A Few Conditions for a 𝐶^∗-Algebra to Be Commutative

Abstract

We present a few characterizations of the commutativity of 𝐶^∗-algebras in terms of particular algebraic properties of power functions, the logarithmic and exponential functions, and the sine and cosine functions.

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Let 𝐺,𝐻 be subsets of groups which are closed under forming the Jordan triple product 𝑎𝑏𝑎 of its elements. That means that for any 𝑎,𝑏 from 𝐺 (resp., from 𝐻) we have 𝑎𝑏𝑎∈𝐺 (resp., 𝑎𝑏𝑎∈𝐻). A map 𝜙∶𝐺→𝐻 is called a Jordan triple map if it is a homomorphism relative to that operation, that is, if 𝜙 satisfies 𝜙(𝑎𝑏𝑎)=𝜙(𝑎)𝜙(𝑏)𝜙(𝑎) for all 𝑎,𝑏∈𝐺. Such maps appear in many areas, in particular in ring theory and, as recent investigations have shown, also in correspondence with surjective isometries of noncommutative metric groups and related structures. Along this latter line of research, we have recently published several results concerning the descriptions of different sorts of surjective isometries of the sets of positive definite matrices and unitary matrices [1, 2]. In the paper [3] we have considered similar but much more general transformations that can be termed as maps preserving generalized distance measures in the context of operator algebras, namely, in that of factor von Neumann algebras. On the way to obtain structural results for those preserver transformations we have faced the following problem. In the case of the positive definite cone of a full matrix algebra or that of a nontrivial von Neumann factor we have shown that if for some nonzero exponent 𝑐 the power function 𝐴↦𝐴^𝑐 is essentially a Jordan triple map, meaning that if (𝐴𝐵𝐴)^𝑐 and 𝐴^𝑐𝐵^𝑐𝐴^𝑐 differ only by a scalar multiplier for any pair 𝐴,𝐵 of positive definite elements, then 𝑐 must be trivial; that is, 𝑐∈{−1,0,1}. For details see the proofs of Theorem 1 in [2] and Theorem 5 in [3]. We have used a similar observation concerning power functions on unitary groups (the exponent being an integer there); see the proofs of Theorem 1 in [1] and Theorem 8 in [3].

In this short note we are going to prove a stronger result for general unital 𝐶^∗-algebras. Namely, we show that if the power function corresponding to a nontrivial exponent is a Jordan triple map up to multiplication by central elements, then the underlying 𝐶^∗-algebra is necessarily commutative. We also present several results of similar spirits concerning algebraic properties of the logarithmic and exponential functions as well as the sine and cosine functions. For example, we will show that it happens only in commutative algebras that the logarithm of the geometric mean of any two positive definite elements differs from the arithmetic means of their logarithms only by central elements.

We recall that in the literature one can find several conditions characterizing commutativity of 𝐶^∗-algebras most of which are related to the order structure; see, for example, [4–8]. A characterization of algebraic character can be found in [9]. In this line we also mention that in the recent paper [10] we have investigated the standard 𝐾-loop structure of the positive definite cone of a 𝐶^∗-algebra and obtained some results showing that the commutativity, associativity (even in certain weak senses), or distributivity of that structure are each equivalent to the commutativity of the underlying 𝐶^∗-algebra.

We now turn to the results of the paper and first fix the notation. Let 𝒜 be a unital 𝐶^∗-algebra and 𝒜_𝑠 stand for the space of its self-adjoint elements. The cone of positive elements of 𝒜, that is, those which are self-adjoint and have nonnegative spectrum, is denoted by 𝒜₊, and 𝒜₊⁻¹ stands for the set of all positive definite elements (in other words, positive invertibles), what we call the positive definite cone of 𝒜. The geometric mean 𝐴#𝐵 (in Kubo-Ando sense) of the elements 𝐴,𝐵∈𝒜₊⁻¹ is defined by 𝐴#𝐵=𝐴^1/2𝐴^−1/2𝐵𝐴^−1/21/2𝐴^1/2. (1) The unitary group in 𝒜 is denoted by 𝒜_𝑢, and, finally, 𝒵_𝒜 stands for the center of 𝒜.

Our main result reads as follows.

