Grothendieck’s inequality
in the noncommutative Schwartz space
Abstract.
In the spirit of Grothendieck’s famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a Fréchet algebra of smooth operators. These hold in nonoptimal form by a simple nuclearity argument. We obtain optimal versions and reformulate the inequalities in several different ways.
Key words and phrases:
noncommutative Grothendieck inequality, noncommutative Schwartz space, mconvex Fréchet algebra, bilinear form, state.2010 Mathematics Subject Classification:
Primary: 47A30, 47A07, 47L10. Secondary: 47A63.1. Introduction
The noncommutative Schwartz space is a weakly amenable mconvex Fréchet algebra whose properties have been investigated in several recent papers, see e.g. [2, 3, 13, 14]. It is not difficult to see that as a Fréchet space, is nuclear. From this, we can easily deduce the following analogue of Grothendieck’s inequality, which we call Grothendieck’s inequality in : there exists a constant so that for any continuous bilinear form and any , there exists such that for every and any , we have
(1) 
The norms on the right hand side arise naturally from the definition of , as explained in Section 2 below. Our goal in this note is to show that in fact always suffices, and that this is best possible.
This appears to be the first result concerning Grothendieck’s inequality in the category of Fréchet algebras; to the best of our knowledge, all previous results along these lines concern Banach spaces (including Calgebras, general Banach algebras and operator spaces). For Fréchet algebras, Grothendieck’s inequality seems to have a specific flavour. Every Fréchet space (and a fortiori, every Fréchet algebra) which appears naturally in analysis is nuclear, meaning that all tensor product topologies are equal. Since Grothendieck’s inequality can be understood as the equivalence of two tensor products, it seems that we can take inequality (1) for granted. The interesting question that remains is then optimality.
This paper is divided into four sections. In the remainder of this section, we recall a Calgebraic version of Grothendieck’s inequality due to Haagerup, and then review the definition and the basic properties of which we require. In Section 2 we explain how nuclearity gives Grothendieck’s inequality in , and we estimate the constants and . Section 3 then settles the optimality question for via a matricial construction. We conclude with a short section containing several reformulations of the inequality.
1.1. Grothendieck’s inequality
Pisier’s survey article [12] is a comprehensive reference for Grothendieck’s inequality. This presents many equivalent formulations and applications of this famous result, and recounts its evolution from ‘commutative’ [5] to ‘noncommutative’. Of these reformulations and extensions, Haagerup’s noncommutative version most closely resembles (1), and we state it here for the convenience of the reader.
1.2. The noncommutative Schwartz space
Let
denote the socalled space of rapidly decreasing sequences. This space becomes Fréchet when endowed with the sequence of norms defined above. The basis of zero neighbourhoods of is defined by . The topological dual of is the socalled space of slowly increasing sequences, namely
where the duality pairing is given by for , .
The noncommutative Schwartz space is the Fréchet space of all continuous linear operators from into , endowed with the topology of uniform convergence on bounded sets. The formal identity map is a continuous embedding and defines a product on by for . There is also a natural involution on given by for , . With these operations, becomes an mconvex Fréchet algebra. The inclusion map is continuous, and in fact it is a spectrumpreserving homomorphism [3]. Moreover [13, Proposition 3], an element is positive (i.e., for some ), if and only if the spectrum of is contained in , or equivalently for all . On the other hand, by [3, Cor. 2.4] and [4, Theorems 8.2, 8.3], the topology of cannot be given by a sequence of Cnorms. This causes some technical inconvenience (e.g. there is no bounded approximate identity in ) meaning we cannot apply Calgebraic techniques directly.
2. The inequality
Let be a nondecreasing sequence of norms which gives the topology of . For a continuous bilinear form, we write
where ; similarly, for a functional , we write
Following Pisier [11, p. 316], for and , we write
Relative to our choice of norms , we have now defined each term in our hopedfor inequality (1). We will now reformulate it using tensor products.
For Calgebras, such a reformulation is standard. Indeed, by [6, Theorem 1.1] (formulated along the lines of [8, Theorem 2.1]), Haagerup’s noncommutative Grothendieck inequality entails the existence of a such that for any Calgebras and in the algebraic tensor product , we have where is the projective tensor norm and is the absolute Haagerup tensor norm [8, p. 164] on , given by
Here for an element of a Calgebra, and the infimum is taken over all representations where .
We proceed similarly for . For , let and consider the sequence of absolute Haagerup tensor norms on the algebraic tensor product given by
where the infimum runs over all ways to represent in . As usual, we write for the sequence of projective tensor norms on .
Just as in the Calgebra case, inequality (1) will follow once we show that the sequences of projective and absolute Haagerup tensor norms are equivalent on . In fact, the equivalence of these norms follows immediately from the nuclearity of (see [9, Theorem 28.15] and [7, Ch. 21, §2, Theorem 1] for details). On the other hand, the optimal values of and (depending on and our choice of norms ) for which (1) hold are not given by such general considerations. These optimal parameters will be denoted by and .
Henceforth, we focus only on the sequence of norms where
and . In other words, is the norm of , considered as a Hilbert space operator from to . This sequence does indeed induce the topology of . In this context, we will estimate and compute the exact values of .
We start with the following result, which can be compared with [13, Lemma 8]. To fix some useful notation, for we define an infinite diagonal matrix which we consider as an isometry and simultaneously as an isometry .
Proposition 2.
Let . We have

for every positive ;

