aa r X i v : . [ m a t h . OA ] J a n AROUND TRACE FORMULASIN NON-COMMUTATIVE INTEGRATION
SHIGERU YAMAGAMI
Graduate School of MathematicsNagoya UniversityNagoya, 464-8602, JAPAN
Abstract.
Trace formulas are investigated in non-commutative integrationtheory. The main result is to evaluate the standard trace of a Takesaki dualand, for this, we introduce the notion of interpolator and accompanied bound-ary objects. The formula is then applied to explore a variation of Haagerup’strace formula.
Introduction
The Haagerup’s trace formula in non-commutative integration is a key to hiswhole theory of non-commutative L p -spaces (see [6] and [9]). Our purpose hereis to analyse it from the view point of modular algebras ([10], [11]), which wasoriginally formulated in terms of Haagerup’s L p -theory itself. So, to circumventtautological faults and also to fix notations, we first describe modular algebras aswell as standard Hilbert spaces in terms of basic ingredients of Tomita-Takesakitheory.The semifiniteness of Takesaki’s duals is then established by constructing relevantHilbert algebras as a collaboration of modular algebras and complex analysis. Notethat the known proofs of the existence of standard traces are not direct; for example,it is usually deduced from the innerness of modular automorphism groups combinedwith a reverse Radon-Nikodym theorem such as Pedersen-Takesaki’s or Connes’.Since our construction of the Hilbert algebras is based on complex analysis,the associated trace can be also described in a calculational way. To make thesetup transparent, we introduce the notion of interpolators together with associ-ated boundary operators and vectors. Viewing things this way, the main traceformula turns out to be just a straightforward consequence of definitions. TheHaagerup’s trace formula is then derived in a somewhat generalized form as a con-crete application of our formula.The Haagerup’s correspondence between normal functionals and relatively invari-ant measurable operators on Takesaki’s duals is also established on our streamlines.Recall that the standard approach to these problems is by the theory of operator-valued weights ([4], [5]) coupled with dual weights ([2], [3]), which is based on ex-tended positive parts, a notion of metaphysical flavor, and somewhat elaborate. Ourmethod may not provide an easy route either but can be applied rather straightfor-wardly; it is just a simple combination of elementary Fourier calculus and complexanalytic nature of modular stuffs. The presentation below originates from the author’s old work in 1990, whichwas addressed on the occasion of a satellite meeting of ICM90 held at NiigataUniversity. The author would like to express hearty gratitude to Kichisuke Saitofor his organization of the meeting and these records.
Notation and Convention:
The positive part of a W*-algebra M (resp. itspredual M ∗ ) is denoted by M + (resp. M + ∗ ).For a positive element p in M + or M + ∗ , its support projection in M is denotedby [ p ].For a functional ϕ ∈ M + ∗ , the associated GNS-vector in the standard Hilbertspace L ( M ) of M is denoted by ϕ / (natural notation though not standard) andthe modular operator by ∆ ϕ so that ∆ ϕ ( aϕ / ) = ϕ / a for a ∈ [ ϕ ] M [ ϕ ].For ϕ, ψ ∈ M + ∗ , σ ϕ,ψt stands for the relative modular group of [ ϕ ] M [ ψ ], which issimply denoted by σ ϕt and expresses a modular automorphism group of the reducedalgebra [ ϕ ] M [ ϕ ] when ϕ = ψ .For convergence in M , w*-topology (resp. s-topology or s*-topology) means weakoperator topology (resp. strong operator topology or *strong operator topology) asa von Neumann algebra on the standard Hilbert space L ( M ).Direct integrals are indicated by H instead of ordinary R ⊕ . This is to avoidduplication of sum meanings.The notion of weights is used in a very restrictive sense: weights are orthogonalsums of functionals in M + ∗ .For an interval I contained in [0 , T I expresses the tubular domain based onan imaginary trapezoid { ( x, y ) ∈ R ; x ≤ , y ≤ , − ( x + y ) ∈ I } : T I = { ( z, w ) ∈ C ; Im z ≤ , Im w ≤ , − ( Im z + Im w ) ∈ I } .A function f : D → M with D ⊂ C is said to be w*-analytic (s*-analytic) if it isw*-continuous (s*-continuous) and holomorphic when restricted to the interior D ◦ .Note that topologies are irrelevant for holomorphicity because weaker one impliespower series expansions in norm.For real numbers α, β , α ∨ β = max { α, β } , α ∧ β = min { α, β. } . Standard Hilbert Spaces
Given a faithful ω ∈ M + ∗ , we denote the associated GNS-vector by ω / and iden-tify the left and right GNS-spaces by the relation ∆ / ω ( xω / ) = ω / x , resultingin an M -bimodule L ( M, ω ) =
M ω / M with the positive cone L ( M, ω ) + and thecompatible *-operation given by L ( M, ω ) + = { aω / a ∗ ; a ∈ M } and ( aω / b ) ∗ = b ∗ ω / a ∗ in such a way that these constitute a so-called standard form of M .The dependence on ω as well as its faithfulness is then removed by the matrixampliation technique: For each ϕ ∈ M + ∗ , let M ⊗ ϕ / ⊗ M be a dummy of thealgebraic tensor product M ⊗ M , which is an M -bimodule in an obvious mannerwith a compatible *-operation defined by the relation ( a ⊗ ϕ / ⊗ b ) ∗ = b ∗ ⊗ ϕ / ⊗ a ∗ .On the algebraic direct sum M ϕ ∈ M + ∗ M ⊗ ϕ / ⊗ M RACE FORMULAS 3 of these *-bimodules, introduce a sesquiliear form by n M j =1 x j ⊗ ω / j ⊗ y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n M k =1 x ′ k ⊗ ω / k ⊗ y ′ k = X j,k ([ ω k ]( x ′ k ) ∗ x j ω / j | ω / k y ′ k y ∗ j [ ω j ]) , which is positive because of X j,k ([ ω k ] x ∗ k x j ω / j | ω / k y k y ∗ j [ ω j ]) = ( Xω / | ω / Y )= ( X / ω / Y / | X / ω / Y / ) ≥ . Here ω = diag( ω , . . . , ω n ) denotes a diagonal functional on the n -th matrix ampli-ation M n ( M ) of M and X = [ ω ] x ∗ ... x ∗ n (cid:0) x . . . x n (cid:1) [ ω ] and Y = [ ω ] y ... y n (cid:0) y ∗ . . . y ∗ n (cid:1) [ ω ]are positive elments in [ ω ] M n ( M )[ ω ]. Recall that [ ω ] = diag([ ω ] , . . . , [ ω n ]).The associated Hilbert space is denoted by L ( M ) and the image of a ⊗ ϕ / ⊗ b in L ( M ) by aϕ / b . Here the notation is compatible with the one for L ( M, ϕ )because [ ϕ ] M [ ϕ ] ⊗ ϕ / ⊗ [ ϕ ] M [ ϕ ] ∋ a ⊗ ϕ / ⊗ b aϕ / b ∈ L ( M, ϕ )gives an isometric map by the very definition of inner products. Similar remarksare in order for left and right GNS spaces.The left and right actions of M are compatible with taking quotients and theyare bounded on L ( M ): For a ∈ M , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)M j ax j ⊗ ω / j ⊗ y j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = ( ω / | ZJY Jω / )with 0 ≤ Z = [ ω ] x ∗ ... x ∗ n a ∗ a (cid:0) x . . . x n (cid:1) [ ω ] ≤ k a k X. Moreover, these actions give *-representations of M : ( aξ | η ) = ( ξ | a ∗ η ) and ( ξa | η ) =( ξ | ηa ∗ ) for ξ, η ∈ L ( M ) and a ∈ M , which is immediate from the definition ofinner product. SHIGERU YAMAGAMI
The *-operation on L ( M ) is also compatible with the inner product: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)M j x j ⊗ ω / j ⊗ y j ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)M j y ∗ j ⊗ ω / j ⊗ x ∗ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = ( Y ω / | ω / X )= (( ω / X ) ∗ | ( Y ω / ) ∗ ) = ( Xω / | ω / Y )= (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)M j x j ⊗ ω / j ⊗ y j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . In this way, we have constructed a *-bimodule L ( M ) of M in such a way that L ( M, ϕ ) ⊂ L ( M ) for each ϕ ∈ M + ∗ and the closed subspaces M ϕ / , ϕ / M in L ( M ) are naturally identified with the left and right GNS spaces of ϕ respectively.Moreover, for ϕ, ψ ∈ M + ∗ , we have [ ϕ ] M ψ / = ϕ / M [ ψ ] in L ( M ), which is justa reflection of the fact that the same identification inside L ( M n ( M ) , ω ) is used inthe definition of inner product.2. Modular Algebras
Recall the definition of (boundary) modular algebra which was introduced in [10]to ‘resolve’ various cocycle relations in modular theory. We shall here describe itwithout essential use of the notion of weights.Let M be a W*-algebra which is assumed to admit a faithful ω ∈ M + ∗ for themoment. The modular algebra M ( i R ) of M is then the *-algebra generated byelements in M and symbols ϕ it for ϕ ∈ M + ∗ and t ∈ R under the conditions that(i) M ( i R ) contains M as a *-subalgebra, (ii) { ϕ it } t ∈ R is a one-parameter groupof partial isometries satisfying ϕ i = [ ϕ ], (iii) ϕ it a = σ ϕ,ψt ( a ) ψ it for ϕ, ψ ∈ M + ∗ , a ∈ [ ϕ ] M [ ψ ] and t ∈ R .By utilizing a faithful ω ∈ M + ∗ , it turns out that M ( i R ) is *-isomorphic to thealgebraic crossed product of M by { σ ωt } and therefore M ( i R ) is an algebraic directsum of M ( it ) = M ω it = ω it M , where M ( it ) = P ϕ ∈ M + ∗ M ϕ it M .Thus the modular algebra M ( i R ) is i R -graded in the sense that M ( it ) ∗ = M ( − it ), M ( is ) M ( it ) = M ( i ( s + t )) and M ( i
0) = M .We say that an element a ∈ M is finitely supported if a = [ ϕ ] a [ ϕ ] for some ϕ ∈ M + ∗ . Let M f be the set of finitely supported elements in M . Lemma 2.1. M f is a w*-dense *-subalgebra of M and closed under sequentialw*-limits in M . Moreover, M f = X ϕ ∈ M + ∗ M [ ϕ ] = X ϕ ∈ M + ∗ [ ϕ ] M. Proof.