Theorem 1. Let 𝒜 be a unital 𝐶^∗-algebra. The following assertions are equivalent. (i)The algebra 𝒜 is commutative.(ii)There is a real number 𝑐∉{−1,0,1} with the property that for any pair 𝐴,𝐵∈𝒜₊⁻¹ of positive invertibles there exists a central element 𝜆∈𝒵_𝒜 such that (𝐴𝐵𝐴)^𝑐=𝜆𝐴^𝑐𝐵^𝑐𝐴^𝑐. (2)(iii)One has log(𝐴𝐵𝐴)−2log𝐴+log𝐵∈𝒵_𝒜,𝐴,𝐵∈𝒜₊⁻¹. (3)(iv)One has log(𝐴#𝐵)−log𝐴+log𝐵2∈𝒵_𝒜,𝐴,𝐵∈𝒜₊⁻¹. (4)(v)For any pair 𝐻,𝐾∈𝒜_𝑠 of self-adjoint elements there exists a central element 𝜆∈𝒵_𝒜 such that exp(2𝐻+𝐾)=𝜆exp(𝐻)exp(𝐾)exp(𝐻). (5)(vi)For any pair 𝐻,𝐾∈𝒜_𝑠 of self-adjoint elements there exists a central element 𝜆∈𝒵_𝒜 such that exp𝐻+𝐾2=𝜆exp(𝐻)#exp(𝐾). (6)

