: ;
concavity: for .
Moreover, if ,
and, in general, for each ,
where and . These properties are useful tools for deriving operator inequalities involving connections. The formulas (▭) and (▭) give a way for computing the formula of connection from its representing function.
The left- and the right-trivial means have representing functions given by and , respectively. The representing measures of the left- and the right-trivial means are given respectively by and where is the Dirac measure at . So, the -weighed arithmetic mean has the representing function and it has as the representing measure.
The geometric mean has the representing function .
The harmonic mean has the representing function while corrsponds to the parallel sum.
Remark 0.31 The map , where is the representing measure of , is not order-preserving in general. Indeed, the representing measure of is given by while the representing measure of is given by . We have but
6. Operator means
According to , a (scalar) mean is a binary operation on such that lies between and for any . For a connection, this property is equivalent to various properties in the next theorem.
Theorem 0.32 The following are equivalent for a connection on :
satisfies the betweenness property, i.e. .
satisfies the fixed point property, i.e. for all .
is normalized, i.e. .
the representing function of is normalized, i.e. .
the representing measure of is normalized, i.e. is a probability measure.
Clearly, (i) (iii) (iv). The implication (iii) (ii) follows from the congruence invariance and the continuity from above of . The monotonicity of is used to prove (ii) (i). Since
we obtain that (iv) (v) (iii).
Definition 0.33 A mean is a connection satisfying one, and thus all, of the properties in the previous theorem.
Hence, every mean in Kubo-Ando's sense satisfies the desired properties (A1)–(A9) in Section 3. As a consequence of Theorem ▭, a convex combination of means is a mean.
Theorem 0.34 Given a Hilbert space , there exist affine bijections between any pair of the following objects:
the means on ,
the operator monotone functions such that ,
the probability Borel measures on .
Moreover, these correspondences between (i) and (ii) are order isomorphic. Hence, there exists an affine order isomorphism between the means on the positive operators acting on different Hilbert spaces.
Example 0.35 The left- and right-trivial means, weighed arithmetic means, the geometric mean and the harmonic mean are means. The parallel sum is not a mean since its representing function is not normalized.
Example 0.36 The function is an operator monotone function on for each by the Löwner-Heinz's inequality. So it produces a mean, denoted by , on . By the direct computation,
i.e. is the -weighed geometric mean on . So the -weighed geometric mean on is really a Kubo-Ando mean. The α-weighed geometric mean on is defined to be the mean corresponding to that mean on . Since has an integral expression
(see ) where denotes the Lebesgue measure, the representing measure of is given by
Example 0.37 Consider the operator monotone function
The direct computation shows that
which is the -weighed harmonic mean. We define the -weighed harmonic mean on to be the mean corresponding to this operator monotone function.
Example 0.38 Consider the operator monotone function for , and . Then it gives rise to a mean, denoted by , on . By the direct computation,
i.e. is the logarithmic mean on . So the logarithmic mean on is really a mean in Kubo-Ando's sense. The logarithmic mean on is defined to be the mean corresponding to this operator monotone function.
Example 0.39 The map is operator monotone for any . This function produces a mean on . The computation shows that
However, the corresponding mean on is not given by the formula
Example 0.40 For each and , the map
is an operator monotone function on . Here, the case is understood that we take limit as . Then
The corresponding mean on is called the quasi-arithmetic power mean with parameter , defined for and by
The class of quasi-arithmetic power means contain many kinds of means: The mean is the -weighed arithmetic mean. The case is the -weighed geometric mean. The case is the -weighed harmonic mean. The mean is the power mean or binomial mean of order . These means satisfy the property that
Moreover, they are interpolated in the sense that for all ,
Example 0.41 If are means such that , then there is a family of means that interpolates between and , namely, for all . Note that the map is increasing. For instance, the Heron mean with weight is defined to be . This family is the linear interpolations between the geometric mean and the arithmetic mean. The representing function of is given by
The case is called the Heronian mean in the literature.
7. Applications to operator monotonicity and concavity
In this section, we generalize the matrix and operator monotonicity and concavity in the literature (see e.g. , ) in such a way that the geometric mean, the harmonic mean or specific operator means are replaced by general connections. Recall the following terminology. A continuous function is called an operator concave function if
for any and Hermitian operators whose spectrums are contained in the interval and for all Hilbert spaces . A well-known result is that a continuous function is operator monotone if and only if it is operator concave. Hence, the maps and are operator concave for . Let and be Hilbert spaces. A map is said to be positive if whenever . It is called unital if . We say that a positive map is strictly positive if when . A map from a convex subset of to is called concave if for each and ,
A map is called monotone if assures . So, in particular, the map is monotone and concave on for each . The map is monotone and concave on .