Clearly M f is closed under the *-operation and M f is a subalgebra in viewof [ ϕ ] ∨ [ ψ ] ≤ [ ϕ + ψ ]. The *-subalgebra M f is then w*-dense in M in view of ∨ ϕ ∈ M + ∗ [ ϕ ] = 1. If a = a [ ϕ ], [ aϕa ∗ ] is the left support of a and [ ϕ + aϕa ∗ ] a [ ϕaϕa ∗ ] = a . Let a be a w*-limit of { a n } n ≥ in M f with [ ϕ n ][ a n ][ ϕ n ] = a n for n ≥
1. Then,for ϕ = P ∞ n =1 − n ϕ n /ϕ n (1) ∈ M + ∗ , [ ϕ ] a [ ϕ ]. (cid:3) RACE FORMULAS 5
Now we relax the existence of faithful functionals in M + ∗ and set M f ( i R ) = [ ϕ ∈ M + ∗ [ ϕ ] M [ ϕ ]( i R ) , where the natural inclusions [ ϕ ] M [ ϕ ]( i R ) ⊂ [ ψ ] M [ ψ ]( i R ) for ϕ, ψ ∈ M + ∗ satisfying[ ϕ ] ⊂ [ ψ ] are assumed in the union.Finally we add formal expressions of the form ω it = P j ∈ I ω itj for families { ω j ∈ M + ∗ } j ∈ I of mutually orthogonal supports and allow products with elements in M to get { M ( it ) } t ∈ R so that M f ( it ) ⊂ M ( it ) and M (0) = M . In what follows, aformal sum ω = P j ∈ I ω j is referred to as a weight of M . A weight ω = P ω j is said to be faithful if 1 = P [ ω j ] in M . Note that any weight is extended to afaithful one and { ω it } is a one-parameter group of unitaries in M ( i R ) = ⊕ M ( it ) fora faithful ω and, for another choice of a faithful weight φ = P k ∈ J φ k and a ∈ M , φ it aω − it = X j,k φ itk aω − itj defines a continuous family of elements in M so that it consists of unitaries when a = 1 and σ ωt ( a ) = ω it aω − it gives an automorphic action of R on M . Remark . Here weights are introduced in a formal and restricted way.At this stage, we introduce two more classes of modular algebraic stuffs: M ( i R + 1 /
2) = X t ∈ R M ( it + 1 / , M ( i R + 1) = X t ∈ R M ( it + 1) , with M ( it + 1 /
2) = X ϕ ∈ M + ∗ M ϕ it +1 / = X ϕ ∈ M + ∗ ϕ it +1 / M and M ( it + 1) = X ϕ ∈ M + ∗ M ϕ it +1 = X ϕ ∈ M + ∗ ϕ it +1 M so that M (1 /
2) = L ( M ) and M (1) = M ∗ .These are i R -graded *-bimodules of M ( i R ) in an obvious way and we have anatural module map M ( i R + 1 / ⊗ M ( i R ) M ( i R + 1 / → M ( i R + 1) which respectsthe grading in the sense that M ( is + 1 / M ( it + 1 /
2) = M ( i ( s + t ) + 1) for s, t ∈ R .In particular, given a weight ω on M , we have M ( it + s ) = M ( s ) ω it = ω it M ( s ) for s = 1 / , ϕ ∈ M ∗ at the unit 1 ∈ M is called the expectation of ϕ anddenoted by h ϕ i . Note that the expectation satisfies the trace property for variouscombinations of multiplications such as h aϕ i = h ϕa i and h ϕ it ξψ − it η i = h ψ − it ηϕ it ξ i for a ∈ M , ϕ, ψ ∈ M + ∗ and ξ, η ∈ L ( M ).The scaling ϕ e − s ϕ on M + ∗ gives rise to a *-automorphic action θ s of s ∈ R (called the scaling automorphisms ) on these modular stuffs: θ s ( xϕ it + r ) = e − ist − sr xϕ it + r for x ∈ M , r ∈ { , / , } and t ∈ R . Remark . Since elements in L ( M ) and M ∗ are always ‘finitely supported’, we candecribe M ( it + s ) ( s = 1 / ,
1) without referring to weights.
SHIGERU YAMAGAMI Analytic Properties
We here collect well-known analytic properties of modular stuffs (proofs can befound in [1] and [8] for example).
Lemma 3.1 (Modular Extension) . For ϕ, ψ ∈ M + ∗ and a ∈ M , R ∋ t ϕ it aψ − it ∈ M ∗ is extended analytically to a norm-continuous function ϕ iz aψ − iz on the strip − ≤ Im z ≤ k ϕ it + r aψ − it +1 − r k ≤ k ϕa k r k aψ k − r (0 ≤ r ≤ ϕ iz aψ − iz ) | z = t − i = ϕ it aψ − it , ( ϕ iz aψ − iz ) | z = t − i/ = ϕ it +1 / aψ − it +1 / . Corollary 3.2 (KMS condition) . Let ϕ, ψ ∈ M ∗ + and a ∈ [ ϕ ] M [ ψ ]. Then thefunction σ ϕ,ψt ( a ) ψ / = ϕ it aψ − it ψ / of t ∈ R is analytically extended to an L ( M )-valued continuous function ϕ iz aψ − iz +1 / of z ∈ R − i [0 , /
2] so that ( ϕ iz aψ − iz +1 / ) z = t − i/ = ϕ / ϕ it aψ − it = ϕ / σ ϕ,ψt ( a ). Lemma 3.3.
Let ω ∈ M + ∗ be faithful and let a ∈ M . Then the following conditionsare equivalent.(i) The inequality a ∗ ωa ≤ ω holds in M + ∗ .(ii) We can find a function a ( z ) ∈ M of z ∈ R − i [0 , /
2] such that a ( t ) = ω it aω − it for t ∈ R , a ( z ) ξ ∈ L ( M ) is norm-analytic in z for any ξ ∈ L ( M )and k a ( − i/ k ≤ b ∈ M satisfying k b k ≤ ω / a = bω / .Moreover, if this is the case, with the notation in (ii), ξa ( z ) ∈ L ( M ) is norm-continuous in z for every ξ ∈ L ( M ). Corollary 3.4.
For ϕ, ψ ∈ M + ∗ , the following conditions are equivalent.(i) The inequality ϕ ≤ ψ holds in M + ∗ .(ii) [ ϕ ] ≤ [ ψ ] and the function ϕ it ψ − it of t ∈ R is analytically extended to an M -valued function ϕ iz ψ − iz of z ∈ R − i [0 , /
2] so that ϕ iz ψ − iz ξ ∈ L ( M )is norm-continuous in z for any ξ ∈ L ( M ) and k ϕ / ψ − / k ≤ c ∈ M satisfying k c k ≤ ϕ / = cψ / .Moreover, if this is the case, ξϕ iz ψ − iz ∈ L ( M ) is norm-continuous in z ∈ R − i [0 , /
2] for any ξ ∈ L ( M ). Remark . Under the above majorization conditions, the relevant analytic exten-sions are norm-bounded as M -valued functions of z ∈ R − i [0 , /
2] thanks to Banach-Steinhaus theorem. 4.
Sectional Continuity
We now describe continuity properties of families { M ( it + r ) } t ∈ R for r = 0 , / ϕ = P ϕ j and ψ = P ψ k be weights on M in our sense. For ξ ∈ L ( M ), ϕ it ξψ − it = X j,k ϕ itj ξψ − itk RACE FORMULAS 7 is norm-continuous in t ∈ R as an orthogonal sum of L ( M )-valued norm-continuousfunctions ϕ itj ξψ − itk . As any φ ∈ M + ∗ has an expression ξη with ξ, η ∈ L ( M ), onesees that ϕ it φψ − it = ( ϕ it ξω − it )( ω it ηψ − it )( ω being an auxiliary faithful weight) is an M ∗ -valued norm-continuous function of t ∈ R as a product of L ( M )-valued norm-continuous functions.The following facts on continuity of sections of { M ( it + r ) } are then more or lessstraightforward from this observation. Lemma 4.1.