for every selfadjoint ; and

for every .
Moreover, inequalities (ii) and (iii) are sharp.
Proof.
(i) Observe that . Furthermore, since is positive, is positive and we have
(ii) For selfadjoint, we have
and by [1, Proposition II.1.4.2],
where denotes the spectral radius. This gives the desired inequality.
(iii) Since , any is also a Hilbert space operator, and the blockmatrix operator is positive in (see e.g. [10, p. 117]). Equivalently,
(2) 
For , let us write where is the identity matrix. Now fix and choose . Then for all and (2) gives
Since is an approximate identity in (see [13, Proposition 2]), we obtain
Taking the supremum over all in the unit ball of we get
Applying (ii) to the positive operators and we conclude that .
For sharpness, observe that if is a diagonal rank one matrix unit then we have equality in both (ii) and (iii). ∎
Proposition 3.
For any and , we have
Proof.
Let and let . We claim that
By the Cauchy–Schwarz inequality and Proposition 2(iii) this will then imply the desired inequality. To establish the claim, let and let us write for the standard basis vectors in . We have
Applying the Cauchy–Schwarz inequality to summation over gives
Since is positive for any , and for positive operators we have (where ), this implies that
As a straightforward consequence of Proposition 3, we obtain:
Theorem 4 (Grothendieck’s inequality in ).
There is a constant such that for any and . Moreover, every continuous bilinear form satisfies inequality (1) with , for any and any . In particular, taking where , we obtain
(3) 
Remark.
This shows that . On the other hand, it is easy to show that . Indeed, if not, then (3) would hold with replaced by some . Take , define and by . Then for we get and . On the other hand, is equivalent (up to a constant) to . Therefore (3) takes the form for some constant (independent of ). Letting tend to infinity, we obtain , a contradiction. Hence .
3. Optimality
We will now show that . For this, we will use the tensor product formulation, noting that
Recall the diagonal operator defined on page 2 above. Since every is an operator on via the canonical inclusions , it is clear that if , then and are both operators on . This leads to the following observation.
Proposition 5.
If , then
Proof.
Write
If and for some , then and
This gives . The reverse inequality is proved similarly. ∎
We also need the following wellknown fact.
Proposition 6.
If is a Hilbert space and , then
Proof.
By [15, Theorem 4.3], the Haagerup norm on the left hand side is equal to the completely bounded norm of the map on given by , which is completely positive, so attains its completely bounded norm at the identity operator. ∎
Theorem 7.
For every , we have .
Proof.
By Theorem 4, it only remains to show that . Choose sufficiently large that for all . This inequality ensures that for every , if we define
then . Denote by the standard matrix units, and for , consider the selfadjoint operators
Let
Since and , by Propositions 5 and 6 we obtain
On the other hand,
Therefore
Hence
by our choice of . So as required. ∎
4. Reformulations of the inequality
Here we give several different ways of stating our inequality; in each case, an analogous result may be found in [12]. The methods here are fairly standard, so full proofs are often omitted. Throughout, we write .
4.1. Grothendieck’s inequality with states
Given , let be given by , . We call an element of the closed convex hull of an state on . Note that by Proposition 2(i), for any positive element we have , where is the set of all states on . The next result may be deduced from Theorem 4 by closely following the Hahn–Banach Separation argument of [12, §23].
Theorem 8.
For any continuous bilinear form and , there are states on with
for all .
4.2. ‘Little’ Grothendieck inequality
As a consequence we obtain the following ‘little’ Grothendieck inequality in . Recall that if is a linear map between Fréchet spaces, then .
Theorem 9.
For any FréchetHilbert space , if are continuous linear maps, and , then
Equivalently, for any there are states such that for all we have
Using the same argument as in the proof of Theorem 8 we can obtain an equivalent version of the ‘little’ Grothendieck inequality.
Theorem 10.
For any FréchetHilbert space , if is a continuous linear map and , then there exist states on such that for all we have
References
 [1] B. Blackadar. Operator algebras, volume 122 of Encyclopaedia of Mathematical Sciences. SpringerVerlag, Berlin, 2006.
 [2] T. Ciaś. On the algebra of smooth operators. Studia Math., 218(2):145–166, 2013.
 [3] P. Domański. Algebra of smooth operators. http://main3.amu.edu.pl/~domanski/salgebra1.pdf, 2012.
 [4] M. Fragoulopoulou. Topological algebras with involution. Elsevier Science B.V., Amsterdam, 2005.
 [5] A. Grothendieck. Résumé de la théorie métrique des produits tensoriels topologiques. Resenhas, 2(4):401–480, 1996. Reprint of Bol. Soc. Mat. São Paulo 8 (1953), 1–79.
 [6] U. Haagerup. The Grothendieck inequality for bilinear forms on algebras. Adv. in Math., 56(2):93–116, 1985.
 [7] H. Jarchow. Locally convex spaces. B. G. Teubner, 1981.
 [8] S. Kaijser and A.M. Sinclair. Projective tensor products of algebras. Math. Scand., 55(2):161–187, 1984.
 [9] R. Meise and D. Vogt. Introduction to functional analysis, volume 2 of Oxford Graduate Texts in Mathematics. Oxford University Press, 1997.
 [10] V. Paulsen. Completely bounded maps and operator algebras. Cambridge University Press, Cambridge, 2002.
 [11] G. Pisier. Introduction to operator space theory, volume 294 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2003.
 [12] G. Pisier. Grothendieck’s theorem, past and present. Bull. Amer. Math. Soc. (N.S.), 49(2):237–323, 2012.
 [13] K. Piszczek. Automatic continuity and amenability in the noncommutative Schwartz space. J. Math. Anal. Appl., 432(2):954–964, 2015.
 [14] K. Piszczek. A Jordanlike decomposition in the noncommutative Schwartz space. Bull. Aust. Math. Soc., 91(2):322–330, 2015.
 [15] R. R. Smith. Completely bounded module maps and the Haagerup tensor product. J. Funct. Anal., 102(1):156–175, 1991.