Proof. The implication (i) ⇒ (ii) is trivial.
Suppose (ii) holds. We first show that the number 𝑐 can be assumed to be positive and less than 1. In fact, if 𝑐 is negative, then taking inverse in (2) we obtain a related property for the positive number −𝑐. So, we can assume that 𝑐 is positive. If 𝑐>1, then plug 𝐴^1/𝑐, 𝐵^1/𝑐 in the place of 𝐴, 𝐵 in (2) and then take 𝑐th root. Hence, we deduce an equality similar to (2) which holds for the number 1/𝑐. Therefore, we may and do assume that the constant 𝑐 is positive and less than 1.
Plugging 𝐴^𝑐, 𝐵^𝑐 in the place of 𝐴, 𝐵 in (2) again we obtain a similar equality with 𝑐² in the place of 𝑐. Repeating the argument we next deduce a similar equality for 𝑐³ and so forth.
Now fix two elements 𝐴,𝐵∈𝒜₊⁻¹. For the complex variable 𝑧 define 𝑓(𝑧)=(𝐴𝐵𝐴)^𝑧𝐴^−𝑧𝐵^−𝑧𝐴^−𝑧 (here we mean 𝐴^𝑧=exp(𝑧log𝐴)). Plainly, 𝑓 is a holomorphic (entire) function with values in 𝒜 such that its values at 𝑐,𝑐²,𝑐³,… belong to the center 𝒵_𝒜 of 𝒜. Since the sequence (𝑐^𝑛) has a limit point in the domain of 𝑓, by the uniqueness theorem of holomorphic functions, it is easy to deduce that the whole range of 𝑓 lies in 𝒵_𝒜. Indeed, for any 𝑇∈𝒜 we have that (𝐴𝐵𝐴)^𝑧𝐴^−𝑧𝐵^−𝑧𝐴^−𝑧𝑇=𝑇(𝐴𝐵𝐴)^𝑧𝐴^−𝑧𝐵^−𝑧𝐴^−𝑧 (7) holds for all 𝑧=𝑐^𝑛, 𝑛∈ℕ. On both sides of this equality we have holomorphic (entire) functions which coincide on a set that has a limit point in ℂ. It follows that they necessarily coincide everywhere which means that the range of 𝑓 is indeed in 𝒵_𝒜. We have that (𝐴𝐵𝐴)^𝑧=𝑓(𝑧)𝐴^𝑧𝐵^𝑧𝐴^𝑧,𝑧∈ℂ. (8) Clearly, 𝑓(0)=𝐼. Differentiating both sides of the above equality at 𝑧=0 we obtain log(𝐴𝐵𝐴)=𝑓^′(0)+2log𝐴+log𝐵. (9) This means that for every 𝐴,𝐵∈𝒜₊⁻¹ we have log(𝐴𝐵𝐴)−2log𝐴+log𝐵∈𝒵_𝒜, (10) which proves (iii).
Suppose that (iii) holds. Clearly, by Gelfand-Naimark theorem we may assume that 𝒜 is a 𝐶^∗-subalgebra of the algebra 𝐵(𝐻) of all bounded linear operators acting on a Hilbert space 𝐻. Denote by ℬ the strong closure of 𝒜 in 𝐵(𝐻) which is a von Neumann algebra. Using Kaplansky density theorem it follows that every element of ℬ₊⁻¹ is the strong limit of a bounded net in 𝒜₊⁻¹ which is bounded away from zero. Since the multiplication is strongly continuous on bounded sets of operators, bounded continuous complex valued functions of a real variable are strongly continuous, and the strong limit of a net of central elements is again central, we obtain that log(𝐴𝐵𝐴)−2log𝐴+log𝐵∈𝒵_ℬ (11) holds for all 𝐴,𝐵∈ℬ₊⁻¹.
Now select projections 𝑃,𝑄 in ℬ. Let 𝐴=𝐼+𝑡𝑃,𝐵=𝐼+𝑡𝑄, (12) where 𝑡 is a real number with |𝑡|<1. Easy computation shows that ()()𝐴𝐵𝐴=𝐼+𝑡𝑃)(𝐼+𝑡𝑄)(𝐼+𝑡𝑃=𝐼+𝑡2𝑃+𝑄+𝑡²(𝑃+𝑃𝑄+𝑄𝑃)+𝑡³(𝑃𝑄𝑃). (13) Recall that in an arbitrary unital Banach algebra, for any element 𝑎 with ‖𝑎‖<1 we have log(1+𝑎)=^∞_𝑛=1(−1)^𝑛+1𝑎^𝑛𝑛. (14) This shows that for a suitable positive number 𝜖(<1), the elements log(𝐴𝐵𝐴), log𝐴, and log𝐵 of ℬ can be expressed by power series of 𝑡 (|𝑡|<𝜖) with coefficients in ℬ. Simple calculation shows that the coefficient of 𝑡³ in the power series of the center valued function −()()𝑔∶𝑡⟼log((𝐼+𝑡𝑃)(𝐼+𝑡𝑄)(𝐼+𝑡𝑃))2log𝐼+𝑡𝑃+log𝐼+𝑡𝑄 (15) is equal to 1𝑃𝑄𝑃−2)+(+1((2𝑃+𝑄)(𝑃+𝑃𝑄+𝑄𝑃𝑃+𝑃𝑄+𝑄𝑃)(2𝑃+𝑄))3(2𝑃+𝑄)³−13()=12𝑃+𝑄3()−1𝑃𝑄𝑃3(𝑄𝑃𝑄). (16) This coefficient could also be obtained by calculating the third derivative of 𝑔 at 𝑡=0. Since 𝑔 is center valued, so are its derivatives. Therefore, we obtain that the element 𝑃𝑄𝑃−𝑄𝑃𝑄 is central, which, in particular, means that it commutes with 𝑃. Clearly, this implies that 𝑄𝑃𝑄 also commutes with 𝑃; that is, we have 𝑃𝑄𝑃𝑄=𝑄𝑃𝑄𝑃. We compute (𝑃𝑄𝑃)³()𝑃()()=𝑃𝑄𝑃𝑄𝑃𝑄𝑃=𝑃𝑄𝑃𝑄𝑃𝑄=𝑃𝑄𝑄𝑃𝑄𝑃𝑃=𝑃𝑄𝑃𝑄𝑃=𝑃𝑄𝑃². (17) It follows from the equality (𝑃𝑄𝑃)³=(𝑃𝑄𝑃)² that the spectrum of the positive element 𝑃𝑄𝑃 is a subset of {0,1} which implies that 𝑃𝑄𝑃 is a projection. But, by Lemma 2 in [10], this happens exactly when 𝑃 and 𝑄 commute (referring to a comment by a referee, the handling editor suggested to include a proof of that observation and proposed the following nice argument for which the author is grateful; in paper [10] a different but also simple proof is presented. Consider the self-adjoint element 𝑋=𝑖(𝑃𝑄−𝑄𝑃). By direct computation we have 𝑋³=0, and thus 𝑋=0 which gives 𝑃𝑄=𝑄𝑃). Consequently, we obtain that the projections in ℬ all commute which immediately implies that the von Neumann algebra ℬ is commutative. This gives us the commutativity of the smaller algebra 𝒜, and hence we obtain (i).
We prove that (iv) implies (iii). We recall the well-known fact that the geometric mean of elements 𝐴,𝐵∈𝒜₊⁻¹ can be characterized as the unique solution 𝑋∈𝒜₊⁻¹ of the Riccati equation 𝑋𝐴⁻¹𝑋=𝐵. For any given 𝐴,𝐵∈𝒜₊⁻¹ let 𝐶=𝐴𝐵𝐴. Then we have 𝐴=𝐵⁻¹#𝐶 and we compute by (iv) log𝐴−log𝐵⁻¹+log(𝐴𝐵𝐴)2∈𝒵_𝒜. (18) It is apparent that this implies log(𝐴𝐵𝐴)−2log𝐴+log𝐵∈𝒵_𝒜, (19) which means that (iii) holds true. Taking into account that (iii) is equivalent to the commutativity of 𝒜, the implication (iii) ⇒ (iv) is clear.
Finally, the assertions (iii) and (v) as well as (iv) and (vi) are obviously equivalent; thus the proof of the theorem is complete.