For a section x = { x ( t ) } of { M ( it ) } , the following conditions areequivalent.(i) There exists a faithful weight ω on M such that ω − it x ( t ) ∈ M is w*-continuous in t ∈ R .(ii) There exists a faithful weight ω on M such that x ( t ) ω − it ∈ M is w*-continuous in t ∈ R .(iii) For any faithful weight ω on M , ω − it x ( t ) ∈ M is w*-continuous in t ∈ R .(iv) For any faithful weight ω on M , x ( t ) ω − it ∈ M is w*-continuous in t ∈ R .(v) For any φ ∈ M + ∗ , φ − it x ( t ) ∈ M is w*-continuous in t ∈ R .(vi) For any φ ∈ M + ∗ , x ( t ) φ − it ∈ M is w*-continuous in t ∈ R .Moreover, if { x ( t ) } satisfies these equivalent conditions, k x ( t ) k is locally boundedin t ∈ R .We say that a section { x ( t ) } is w*-continuous if it satisfies any of these equiv-alent conditions.We here introduce the *-operation on sections by x ∗ ( t ) = x ( − t ) ∗ ∈ M ( it + s ) for a section { x ( t ) ∈ M ( it + r ) } .As a consequence of the above lemma, for a section x ( t ) ∈ M ( it ), x ∗ ( t ) as well as ax ( t ) b with a, b ∈ M are w*-continuous if so is x ( t ). Lemma 4.2.
Let p = 1 or 2 with the notation L ( M ) = M ∗ for p = 1. Then thefollowing conditions on a section { ξ ( t ) } of { M ( it + 1 /p ) } are equivalent.(i) There exists a faithful weight ω on M such that ω − it ξ ( t ) ∈ L p ( M ) isnorm-continuous in t ∈ R .(ii) There exists a faithful weight ω on M such that ξ ( t ) ω − it ∈ L p ( M ) isnorm-continuous in t ∈ R .(iii) For any faithful weight ω on M , ω − it ξ ( t ) ∈ L p ( M ) is norm-continuous in t ∈ R .(iv) For any faithful weight ω on M , ξ ( t ) ω − it ∈ L p ( M ) is norm-continuous in t ∈ R .(v) For any φ ∈ M + ∗ , φ − it ξ ( t ) ∈ L p ( M ) is norm-continuous in t ∈ R .(vi) For any φ ∈ M + ∗ , ξ ( t ) φ − it ∈ L p ( M ) is norm-continuous in t ∈ R .We say that a section { ξ ( t ) } is norm-continuous if it satisfies any of these equiv-alent conditions. Notice here that ξ ∗ ( t ) = ξ ( − t ) ∗ is norm-continuous if so is ξ ( t ). Definition 4.3.
A section { x ( t ) } of { M ( it ) } t ∈ R is said to be s*-continuous if x ( t ) ξ and ξx ( t ) are norm-continuous for any ξ ∈ L ( M ). Notice that x ∗ ( t ) is s*-continuous if and only if so is x ( t ) in view of x ∗ ( t ) ξ = ( ξ ∗ x ( − t )) ∗ and ξx ∗ ( t ) =( x ( − t ) ξ ∗ ) ∗ . Clearly s*-continuous sections are w*-continuous. SHIGERU YAMAGAMI
To control the norm of a w*-continuous section x = { x ( t ) ∈ M ( it ) } , two normsare introduced by k x k ∞ = sup {k x ( t ) k ; t ∈ R } , k x k = Z R k x ( t ) k dt and x ( t ) is said to be bounded if k x k ∞ < ∞ and integrable if k x k < ∞ . Notehere that k x ( t ) k is locally bounded and lower-semicontinuous. Lemma 4.4.
The following conditions on a section { x ( t ) } of { M ( it ) } are equiva-lent. (i) For any ξ ∈ L ( M ), { x ( t ) ξ } is a norm-continuous section of { M ( it +1 / } .(ii) For any ϕ ∈ M + ∗ and any ξ ∈ L ( M ), x ( t ) ϕ − it ξ ∈ L ( M ) is norm-continuous in t ∈ R .(iii) The norm function k x ( t ) k is locally bounded and, for a sufficiently large φ ∈ M + ∗ , x ( t ) φ − it +1 / ∈ L ( M ) is norm-continuous in t ∈ R , i.e., given any ϕ ∈ M + ∗ , we can find φ ∈ M + ∗ such that ϕ ≤ φ and x ( t ) φ − it +1 / ∈ L ( M )is norm-continuous in t ∈ R . Corollary 4.5.
A section x ( t ) ∈ M ( it ) is s*-continuous if and only if k x ( t ) k islocally bounded and L ( M )-valued functions x ( t ) φ − it +1 / , φ − it +1 / x ( t ) are norm-continuous for a sufficiently large φ ∈ M + ∗ .A section { x ( t ) ∈ M ( it ) } t ∈ R is said to be finitely supported if we can find φ ∈ M + ∗ so that x ( t ) = [ φ ] x ( t )[ φ ] for every t ∈ R . We say that { x ( t ) } is locallybounded (bounded) if so is the function k x ( t ) k of t .5. Convolution Algebra
Consider a bounded, s*-continuous and integrable section { f ( t ) ∈ M ( it ) } andidentify it with a formal expression like Z R f ( t ) dt , which is compatible with the*-operation by (cid:18)Z R f ( t ) dt (cid:19) ∗ = Z R f ( t ) ∗ dt = Z R f ( − t ) ∗ dt = Z R f ∗ ( t ) dt. Moreover, a formal rewriting Z R f ( s ) ds Z R g ( t ) dt = Z R (cid:18)Z R f ( s ) g ( t − s ) ds (cid:19) dt suggests to define a product of f and g by( f g )( t ) = Z R f ( s ) g ( t − s ) ds = Z R f ( t − s ) g ( s ) ds. It is then a routine work to check that the totality of such sections constitutes anormed *-algebra in such a way that k f g k ∞ ≤ ( k f k k g k ∞ ) ∧ ( k f k ∞ k g k ) , k f g k ≤ k f k k g k . We notice that the scaling automorphism θ s on { M ( it ) } induces a *-automorphicaction on the *-algebra of sections by ( θ s f )( t ) = e − ist f ( t ).Here we shall apply formal arguments to illustrate how tracial functionals canbe associated to this kind of *-algebras. RACE FORMULAS 9
Formal manipulation is an easy business : Imagine that a section f ( t ) has ananalytic extension to the region − ≤ Im z ≤ f ∗ ( z ) = f ( − z ) ∗ and define a linear functional by τ (cid:18)Z R f ( t ) dt (cid:19) = h f ( − i ) i . Note that f ( − i ) in the right hand side belongs to M (1) = M ∗ . We then have τ ( f ∗ f ) = Z R h f ∗ ( s ) f ( − i − s ) i ds = Z R h f ∗ ( s − i/ f ( − s − i/ i ds = Z R h f ( − s − i/ ∗ f ( − s − i/ i ds ≥ , where Cauchy’s integral theorem is formally used in the first line. The trace prop-erty is seen from τ ( f g ) = Z h f ( s ) g ( − i − s ) i ds = Z h f ( t − i ) g ( − t ) i dt = Z h g ( − t ) f ( t − i ) i dt = Z h g ( t ) f ( − t − i ) i dt = τ ( gf ) , where Cauchy’s integral theorem is again used formally in the first line.Going back to the sane track, it turns out that it is not easy to make all of theabove formal arguments rigorous at least in a reference-weight-free fashion. Insteadwe shall construct a Hilbert algebra as a halfway business in what follows, which isenough to extract the tracial functional.6. Hilbert Algebras
Definition 6.1.
