Generalizeds-numbers ofτ-measurable operators

Thierry Fack, Hideki Kosaki
1986 Pacific Journal of Mathematics  
We give a self-contained exposition on generalized s-numbers of τ-nieasurable operators affiliated with a semi-finite von Neumann algebra. As applications, dominated convergence theorems for a gage and convexity (or concavity) inequalities are investigated. In particular, relation between the classical //-norm inequalities and inequalities involving generalized s-numbers due to A. Grothendieck, J. von Neumann, H. Weyl and the first named author is clarified. Also, the Haagerup L pspaces
more » ... ted with a general von Neumann algebra) are considered. 0. Introduction. This article is devoted to a study of generalized 5-numbers of immeasurable operators affiliated with a semi-finite von Neumann algebra. Also dominated convergence theorems for a gage and convexity (or concavity) inequalities are investigated. In the "hard" analysis of compact operators in Hubert spaces, the notion of ^-numbers (singular numbers) plays an important role as shown in [10], [24] . For a compact operator A, its nth ^-number μ n (A) is defined as the nth largest eigenvalue (with multiplicity counted) of \A\ = (A*A) 1/2 . The following expression is classical: where P x denotes the projection onto X. In the present article, we will study the corresponding notion for a semi-finite von Neumann algebra. More precisely, let Jί be a semi-finite von Neumann algebra with a faithful trace r. For an operator A in Jί, the "/th" generalized ^-number μ t (A) is defined by Notice that the parameter / is no longer discrete corresponding to the fact that T takes continuous values on the projection lattice. Actually this notion has already appeared in the literature in many contexts ([8], [11], [25], [33]). In fact, Murray and von Neumann used it (in the Π Γ case), [18]. We will consider generalized ^-numbers of r-measurable operators in the sense of Nelson [19]. This is indeed a correct set-up to consider generalized ^-numbers. In fact, the τ-measurability of an operator A 269 270 THIERRY FACK AND HIDEKI KOSAKI exactly corresponds to the property μ t (A) < + oo, / > 0 and the measure topology ( [19], [27]) can be easily and naturally expressed in terms of μ r When Jί is commutative, Jΐ -U°{X\ m), τ( ) = j x dm, the generalized s-number μ t (A) of A = /, a function on X, is exactly the non-increasing rearrangement f*(t) of / in classical analysis. (See [26] for example.) Therefore, through the use of μ n one can employ many classical analysis techniques (such as majorization arguments) in our non-commutative context (as shown in §4). §1 consists of some preliminaries. §2 is expository and we give a self-contained and unified account on the theory of generalized ^-numbers of τ-measurable operators. In §3, we prove certain dominated convergence theorems for a trace (i.e. gage, [23] ). In the literature (see [27] for example), Fatou's lemma for a trace was emphasized. Instead, we will show Fatou's lemma for generalized ^-numbers. Although its proof is simple, it will prove extremely useful. In fact, based on this, we will prove dominated convergence theorems unknown previously. Also, this Fatou's lemma is useful to extend known estimates (involving μ t ) for bounded operators to (unbounded) τ-measurable operators. In §4, we will study convexity (and concavity) inequalities involving μ r For applications and for the sake of completeness, we will prove classicial norm inequalities such as the Holder and Minkowsky inequalities. However, our main emphasis here is to compare carefully the above classical norm inequalities with inequalities due to A. Grothendieck, J. von Neumann, H. Weyl and the first named author. We will also show that these "semi-finite techniques" are useful to derive the corresponding results for the Haagerup L ^-spaces, [12]. This is possible because μ t {A) (with respect to the canonical trace on the crossed product, [28]) is particularly simple in this case. In the final §5, we prove the Clarkson-McCarthy inequalities for the Haagerup L^-spaces "from scratch." Proofs are known, but it may not be without interest. In fact, some false proofs exist in the literature.
doi:10.2140/pjm.1986.123.269 fatcat:5trfwf3kn5eqxexb5lx2dfwvyq