Remark 2. In relation with the equivalence of (iii) and (iv) to (i) we make the following simple observation. Let 𝒜 be a unital 𝐶^∗-algebra and (𝐺,+) a commutative group in which every element is uniquely 2-divisible. We assert that if there is an injective transformation 𝜙∶𝒜₊⁻¹→𝐺 such that either 𝜙𝐴^1/2𝐵𝐴^1/2=𝜙(𝐴)+𝜙(𝐵),𝐴,𝐵∈𝒜₊⁻¹ (20) or 𝜙()=𝜙(𝐴)(𝐵)𝐴#𝐵+𝜙2,𝐴,𝐵∈𝒜₊⁻¹, (21) then 𝒜 is necessarily commutative. In fact, by the commutativity of 𝐺, in the first case we immediately have 𝐴^1/2𝐵𝐴^1/2=𝐵^1/2𝐴𝐵^1/2 which is equivalent to 𝐴𝐵=𝐵𝐴 as it has been shown, for example, in Proposition 1 in [10]. Concerning the second case, considering the map 𝜙(⋅)−𝜙(𝐼) we can clearly assume that 𝜙(𝐼)=0. Then applying the argument we have followed in the proof of the implication (iv) ⇒ (iii) above, we obtain 𝜙(𝐴)=𝜙𝐵⁻¹+𝜙(𝐴𝐵𝐴)2. (22) For any 𝐷∈𝒜₊⁻¹ we have 0=𝜙(𝐼)=𝜙𝐷#𝐷⁻¹=𝜙(𝐷)+𝜙𝐷⁻¹2 (23) yielding 𝜙(𝐷⁻¹)=−𝜙(𝐷). Hence from (22) we deduce 2𝜙(𝐴)+𝜙(𝐵)=𝜙(𝐴𝐵𝐴). (24) Moreover, for any 𝐷∈𝒜₊⁻¹ we have 𝜙𝐷^1/2()=𝜙(𝐷)=𝜙𝐷#𝐼+02. (25) Hence, from (24) we infer 𝜙(𝐴)+𝜙(𝐵)=𝜙𝐴^1/2𝐵𝐴^1/2 (26) and this, as we have seen above, already implies 𝐴𝐵=𝐵𝐴, 𝐴,𝐵∈𝒜₊⁻¹. Consequently, in both cases we deduce that 𝒜 is necessarily commutative.

In the next corollary we present a characterization of the commutativity of 𝐶^∗-algebras concerning the unitary group which is similar to (ii) in Theorem 1.

Corollary 3. Let 𝒜 be a unital 𝐶^∗-algebra and assume that 𝑚 is an integer, 𝑚∉{−1,0,1}, with the property that for any pair 𝑈,𝑉∈𝒜_𝑢 of unitaries there is a central element 𝜆∈𝒵_𝒜 such that (𝑉𝑊𝑉)^𝑚=𝜆𝑉^𝑚𝑊^𝑚𝑉^𝑚. (27) Then 𝒜 is commutative.

Proof. Taking inverses in equality (27) if necessary, we can obviously assume that the integer 𝑚 is positive. For any 𝑉,𝑊∈𝒜_𝑢 we know that (𝑉𝑊𝑉)^𝑚𝑉^−𝑚𝑊^−𝑚𝑉^−𝑚 is central. Inserting one-parameter groups of unitaries in the places of 𝑉 and 𝑊 we deduce that for any given pair 𝐻,𝐽∈𝒜_𝑠 the element 𝑒^𝑖𝑡𝐻𝑒^𝑖𝑠𝐽𝑒^{𝑖𝑡𝐻𝑚}𝑒^{−𝑖𝑚𝑡𝐻}𝑒^{−𝑖𝑚𝑠𝐽}𝑒^{−𝑖𝑚𝑡𝐻} (28) is central for all 𝑡,𝑠∈ℝ. Fixing the real variable 𝑠 and putting the complex variable 𝑧 in the place of 𝑖𝑡 we obtain that the holomorphic (entire) function 𝑧⟼𝑒^𝑧𝐻𝑒^𝑖𝑠𝐽𝑒^𝑧𝐻𝑚𝑒^{−𝑚𝑧𝐻}𝑒^{−𝑖𝑚𝑠𝐽}𝑒^{−𝑚𝑧𝐻} (29) has central values along the 𝑦-axis. Just as in the proof of Theorem 1 we deduce that the same holds for every complex number 𝑧 too. Now fixing 𝑧 and inserting the complex variable 𝑤 in the place of 𝑖𝑠, the same reasoning yields that 𝑒^𝑧𝐻𝑒^𝑤𝐽𝑒^𝑧𝐻𝑚𝑒^{−𝑚𝑧𝐻}𝑒^{−𝑚𝑤𝐽}𝑒^{−𝑚𝑧𝐻} (30) is central for all 𝑧,𝑤 in ℂ. In particular, we obtain that for any 𝑡,𝑠∈ℝ the element 𝑒^𝑡𝐻𝑒^𝑠𝐽𝑒^𝑡𝐻𝑚𝑒^{−𝑚𝑡𝐻}𝑒^{−𝑚𝑠𝐽}𝑒^{−𝑚𝑡𝐻} (31) is central. Since every element of 𝒜₊⁻¹ is the exponential of an element of 𝒜_𝑠, we deduce that (𝐴𝐵𝐴)^𝑚𝐴^−𝑚𝐵^−𝑚𝐴^−𝑚 is central for all 𝐴,𝐵∈𝒜₊⁻¹. By Theorem 1 this implies that 𝒜 is commutative and the proof is complete.