A section { f ( t ) ∈ M ( it ) } is said to be half-analytic if, fora sufficiently large φ ∈ M + ∗ , the function f φ ( t ′ , t ′′ ) = φ − it ′ f ( t ′ + t ′′ ) φ − it ′′ of( t ′ , t ′′ ) ∈ R is analytically extended to a bounded M -valued s*-continuous function f φ ( z ′ , z ′′ ) = φ − iz ′ f ( z ′ + z ′′ ) φ − iz ′′ of ( z ′ , z ′′ ) ∈ T [0 , / .Note here that sufficient largeness in the condition has a meaning: For a φ majorized by ω ∈ M + ∗ , φ it ω − it is analytically extended to a s*-continuous function φ iz ω − iz of z ∈ R − i [0 , /
2] (Corollary 3.4) and therefore ω − it ′ f ( t ′ + t ′′ ) ω − it ′′ has ananalytic extension of the form ( ω − iz ′ φ iz ′ )( φ − iz ′ f ( z ′ + z ′′ ) φ − iz ′′ )( φ iz ′′ ω − iz ′′ ), whichis s*-continuous as a product of s*-continuous locally bounded operator-valuedfunctions.Note also that the s*-continuity of φ − iz ′ f ( z ′ + z ′′ ) φ − iz ′′ is equivalent to the norm-continuity of L -valued functions ( φ − iz ′ f ( z ′ + z ′′ ) φ − iz ′′ ) φ / and φ / ( φ − iz ′ f ( z ′ + z ′′ ) φ − iz ′′ ) (Lemma 4.4). These are analytic extensions of φ − it ′ f ( t ′ + t ′′ ) φ / − it ′′ and φ / − it ′ f ( t ′ + t ′′ ) φ − it ′′ , whence simply denoted by φ − iz ′ f ( z ′ + z ′′ ) φ / − iz ′′ and φ / − iz ′ f ( z ′ + z ′′ ) φ − iz ′′ respectively. Warning : No separate meaning of f ( z ) is assigned here.It is immediate to see that f ( t ) is half-analytic if and only if so is f ∗ ( t ) = f ( − t ) ∗ in such a way that(1) φ − iz ′ f ∗ ( z ′ + z ′′ ) φ − iz ′′ = (cid:0) φ iz ′′ f ( − z ′′ − z ′ ) φ iz ′ (cid:1) ∗ . To get the convolution product in a manageable way, we impose the followingdecaying condition. For a half-analytic section f ( t ) ∈ M ( it ), the obvious identity f φ ( z ′ + s ′ , z ′′ + s ′′ ) = φ − is ′ f φ ( z ′ , z ′′ ) φ − is ′′ shows that k f φ ( z ′ , z ′′ ) k depends only on r ′ = − Im z ′ , r ′′ = − Im z ′′ and t = Re ( z ′ + z ′′ ), which enables us to introduce | f | φ ( t ) = sup {k f φ ( z ′ , z ′′ ) k ; r ′ ≥ , r ′′ ≥ , r ′ + r ′′ ≤ / } . A half-analytic section f ( t ) is said to be of Gaussian decay if, for a sufficientlylarge φ ∈ M + ∗ , we can find δ > | f | φ ( t ) = O ( e − δt ).Now let N be the vector space of half-analytic sections of Gaussian decay,which is closed under taking the *-operation by (1). It is immediate to see thatthe scaling automorphisms leave N invariant so that φ − iz ′ ( θ s f )( z ′ + z ′′ ) φ − iz ′′ = e − is ( z ′ + z ′′ ) φ − iz ′ f ( z ′ + z ′′ ) φ − iz ′′ .Let f, g ∈ N . Thanks to the Gaussian decay assumption, the convolution product f g has a meaning and ( f g )( t ) is an s*-continuous section. To see f g ∈ N , wetherefore need to check that it admits a half-analytic extension of Gaussian decay.Choose an auxiliary weight ω which supports both f and g . Then φ − iz ′ (( f g )( z ′ + z ′′ ) φ − iz ′′ )= Z R φ − iz ′ f ( z ′ + s ) ω − is σ ωs ( g ( z ′′ − s ) φ − i ( z ′′ − s ) ) ω is φ − is ds gives the s*-continuous analytic extension with its norm estimated by k φ − iz ′ ( f g )( z ′ + z ′′ ) φ − iz ′′ k≤ Z R | f | φ ( Re z ′ + s ) | g | φ ( Re z ′′ − s ) ds = O ( e − ǫδt / ( ǫ + δ ) )for t = Re ( z ′ + z ′′ ) if | f | φ ( t ) = O ( e − ǫt ) and | g | φ ( t ) = O ( e − δt ).So far N is shown to be a *-algebra with an automorphic action of R by scalingautomorphisms. We next introduce an inner product which makes N into a Hilbertalgebra. Lemma 6.2.
The following identity holds for f ∈ N and sufficiently large φ, ϕ ∈ M + ∗ . [ ϕ ] (cid:0) f ( t − i/ φ − it − / (cid:1) φ it +1 / = ϕ it +1 / (cid:0) ϕ − it − / f ( t − i/ (cid:1) [ φ ](the left hand side is therefore depends only on [ ϕ ] while the right hand side dependsonly on [ φ ] and the common element in M ( it + 1 /
2) is reasonably denoted by[ ϕ ] f ( t − i/ φ ]). Proof.
For a ∈ M , the identity h ( f ( t ) φ − it ) φ / σ φ,ϕt ( a ) ϕ / i = h ϕ / ( ϕ − it f ( t )) φ / a i is analytically continued from t to t − i/ h ( f ( t − i/ φ − it − / ) φ / φ / σ φ,ϕt ( a ) i = h ϕ / ( ϕ − it − / f ( t − i/ φ / a i (use the KMS-condition at σ φ,ϕt ( a ) ϕ / ) and, after a simple rewriting, h ( f ( t − i/ φ − it − / ) φ it +1 / φ / aϕ − it i = h ϕ it +1 / ( ϕ − it − / f ( t − i/ φ / aϕ − it i . (cid:3) RACE FORMULAS 11
Since [ ϕ ] f ( t − i/ φ ] = [ ϕ ]([ ϕ ′ ] f ( t − i/ φ ′ ])[ φ ] whenever [ ϕ ] ≤ [ ϕ ′ ] and [ φ ] ≤ [ φ ′ ], k [ ϕ ] f ( t − i/ φ ] is increasing in [ ϕ ] and [ φ ]. We claim that f ( t − i/
2) = lim [ ϕ ] → φ ] → [ ϕ ] f ( t − i/ φ ]exists in M ( it + 1 / ϕ n and φ n in M + ∗ so that lim n →∞ k [ ϕ n ] f ( t − i/ φ n ] k = ∞ , which contradicts with k [ ϕ n ] f ( t − i/ φ n ] k ≤ k [ ϕ ] f ( t − i/ φ ] k < ∞ for the choice ϕ = P ϕ n / n k ϕ n k , φ = P φ n / n k φ n k .Moreover, the same reasoning reveals that we can find ϕ, φ ∈ M + ∗ so that f ( t − i/
2) = [ ϕ ] f ( t − i/
2) = f ( t − i/ φ ]. Consequently, { f ( t − i/ ∈ M ( it + 1 / } is anorm-continuous section of Gaussian decay from the expression f ( t − i/
2) = f ( t − i/ φ ] = (cid:0) f ( t − i/ φ − it − / (cid:1) φ it +1 / which is valid for a sufficiently large φ . Remark . By an analytic continuation, one sees that any half-analytic section { f ( t ) } of { M ( it ) } is finitely supported in the sense that there exists φ ∈ M + ∗ satisfying f ( t ) = [ φ ] f ( t )[ φ ] for every t ∈ R . Example 6.3.
Let ϕ ∈ M + ∗ and a, b ∈ [ ϕ ] M [ ϕ ] be entirely analytic for σ ϕt . Then,for α > β ∈ C , f ( t ) = e − αt + βt aϕ it b belongs to N and its boundary sectionis f ( t − i/
2) = e − α ( t − i/ + β ( t − i/ aϕ it +1 / b .The inner product is now introduced by( f | g ) = Z R ( f ( t − i/ | g ( t − i/ dt = Z R h f ( t − i/ ∗ g ( t − i/ i dt, which is clearly positive-definite and the completed Hilbert space H is naturallyidentified with the direct integral H = I R M ( it + 1 / dt because N provides a dense set of measurable sections in the right hand side. TheHilbert space H is then made into a *-bimodule of M ( i R ) by aω is I R ξ ( t ) dt = I R aω is ξ ( t − s ) dt and (cid:18)I ξ ( t ) dt (cid:19) ∗ = I R ξ ( − t ) ∗ dt in such a way that actions of M ( it ) on H are s*-continuous.Since the family { M ( it +1 / } is trivialized by obvious isomorphisms L ( M ) ω it ∼ = L ( M ) ∼ = ω it L ( M ) in terms of a faithful weight ω on M , we have identifications H ∼ = L ( M ) ⊗ L ( R ) in two ways, which transforms left and right multiplicationsof ω it into a translational unitary by t ∈ R . Recall that our weights are orthogo-nal direct sums of bounded functionals and the multiplication of ω is on H gives acontinuous one-parameter group of unitaries.With these observations in mind, it is immediate to check the axioms of Hilbertalgebra: the left and right multiplications are bounded with respect to the innerproduct, N is dense in H and ( f ∗ | g ∗ ) = ( g | f ) for f, g ∈ N . Remark . Note that the scaling automorphism θ s satisfies ( θ s f )( t − i/
2) = e − ist − s/ f ( t − i/
2) and hence scales the inner product: ( θ s f | θ s g ) = e − s ( f | g ) for f, g ∈ N .In this way, we have constructed a Hilbert algebra N . The associated von Neu-mann algebra is denoted by N = M ⋊R and referred to as the Takesaki dual of M in what follows. The scaling automorphisms θ s of N induce a *-automorphic action(also denoted by θ s ) of R on N by θ s ( l ( f )) = l ( θ s f ), which is referred to as the dual action . Here l ( f ) denotes a bounded operator on H defined by l ( f ) g = f g for g ∈ N .Let ω be a faithful weight on M . From the convolution form realization of N on H , one sees that N contains M as well as ω it as operators by left multiplication andthese in turn generates N . Likewise right multiplications of M and ω it generatesthe right action of N on H . Thus the Takesaki dual of M is isomorphic to thecrossed product of M with respect to the modular automorphism group { σ ωt } ,which justifies our notation M ⋊ R for N .We record here the following well-known fact for later use together with a proof toillustrate how the essence can be easily captured in the modular algebra formalism. Theorem 6.4 (Takesaki) . The fixed-point algebra N θ of N under the dual action θ is identified with M . Proof.