Note first that, from the previous section, the quasi-arithmetic power mean is monotone and concave for any and . In particular, the following are monotone and concave:
any weighed arithmetic mean,
any weighed geometric mean,
any weighed harmonic mean,
the logarithmic mean,
any weighed power mean of order .
Recall the following lemma from .
Lemma 0.42 (Choi's inequality) If is linear, strictly positive and unital, then for every , .
Proposition 0.43 If is linear and strictly positive, then for any
For each , set Then is a unital strictly positive linear map. So, by Choi's inequality, for all . For each , we have by Lemma ▭ that
So, we have the claim.
Theorem 0.44 If is a positive linear map which is norm-continuous, then for any connection on and for each ,
If, addition, is strongly continuous, then (▭) holds for any .
First, consider . Assume that is strictly positive. For each , set
Then is a unital strictly positive linear map. So, by Choi's inequality, for all . For each , put . We obtain from the previous proposition that
For general case of , consider the family where . Since the map is norm-continuous, we arrive at
For each connection , since is a bounded linear operator, we have
Suppose further that is strongly continuous. Then, for each ,
The proof is complete.
As a special case, if
In particular, for any and .
Theorem 0.45 If are concave, then the map
is concave for any connection on .
Let and . The concavity of and means that for
It follows from the monotonicity and concavity of that
This shows the concavity of the map .
In particular, if and are concave, then so is for and .
Corollary 0.46 Let be a connection. Then, for any operator monotone functions , the map is concave. In particular,
the map is concave on for any ,
the map is concave on .
Theorem 0.47 If are monotone, then the map
is monotone for any connection on .
Let and . Then and by the monotonicity of and . Now, the monotonicity of forces .
In particular, if and are monotone, then so is for and .
Corollary 0.48 Let be a connection. Then, for any operator monotone functions , the map is monotone. In particular,
the map is monotone on for any ,
the map is monotone on .
Corollary 0.49 Let be a connection on . If is monotone and strongly continuous, then the map
is a connection on . Hence, the map
is a connection for any operator monotone functions .
The monotonicity of this map follows from the previous result. It is easy to see that this map satisfies the transformer inequality. Since and strongly continuous, this binary operation satisfies the (separate or joint) continuity from above. The last statement follows from the fact that if , then for all and hence .
8. Applications to operator inequalities
In this section, we apply Kubo-Ando's theory in order to get simple proofs of many classical inequalities in the context of operators.
Theorem 0.50 (AM-LM-GM-HM inequalities) For , we have
It is easy to see that, for each ,
Now, we apply the order isomorphism which converts inequalities of operator monotone functions to inequalities of the associated operator connections.
Theorem 0.51 (Weighed AM-GM-HM inequalities) For and , we have
Apply the order isomorphism to the following inequalities:
The next two theorems are given in .
Theorem 0.52 For each , let . Then for each connection
Use the concavity of together with the induction.
By replacing with appropriate connections, we get some interesting inequalities.
(1) Cauchy-Schwarz's inequality: For ,
(2) Hölder's inequality: For and such that ,
(3) Minkowski's inequality: For ,
Theorem 0.53 Let , , be such that
Substitute to and to in (▭).
Here are consequences.
(1) Aczél's inequality: For , if
(2) Popoviciu's inequality: For and such that , if are such that and
(3) Bellman's inequality: For , if
The mean-theoretic approach can be used to prove the famous Furuta's inequality as follows. We cite  for the proof.
Theorem 0.54 (Furuta's inequality) For , we have
where and .
By the continuity argument, assume that . Note that (▭) and (▭) are equivalent. Indeed, if (▭) holds, then (▭) comes from applying (▭) to and taking inverse on both sides. To prove (▭), first consider the case . We have and the Löwner-Heinz's inequality (LH) implies the desired result. Now, consider the case and , since (▭) for can be obtained by (LH). Let and let be the associated connection (in fact, ). Must show that, for any ,
For , we have by (LH) that and
Now, set and . Let and consider the associated connection . The previous step, the monotonicity and the congruence invariance of connections imply that
Note that the above result holds for via the continuity of a connection. The desired equation (▭) holds for all by repeating this process.
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