Concerning our final proposition we recall that in [9] Jeang and Ko presented the following result. Assume 𝒜 is a 𝐶^∗-algebra and there are nonconstant continuous functions 𝑓 and 𝑔 defined on intervals 𝐼,𝐽 in the real line such that for any pair 𝑋,𝑌∈𝒜_𝑠 of self-adjoint elements with 𝜎(𝑋)⊂𝐼, 𝜎(𝑌)⊂𝐽 (𝜎(⋅) stands for the spectrum) we have 𝑓(𝑋)𝑔(𝑌)=𝑔(𝑌)𝑓(𝑋). Then it follows that 𝒜 is necessarily commutative. We now give a simple extension of that result.

Proposition 4. Assume 𝒜 is a unital 𝐶^∗-algebra and there are nonconstant complex-valued continuous functions 𝑓,𝑔 defined on some intervals 𝐼,𝐽 in the real line such that for any pair 𝑋,𝑌∈𝒜_𝑠 of self-adjoint elements with 𝜎(𝑋)⊂𝐼, 𝜎(𝑌)⊂𝐽 one has 𝑓(𝑋)𝑔(𝑌)−𝑔(𝑌)𝑓(𝑋)∈𝒵_𝒜. Then 𝒜 is commutative.

Proof. We can trivially assume that 𝐼=𝐽=[0,1]. Next, identify 𝒜 with a 𝐶^∗-algebra of operators acting on a Hilbert space and consider its strong closure ℬ which is a von Neumann algebra. Similarly to the proof of the implication (iii) ⇒ (i) in Theorem 1, referring to Kaplansky’s density theorem, we easily see that the relation 𝑓(𝑋)𝑔(𝑌)−𝑔(𝑌)𝑓(𝑋)∈𝒵_ℬ holds for all pairs 𝑋,𝑌 of self-adjoint elements of ℬ with 𝜎(𝑋),𝜎(𝑌)⊂[0,1]. Considering operators 𝑋,𝑌 of the form 𝑋=𝑡𝐼+𝑠𝑃,𝑌=𝑡^′𝐼+𝑠^′𝑃^′, where 𝑃,𝑃^′∈ℬ are arbitrary projections and 𝑡,𝑡^′,𝑠,s^′ are properly chosen real numbers, we deduce that 𝑃𝑃^′−𝑃^′𝑃∈𝒵_ℬ. It follows that 𝑃𝑃^′−𝑃^′𝑃 commutes with 𝑃. It is then easy to verify that 𝑃,𝑃^′ commute. Therefore, all projections in ℬ commute which implies that ℬ is commutative, and hence we obtain that 𝒜 is commutative too.

In accordance with Example 5 in [9], as an immediate application of the above result, we mention that a unital 𝐶^∗-algebra 𝒜 is commutative if and only if any of the following relations sin(𝑋+𝑌)−(sin𝑋cos𝑌+cos𝑋sin𝑌)∈𝒵_𝒜,sin(𝑋−𝑌)−(sin𝑋cos𝑌−cos𝑋sin𝑌)∈𝒵_𝒜,cos(𝑋+𝑌)−(cos𝑋cos𝑌−sin𝑋sin𝑌)∈𝒵_𝒜,cos(𝑋−𝑌)−(cos𝑋cos𝑌+sin𝑋sin𝑌)∈𝒵_𝒜 (32) hold true for all pairs 𝑋,𝑌 of self-adjoint elements in 𝒜.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author was supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences. This work was partially supported also by the European Union and the European Social Fund through Project Supercomputer, the national virtual lab (Grant no. TAMOP-4.2.2.C-11/1/KONV-2012-0010).