Through H ∼ = L ( M ) ⊗ L ( R ) adapted to the trivialization M ( it +1 / ω − it = L ( M ) of M ( it + 1 / ω is is realized on L ( R ) by translationswhereas θ s by multiplication of e − ist on L ( R ). Since these generate B ( L ( R ))(Stone-von Neumann), N θ is identified with ( B ( L ( M )) ⊗ ∩ End( H M ). Let a ∈ M and f, g ∈ L ( R ). For ξ, η ∈ L ( M ),( η ⊗ g | ( ξ ⊗ f ) a ) = ( η | ξσ ωgf ) with σ ωgf = Z R g ( t ) f ( t ) ω it aω − it ∈ M shows that T ∈ B ( L ( M )) belongs to N θ if and only if it is in the commutant ofthe right action of { σ ωh ( a ); h ∈ L ( R ) } on L ( M ). Since { σ ωh ( a ); a ∈ M, h ∈ L ( R ) } generates M , this implies N θ ⊂ M . (cid:3) Now we introduce some notations and conventions in connection with our Hilbertalgebra: N is regarded as a *-subalgebra of N and we write N τ / = τ / N toindicate the corresponding subspace in H = I R M ( it + 1 / dt , where τ / is just adummy symbol but its square τ will be soon identified with the standard trace on N . Thus h ∈ N is identified with an operator on H satisfying h ( f τ / ) = ( hf ) τ / for f ∈ N , whereas f τ / = τ / f = I R f ( t − i/ dt .Let B ⊃ N be a dense *-ideal of N such that Bτ / = τ / B is the set ofbounded vectors in H ; y ∈ N belongs to B if and only if there exists a vector η ∈ H satisfying ηf = y ( f τ / ) = y ( τ / f ) for any f ∈ N and, if this is the case,we write η = yτ / = τ / y . Recall that the standard trace τ on N + is defined by τ ( y ∗ y ) = ( yτ / | yτ / ) if y ∈ B and τ ( y ∗ y ) = ∞ otherwise. Note that, for f, g ∈ N , f ∗ g ∈ N is in the trace class and its trace is calculated by τ ( f ∗ g ) = Z R ( f ( t − i/ | g ( t − i/ dt = ( f τ / | gτ / ) , which justifies our notation f τ / . RACE FORMULAS 13
From the scaling relation ( θ s f )( t − i/
2) = e − ist − s/ f ( t − i/ e − s under the *-automorphism of N and hence the associatedtrace τ scales like τ ( θ s ( y ∗ y )) = e − s τ ( y ∗ y ) for y ∈ N .To each ξ, η ∈ H , a sesquilinear element ξ ∗ η ∈ N ∗ is associated by h ξ ∗ η, x i =( ξ | ηx ) and a ∗ bτ = τ a ∗ b ∈ N ∗ is defined to be ( aτ / ) ∗ ( bτ / ) for a, b ∈ B .As a square root of this correspondence, we have a unitary map H → L ( N ) insuch a way that | a | τ / ( a ∗ aτ ) / for a ∈ B . Therefore, if we set B + = B ∩ N + ,the closure of B + τ / = τ / B + in H corresponds to the positive cone L ( N ) + .Related to these, we recall the following well-known and easily proved fact (cf. [7]Corollary 19.1). Lemma 6.5.
The Hilbert space H is canonically isomorphic to the vector spaceof Hilbert-Schmidt class operators with respect to τ in such a way that τ ( y ∗ y ) =( yτ / | yτ / ). Note that a closed operator y affiliated to N is in the Hilbert-Schmidtclass if and only if τ ( y ∗ y ) < ∞ .7. Trace Formula
We shall now utilize the Hilbert algebra structure behind N to set up a methodmodeled after N to calculate the standard trace τ on N .Given an open interval I ⊂ [0 , / e F I be the set of M -valued analyticfunctions of z ∈ R − iI and set F I = ∪ φ ∈ M + ∗ [ φ ] e F I [ φ ]. We write f φ ( z ) φ iz for φ ∈ M + ∗ and f φ ∈ [ φ ] F I [ φ ] to indicate dummies of elements in F I . All such dummies are thenidentified by the relation ϕ iz = ( ϕ iz ψ − iz ) ψ iz whenever ϕ ≤ ψ and the obtainedquotient set (which is a kind of inductive limit of dummy elements) is denoted by LI I and an element in LI I is called a left interpolator on I .Thus each left interpolator is of the form f ( z ) = f ϕ ( z ) ϕ iz and we say that f ( z )is supported by ϕ . Then, for φ ∈ M + ∗ mojorizing ϕ , f ( z ) is supported by φ and f φ ( z ) = f ϕ ( z )( ϕ iz φ − iz ), which is also denoted by f ( z ) φ − iz .Clearly we have a similar notion of right interpolators with the obvious notationsfor them. These are related by the *-operation defined by f ∗ ( z ) = f ( − z ) ∗ : If f ∈ LI I , f ∗ ∈ RI I so that φ − iz f ∗ ( z ) = ( f ( − z ) φ iz ) ∗ .A pair ( l ( z ) , r ( z )) of left and right interpolators on I is called an interpolator if one can find φ ∈ M + ∗ which supports l , r and interrelates them in the followingsense: For each w ∈ R − iI , the function σ φt ( φ − iw r ( w )) of t ∈ R is analyticallyextended up to the horizontal line w + R so that the function σ φz ( φ − iw r ( w )) isw*-analytic on D = { ( z, w ) ∈ C ; w ∈ R − iI, Im w ≤ Im z ≤ } and satisfies σ φw ( φ − iw r ( w )) = l ( w ) φ − iw . Here, for z ∈ C \ R and a ∈ M , σ z ( a ) means that σ t ( a )( t ∈ R ) is analytically extended to a w*-continuous function of ζ ∈ R + i Im z [0 , ζ = z .Since analytical extensions are moved back to the starting horizontal lines, thecondition is symmetrical in the left-and-right: σ φ − t ( l ( w ) φ − iw ) is analytically ex-tended to σ φw − z ( φ − iw r ( w )), which is w*-continuous in ( z, w ) ∈ D . For ( z, w ) ∈ T I ,the relation σ φz + w ( φ − i ( z + w ) r ( z + w )) = l ( z + w ) φ − i ( z + w ) is then rewritten into σ φw ( φ − i ( z + w ) r ( z + w )) = σ φ − z ( l ( z + w ) φ − i ( z + w ) ), which is a w*-analytic functionof ( z, w ) ∈ T I and denoted by φ − iz f ( z + w ) φ − iw when ( l ( z ) , r ( z )) is symbolicallyexpressed by f ( z ). Moreover, the interrelating condition is compatible with the majorization changes:Let φ ≤ ω and z ∈ R − iI . Then σ ωt ( ω − iz r ( z )) = ( ω − i ( z − t ) φ i ( z − t ) ) σ φt ( φ − iz r ( z )) φ it ω − it is analytically continued from t to z to get ( l ( z ) φ − iz )( φ iz ω − iz ) = l ( z ) ω − iz .We say that an interpolator f ( z ) = ( l ( z ) , r ( z )) is supported by φ ∈ M + ∗ if both l ( z ) and r ( z ) are supported by φ and, in that case, we write φ f ( z ) = φ − iz f ( z ) = φ − iz r ( z ) and f φ ( z ) = f ( z ) φ − iz = l ( z ) φ − iz .Let I I be the set of interpolators on I . By restriction or extension, I J ⊂ I I if I ⊂ J ⊂ (0 , / I I is defined by ( l ( z ) , r ( z )) ∗ = ( r ∗ ( z ) , l ∗ ( z ))so that it is compatible with the inclusions I J ⊂ I I . Notice that N can be regardedas a *-subspace of I (0 , / .Given an asymptotic function ρ : R \ [ − R, R ] → [0 , ∞ ) with R > f ( z ) on I is said to have a ρ -growth and denoted by f ( z ) = O ( ρ ( Re z )) if we can find C > k φ − iz f ( z + w ) φ − iw k ≤ Cρ ( Re ( z + w )) forany ( z, w ) ∈ T I satisfying z + w ∈ R \ [ − R, R ] − iI . Note that the growth conditionis well-defined thanks to the half-power analyticity for majorization.An interpolator f is said to be of sub-gaussian growth if, for any small ǫ > f ( z ) φ − iz = O ( e ǫ ( Re z ) ). Let I gI be the set of interpolators of sub-gaussian growth.For f ∈ I gI with I = ( α, β ) ⊂ [0 , / N as follows. Continuous functions F ( s, t ) = (cid:0) h ( t − i/ | f φ ( s − ir ) φ it +1 / φ g ( t − s + ir − i/ (cid:1) of ( s, t ) ∈ R parametrized by r ∈ I are of Gaussian decay with their absolutelyconvergent integrals independent of r ∈ I owing to Cauchy’s integral theorem.Moreover F ( s, t ) does not depend on the choice of supporting φ either.Thus a sesqui-linear form h | i f on N is well-defined by h h | g i f = Z R dsdt (cid:0) h ( t − i/ | f φ ( s − ir ) φ it +1 / φ g ( t − s + ir − i/ (cid:1) = Z R dsdt (cid:0) h ( t − i/ g ∗ ) φ ( − t + s + ir − i/ φ i ( s − t ) | f φ ( s − ir ) φ is +1 / (cid:1) as far as r ∈ I and φ ∈ M + ∗ supports f and g , which behaves well under the*-operation: h g | h i f ∗ = h h | g i f . Notice that, when f ∈ N , h h | g i f is reduced to( hτ / | f gτ / ).We interprete the sequilinear form h | i f as defining an operator K in a kernelform by ( hτ / | K ( gτ / )) = h h | g i f , which is referred to as the virtual operator of f ( z ) and denoted by f itself.Note that the *-operation on interpolators is compatible with the associated vir-tual operators; ( hτ / | f ( gτ / )) = ( gτ / | f ∗ ( hτ / )) for g, h ∈ N , and virtual op-erators are affiliated to N in the sense that ( hk ∗ τ / | f ( gτ / )) = ( hτ / | f ( gkτ / ))for g, h, k ∈ N .Let D ( f ) be the set of vectors gτ / ∈ N τ / which makes the conjugate-linearfunctional hτ / ( hτ / | f ( gτ / )) bounded. For gτ / ∈ D ( f ), if the vector ξ ∈ H satisfying ( hτ / | ξ ) is denoted by f ( gτ / ), then we obtain a linear operatoron H by D ( f ) ∋ gτ / f ( gτ / ) ∈ H .A virtual operator is said to be densely defined if D ( f ) is dense in H . When thesesqui-linear form h | i f itself is bounded, D ( f ) = N τ / and the associated linear RACE FORMULAS 15 operator N τ / → H is bounded and identified with an element y ∈ N in such away that h h | g i f = (cid:0) hτ / | y ( gτ / ) (cid:1) for g, h ∈ N .We next introduce the virtual vector as a conjugate-linear form on N τ / . Lemma 7.1. If φ ∈ M + ∗ supports g, h ∈ N , then vector-valued functions ( hg ∗ ) φ ( s ) φ / and φ / φ ( hg ∗ )( s ) of s ∈ R are analytically continued to L ( M )-valued norm-continuous functions ( hg ∗ ) φ ( z ) φ / and φ / φ ( hg ∗ )( z ) of z ∈ R − i [0 ,
1] so thatthese are of Gaussian decay and, for 0 ≤ r ≤ /
2, satisfy( hg ∗ ) φ ( s − i (1 − r )) φ / = Z R h ( t − i/ g ∗ ) φ ( − t + s + ir − i/ φ − it dt,φ / φ ( hg ∗ )( s − i (1 − r )) = Z R φ itφ h ( t + s + ir − i/ g ∗ ( − t − i/ dt respectively. Proof.
We already know that ( hg ∗ ) φ ( s ) has an s*-continuous analytic extension( hg ∗ ) φ ( z ) ∈ M to z ∈ R − i [0 , /
2] so that ( hg ∗ ) φ ( s − i/ φ / = f ( s − i/ φ − is ,whereas ( hg ∗ ) φ ( s − i/ φ / = Z h ( t − i/ g ∗ φ ( − t + s ) φ − it dt is analytically continued to the norm-continuous function Z h ( t − i/ g ∗ φ ( − t + z ) φ − it dt of z ∈ R − i [0 , / (cid:3) The sesqui-linear form h h | g i f is now expressed by h h | g i f = Z R ds (cid:0) ( hg ∗ ) φ ( s − i (1 − r )) φ is +1 / | f φ ( s − ir ) φ is +1 / (cid:1) , whenever 0 < r < φ supports g , h as well as f , which reveals that a conjugate-linear form f τ / on N τ / is well-defined by the relation( hg ∗ τ / | f τ / ) = h h | g i f and called the vitual vector of f .Note that the virtual vector of f ∗ is given by ( f τ / ) ∗ which is defined by( ξ | ( f τ / ) ∗ ) = ( ξ | f τ / ) for ξ ∈ N τ / : h h | g i f ∗ = h g | h i f = ( gh ∗ τ / | f τ / ) = ( hg ∗ τ / | ( f τ / ) ∗ ) . These are also referred to as a boundary operator and a boundary vector for I = (0 , ν ) and I = ( ν, /
2) with additional notations R f ( t ) dt and H f ( t − i/ dt respectively. We now focus on these. Boundary Operator:
In extracting linear operators from the kernel form ofboundary operators, the following illustrates the meaning of boundary (limit).Let D be the set of s*-continuous sections of { M ( it + 1 / } of Gaussian de-cay, which is a topological vector space of inductive limit of Banach spaces D δ = {{ ξ ( t ) } ∈ { M ( it + 1 / } ; k ξ k δ < ∞} with k ξ k δ = sup { e δt k ξ ( t ) k ; t ∈ R } . The embedding D δ → H is norm-continuous and therefore so is D → H . For f ∈ I gI with I = (0 , ν ) and ξ ∈ D δ Z R f ϕ ( s − ir ) ϕ it ξ ds is norm-convergent in D δ ′ for any δ ′ < δ and gives a bounded linear map D δ → D δ ′ ,which depends continuously on r ∈ I in the norm-topology of B ( D δ , D δ ′ ). Theinduced continuous linear operator on D is then denoted by R R f ϕ ( s − ir ) ϕ is ds .We say that R R f ϕ ( s − ir ) ϕ is ds is bounded if it is bounded as a densely definedlinear operator on H .Note that, if R R f ϕ ( s − ir ) ϕ is ds ∈ B ( H ) is locally norm-bounded for r ∈ I , it iss-continuous in r ∈ I by the density of D in H . Lemma 7.2.
Let f ∈ I gI be supported by ϕ ∈ M + ∗ . Assume that D ∋ ξ Z R f ϕ ( s − ir ) ϕ is ξ ds ∈ H gives rise to a bounded linear operator y r = R R f ϕ ( s − ir ) ϕ is on H and y = Z R f ( s − i ds = lim r → +0 Z R f ϕ ( s − ir ) ϕ is ds exists in the w*-topology of N .Then the boundary operator of f ( z ) is bounded and given by the above limit. Proof.
Given g ∈ N and ϕ ∈ M + ∗ , choose φ ∈ M + ∗ so that it supports g andmajorizes ϕ . Then, R ∋ ( s, t ) ϕ is f ( t − s − i/ φ − it ∈ L ( M )is analytically extended to an L ( M )-valued norm-continuous function ( ϕ iz φ − iz ) (cid:0) φ iz g ( t − z − i/ (cid:1) of z ∈ R − i [0 , /
2] and t ∈ R , which is denoted by ϕ iz g ( t − z − i/ ϕ iz g ( t − z − i/
2) = ( ϕ iz φ − iz ) φ it +1 / φ g ( t − z − i/ ϕ iz g )( t − i/
2) = ϕ iz g ( t − z − i/
2) belongs to D δ as a function of t ∈ R if | g | φ ( t ) = O ( e − δt ). Thus, ξ r ( t ) = ϕ r g ( t + ir − i/
2) is a D δ -valued norm-analytic function of r .By our assumptions, s*-continuous family { y r } r ∈ I in N converges to y in w*-topology as r → +0, whence the operator norm k y r k is bounded in a neighborhoodof r = 0 and we see that y r ξ r = lim r → y r ξ r = lim ( r ′ ,r ′′ ) → (0 , y r ′ ξ r ′′ = lim r ′ → y r ′ ξ = yξ . Now the identity Z f φ ( s − ir ) φ it +1 / φ g ( t − s + ir − i/ ds = ( y r ξ r )( t )is used to get h h | g i f = ( hτ / | y r ξ r ) = ( hτ / | yξ ) = ( hτ / | y ( gτ / )) . (cid:3) Corollary 7.3.
Let f ( z ) be an interpolator on I = (0 , ν ) (0 < ν ≤ /
2) andsuppose that f is supported by a φ ∈ M + ∗ so that f φ ( z ) = f ( z ) φ − iz is a scalaroperator of polynomial growth with its horizontal Fourier transform R R f φ ( s − RACE FORMULAS 17 ir ) e isλ ds being in L ∞ ( R ) for a small r > c f φ ∈ L ∞ ( R ) as r →
0, then the boundary operator of f ( z ) is a bounded operator c f φ (log φ ) = Z R c f φ ( λ ) E ( dλ ) ∈ N. Here E ( · ) denotes the spectral measure of φ it : φ it = R R e itλ E ( dλ ). Proof.
Due to the left trivialization [ φ ] L ( N ) ∼ = L ( R ) ⊗ [ φ ] L ( M ), the whole thingis reduced to L ∞ ( R ) on L ( R ) and the classical harmonic analysis on the real lineworks. (cid:3) Example 7.4. If f ( z ) φ − iz extends to a bounded w*-continuous M -valued functionof z ∈ R − i [0 , ν ) in such a way that there exists an integrable function ρ ( t ) satisfying k f ( t − ir ) φ − i ( t − ir ) k ≤ ρ ( t ) for t ∈ R and 0 ≤ r < ν , then the boundary operator isbounded and hence belongs to N . Example 7.5.
For φ ∈ M + ∗ and µ ∈ C , consider an interpolator f ( z ) = µ + iz φ iz on I with I specified according to µ as follows:(i) I = (0 , /
2) ( Re µ ≥ π (1 ∨ φ ) − µ .(ii) Either I = (0 , − Re µ ) ( − / < Re µ <
0) or I = (0 , /
2) ( Re ≤ − / − π (1 ∧ φ ) − µ for Re µ < φ it = R R e itλ E ( dλ ),(1 ∨ φ ) − µ = Z ∞ e − µλ E ( dλ ) , (1 ∧ φ ) − µ = Z −∞ e − µλ E ( dλ ) . Boundary Vector:
We next look into boundary vectors. Let f ( z ) ∈ I gI with I = ( ν, /
2) and g, h ∈ N . In the expression( hg ∗ τ / | f τ / ) = Z R ds (cid:0) ( hg ∗ ) ϕ ( s − i (1 − r )) ϕ is +1 / | f ϕ ( s − ir ) ϕ is +1 / (cid:1) ( g , h and f ( z ) being supported by ϕ ∈ M + ∗ ), notice that the norm-convergencelim r → / ( hg ∗ ) ϕ ( s − i (1 − r )) ϕ / = ( hg ∗ )( s − i/ ϕ − is in L ( M ) is uniformly in s ∈ R and the domination k ( hg ∗ ) ϕ ( s − i (1 − r )) ϕ / k ≤ Ce − δs holds uniformly in r ,whereas k f ϕ ( s − ir ) k = O ( e ǫs ) uniformly in r for any ǫ > f ( z ) satisfies the condition that(i) ρ ϕ ( s ) = sup {k f ϕ ( s − ir ) k ; r ∈ ( ν, / } is a locally integrable function of s ∈ R for some supporting ϕ and(ii) we can find a locally integrable measurable section η ( s ) ∈ M ( is + 1 / φ and for almost all s , f φ ( s − ir ) φ is +1 / converges weakly to η ( s ) in M ( is + 1 /
2) as r → / hg ∗ τ / | f τ / ) = Z R (cid:0) ( hg ∗ )( s − i/ | η ( s ) (cid:1) ds, which shows that the boundary vector of f ( z ) is represented by the measurablesection η ( s ) ∈ M ( is + 1 / k η ( s ) k is of sub-gaussian growth. Example 7.6. If f φ ( z ) is extended to an M -valued w*-continuous function of z ∈ R − i ( ν, / ρ φ ( s ) is locally bounded and η ( s ) = f φ ( s − i/ φ is +1 / = f ( s − i/
2) meets the requirements.
Example 7.7.
Again consider f ( z ) = µ + iz ϕ iz on I but this time I = ( − Re µ, / − / < Re µ < I = (0 , /
2) otherwise.Then, for Re µ = − /
2, the boundary vector of f belongs to H and is given by f ( t − i/
2) = ( µ + it + 1 / − ϕ it +1 / .When Re µ [ − , − / kτ / | f τ / ) = Z R it + µ + 1 / h k ∗ ( − t − i/ ϕ it +1 / i dt for k ∈ N is analytically changed in the integration variable to get( kτ / | f τ / ) = Z R φ ( k ∗ ( − t ) ϕ it ) it + µ + 1 dt. Thus the parametric limit of f τ / exists in simple convergence as µ approachesto a point in Re µ = − / µ > − / µ = im − / m ∈ R ) be on the critical line Re µ = − / ǫ = 1 / − r . By Lemma 7.1, we have h h | g i f = Z R i ( s + m ) + 1 / φ (cid:0) φ is ( gh ∗ )( − s ) (cid:1) ds, which reveals that the boundary vector of f ( z ) coincides withlim ǫ → +0 I R i ( t + m ) + ǫ φ it +1 / dt. We now generalize the notion of interpolators on I = (0 , /
2) so that f ( z ) isallowed to be not defined on a compact subset K of R − i (0 , / K . Since the growth condition is abouthorizontal asymptotics, it remains having a meaning as well.We introduce the residue operator R f = H K f ( z ) dz : N τ / → H by R f ( gτ / ) = I R (cid:18)I f ( z ) g ( t − z − i/ dz (cid:19) dt. Here f ( z ) g ( t − z − i/
2) = f φ ( z ) φ it +1 / φ g ( t − z − i/
2) is an M ( it + 1 / z ∈ ( R − i [0 , / \ K and the coutour integral is performedby surrounding K . Theorem 7.8 (Trace Formula) . Let f ( z ) be an interpolator on (0 , /
2) of sub-gaussian growth and assume that the boundary vector f τ / = H R f ( t − i/ dt exists in H .Then the sum of the boundary operator f and the residue operator R f is τ -measurable and we have τ (( f + R f ) ∗ ( f + R f )) = ( f τ / | f τ / ) = Z R (cid:0) f ( t − i/ | f ( t − i/ (cid:1) dt. Proof.
Let V f be the virtual operator of f ( z ) ( z ∈ R − i (1 / − ǫ, / V f = f + R f and, for g, h ∈ N ,( hτ / | V f ( gτ / )) = ( hτ / | ( f τ / ) g ) = ( h / τ / | l ( f τ / )( gτ / ))shows that the virtual operator V f is closable with its closure given by l ( f τ / ).Lemma 6.5 is then the applied to get the assertion. (cid:3) RACE FORMULAS 19
Corollary 7.9. If f ( z ) is analytic on the whole R − i (0 , /
2) additionally, then theboundary operator f is τ -measurable and we have τ ( f ∗ f )) = ( f τ / | f τ / ) = Z R (cid:0) f ( t − i/ | f ( t − i/ (cid:1) dt. Example 7.10.
Let G ∈ L ( R ) and suppose that its Fourier transform b G ( λ ) = R R G ( t ) e − iλt dt is integrable and satisfies R ∞ | b G ( λ ) | e λ dλ < ∞ .Then the inverse Fourier transform G w of b G ( λ ) e iwλ belongs to L ( R ) ∩ C ( R )and depends on w ∈ R − i [0 , /
2] norm-continuously for both k ·k ∞ and k ·k . Since,for F ∈ L ( R ), ( F | G w ) = 12 π Z R b F ( λ ) b G ( λ ) e iwλ dλ is analytic in w and G s is reduced to the translation G ( t + s ) of G ( t ), F ( t ) isanalytically extended to F ( z ) so that G w ( t ) = G ( t + w ) for w ∈ R − i [0 , /
2] and t ∈ R .Now, for φ ∈ M + ∗ , g ( z ) = G ( z ) φ it defines an interpolator on (0 , /
2) whichvanishes at Re z = ±∞ . Since φ it on H is given by translation on L ( R ) ⊗ [ φ ] L ( M )[ φ ] ∼ = [ φ ] H [ φ ], the associated boundary operator is bounded and theboundary vector is given by H R G ( t − i/ φ it +1 / dt so that τ ( g ∗ f g ) = Z R ( g ( t − i/ | f ( s ) g ( t − s − i/ dsdt = φ (1) Z R G ( t − i/ F ( s ) G ( t − s − i/ dsdt = φ (1)2 π Z R b F ( λ ) | b G ( λ ) | e λ dλ. Here, for F ∈ L ( R ), an L -section { f ( t ) } of { M ( it ) } is defined by f ( t ) = F ( t ) φ it and f = R R f ( t ) dt ∈ N .Thus, letting A be the W*-subalgebra of [ φ ] N [ φ ] generated by { φ it ; t ∈ R } , L ( A, τ ) is identified with L ( R , e λ dλ ) by a unitary map U φ : L ( A, τ ) ∋ gτ / ) r φ (1)2 π b G ( λ ) ∈ L ( R , e λ dλ )so that φ is on L ( A, τ ) is realized by a multiplication of the function e − isλ of λ ∈ R . Example 7.11.
For − / < Re β < φ ∈ M + ∗ , the interpolator f ( z ) = β + iz φ iz has − π (1 ∧ φ ) − β ∈ N as the boundary operator. The residue operator iscalculated by the realization L ∞ ( A ) on L ( A ) as Z | z − iβ | = ǫ β + iz e iλz dz = 2 πe − βλ , which is therefore 2 πφ − β . Adding these, we see that (1 ∨ φ ) − β is in the Hilbert-Schmidt class and hence, for x ∈ M and µ = − r + is ∈ − (0 ,
1) + i R , x (1 ∨ φ ) − µ = x (1 ∨ φ ) r/ − is (1 ∨ φ ) r/ is in the trace class with2 πτ ( x (1 ∨ φ ) − µ ) = φ ( x ) Z R − it + (1 − r ) / i ( t + s ) + (1 − r ) / dt = φ ( x ) is − r + 1 = φ ( x ) µ + 1 . Although Haagerup deals only with the case µ = 0 and its scaled variation, thefollowing generalization should also be attributed to him. Theorem 7.12 (Haagerup’s Trace Formula) . Let ω be a weight on M in our sense.The trace of a positive operator (1 ∨ ω ) − µ with µ ∈ R , which belongs to N for µ ≥ N for µ <
0, is given by τ ((1 ∨ ω ) − µ ) = ( ω (1)2 π ( µ +1) if µ > − ∞ otherwise.Moreover, when ω ∈ M + ∗ , for any x ∈ M and µ ∈ ( − , ∞ )+ i R , the τ -measurableoperator x (1 ∨ ω ) − µ is in the trace class and we have τ ( x (1 ∨ ω ) − µ ) = ω ( x )2 π ( µ + 1) . Proof.
Assume ω ∈ M + ∗ . Then ω it is realized as a multiplication operator on L ( R , e λ dλ ) by a function e − itλ of λ ∈ R . Consequently (1 ∨ ω ) − µ is representedby the function 1 ( −∞ , ( λ ) e λµ of λ , which is integrable relative to the measure e λ dλ if and only if Re µ > − Z −∞ e λµ e λ dλ = 1 µ + 1 . Since our weights are orthogonal sums of elements in M + ∗ , the formula for ω ∈ M + ∗ remains valid for weights.The remaining part is already covered in Example 7.11. (cid:3) Remark . (i) By the integral expression Z R µ + it ω it dt of 2 π (1 ∨ ω ) − µ , the formulacoincides with the one obtained from the formal argument.(ii) The normalization of our trace is different from that in [6] and [9] by afactor 2 π .Thus, for ω ∈ M + ∗ , the analytic generator h of ω it as a positive operator on H , which satisfies θ s ( h ) = e − s h (called relative invariance of degree − τ -measurable in the sense that lim r →∞ τ ([ r ∨ h ]) = 0. Haagerup’s ingeneous ob-servation is that the whole L p ( M )’s are captured as measurable operators on H satisfying relative invariance of degree − /p .We now go into the reverse problem of characterizing τ -measurable positive op-erators satisfying relative invariance of degree −
1, which is the heart of Haagerup’scorrespondence.Recall the original approach to this problem: First establish a one-to-one cor-respondence between normal weights on M and θ -invariant normal weights on N .Second the latter is then paraphrased into positive operators of relative invarianceof degree − τ . Finally, posi-tive operators associated to M + ∗ are characterised as τ -measurable operators amongthese.Formally the whole processes look natural and seem harmless but it is in factsupported by clever and effective controls over infinities based on extended positiveparts.We shall here present an inelegant but down-to-earth proof by continuing ele-mentary Fourier calculus. RACE FORMULAS 21 Haagerup Correspondence
Let h ≥ τ -measurable operator on H satisfying θ s ( h ) = e − s h for s ∈ R .Our first task here is to identify h it with ϕ it for some ϕ ∈ M + ∗ .Let e = [1 ∨ h ] be the support projection of 1 ∨ h . By the relative invarianceof h , θ s ( e ) is the support projection of e s ∨ h and we have a Stieltjes integralrepresentation of h h = − Z ∞−∞ e s dθ s ( e ) = Z ∞−∞ e − s dθ − s ( e )and set (1 ∨ h ) − µ = − Z ∞ e − µs dθ s ( e ) , which is τ -measurable for any µ ∈ C in view of τ ( e ) < ∞ . Notice that θ s ( e ) iscontinuous in s ∈ R and dθ s has no spectral jumps.Let x ∈ M and start with the computation τ ( hx (1 ∨ h ) − µ ) = τ ( x (1 ∨ h ) − µ h ) = τ ( x (1 ∨ h ) − µ )= − Z ∞ e (1 − µ ) s dτ ( xθ s ( e )) = − Z ∞ e (1 − µ ) s d ( e − s ) τ ( xe )= τ ( xe ) Z ∞ e − µs ds = 1 µ τ ( xe ) , which is valid for Re µ > t ∈ R , σ t ( x ) = h it xh − it ( x ∈ M ) defines an automorphic action of R on M because h it xh − it is θ -invariant in view of θ s ( h it ) = e − ist h it . We claim that ϕ ( x ) = 2 πτ ( xe ) satisfies the KMS-condition for the automorphic action σ t .First notice that [ h ] = [ ϕ ]. In fact, from the definition of ϕ and the faithfulnessof the standard trace, (1 − [ ϕ ]) e = 0, which means that e ≤ [ ϕ ] and then [ h ] =lim s →−∞ θ s ( e ) ≤ θ s ([ ϕ ]) = [ ϕ ]. Conversely, from (1 − [ h ]) e = 0, 1 − [ h ] ≤ − [ ϕ ]gives the reverse inequality.Now consider ϕ ( x ∗ σ t ( x )) = τ ( x ∗ h it xh − it e ) with x ∈ M . If the Stieltjes integralexpression for h is used as in xh − it e = − Z ∞ e − ist dθ s ( xe ), we have − π ϕ ( x ∗ σ t ( x )) = Z ∞ e − ist dτ ( x ∗ h it θ s ( xe ))= Z ∞ e − ist dτ (cid:16) θ s (cid:0) x ∗ θ − s ( h it ) xe (cid:1)(cid:17) = Z ∞ e − ist d ( e − s e ist ) τ ( x ∗ h it xe ) = ( it − τ ( ∗ h it xe )and then − τ ( x ∗ h it xe ) = Z −∞ e ist dτ (cid:16) x ∗ θ s ( e ) xe (cid:17) + Z ∞ e ist dτ (cid:16) x ∗ θ s ( e ) xe (cid:17) , together with Z ∞ e ist dτ (cid:16) x ∗ θ s ( e ) xe (cid:17) = Z ∞ e ist d (cid:16) e − s τ (cid:0) x ∗ exθ − s ( e ) (cid:1)(cid:17) = Z ∞ e ist e − s dτ (cid:16) x ∗ exθ − s ( e ) (cid:17) − Z ∞ e ist e − s τ (cid:0) x ∗ exθ − s ( e ) (cid:1) ds, reveals that − τ ( x ∗ h it xe ) is analytically extended to a bounded continuous function − τ ( x ∗ h iz xe ) = Z ∞−∞ e isz dτ (cid:16) x ∗ θ s ( e ) xe (cid:17) = Z ∞−∞ e isz d (cid:16) e − s τ ( x ∗ exθ − s ( e )) (cid:17) = Z ∞−∞ e isz e − s dτ (cid:16) x ∗ exθ − s ( e ) (cid:17) − Z ∞−∞ e isz e − s τ (cid:0) x ∗ exθ − s ( e ) (cid:1) ds. of z = t − ir ∈ R − i [0 , τ ( x ∗ exθ − s ( e )) ( τ ( x ∗ θ s ( e ) xe )) ispositive, increasing (decreasing) and continuous in s ∈ R , whence both dτ (cid:16) x ∗ exθ − s ( e ) (cid:17) and − dτ (cid:16) x ∗ θ s ( e ) xe (cid:17) give rise to positive finite measures on R .Consequently, with the notation ϕ ( x ∗ σ z ( x )) for the analytic continuation of ϕ ( x ∗ σ t ( x )) and, with the help of integration-by-parts, we get the expression12 π ϕ ( x ∗ σ t − ir ( x )) = ( it + r − Z ∞−∞ e ( it + r − s dτ (cid:16) x ∗ exθ − s ( e ) (cid:17) − ( it + r − Z ∞−∞ e ( it + r − s τ (cid:0) x ∗ exθ − s ( e ) (cid:1) ds = ( it + r ) Z ∞−∞ e ( it + r − s dτ (cid:16) x ∗ exθ − s ( e ) (cid:17) − h e ( it + r − s τ (cid:0) x ∗ exθ − s ( e ) (cid:1)i ∞−∞ . For 0 < r <
1, we see lim s →∞ e ( it + r − s τ (cid:0) x ∗ exθ − s ( e ) (cid:1) = 0 andlim s →−∞ e ( it + r − s τ (cid:0) x ∗ exθ − s ( e ) (cid:1) = lim s →−∞ e ( it + r ) s τ (cid:0) x ∗ θ s ( e ) xe (cid:1) = 0at the boundary values and therefore12 π ϕ ( x ∗ σ t − ir ( x )) = ( it + r ) Z ∞−∞ e ( it + r − s dτ (cid:16) x ∗ exθ − s ( e ) (cid:17) . Since both sides are continuous in r ∈ [0 , π ϕ ( σ t ( x ) x ∗ ) = τ ( eh it xh − it x ∗ ) = − Z ∞ e ist dτ (cid:16) θ s ( e ) xh − it x ∗ (cid:17) = − Z ∞ e ist dτ (cid:16) θ s (cid:0) exθ − s ( h − it ) x ∗ (cid:1)(cid:17) = − Z ∞ e ist d ( e − s − ist ) τ ( exh − it x ∗ ) = ( it + 1) τ ( exh − it x ∗ )= ( it + 1) Z ∞−∞ e ist dτ (cid:16) exθ − s ( e ) x ∗ (cid:17) to conclude that ϕ ( x ∗ σ t − i ( x )) = ϕ ( σ t ( x ) x ∗ ) for t ∈ R .So far we have checked that h it xh − it = ϕ it xϕ − it for x ∈ [ ϕ ] M [ ϕ ]. Then u ( t ) = h it ϕ − it is a unitary in the center of [ ϕ ] M [ ϕ ]. Since each ϕ it commutes with thereduced center, { u ( t ) } is a one-parameter group of unitaries in the reduced algebra.Let u ( t ) = R R e ist E ( ds ) be the spectral decomposition in [ ϕ ] M [ ϕ ]. Then a n = R [ − n,n ] e s/ E ( ds ) is an increasing sequence of positive elements in the reduced centerand ϕ n = a n ϕa n ∈ M + ∗ satisfies ϕ itn = h it [ a n ] = [ a n ] h it for t ∈ R . Set h n = h [ a n ] = RACE FORMULAS 23 [ a n ] h , which is also τ -measurable and satisfies θ s ( h n ) = e − s h n . From the equalities ϕ n ( x )2 πµ = τ ( x (1 ∨ ϕ n ) − µ ) = τ ( x (1 ∨ h n ) − µ ) = τ ( x [ a n ](1 ∨ h ) − µ ) = ϕ ( x [ a n ])2 πµ for x ∈ M and µ ≥
1, one sees that ϕ n = ϕ [ a n ] = [ a n ] ϕ and then ϕ itn = ϕ it [ a n ] for t ∈ R . Finally we have h it = lim n →∞ h itn = lim n →∞ ϕ it [ a n ] = ϕ it .We next check the additivity of the correspondence h ϕ ↔ ϕ . To see this, we firstestablish the following relation. Lemma 8.1.
Let ω ∈ M + ∗ and µ >
0. Then(1 ∨ ω ) − µ = 12 π Z R µ + it ω it dt is in the τ -trace class and, for x ∈ [ ϕ ] M , we have τ ( hx ∗ (1 ∨ ω ) − µ x ) = 12 πµ ϕ ( x ∗ x ) . Recall here that (1 ∨ ω ) − µ/ = 12 π Z R it + µ/ ω it dt belongs to B + in such away that (1 ∨ ω ) − µ/ τ / = 12 π I R it + ( µ + 1) / ω it +1 / dt. The identity is checked as follows: Letting y = x ∗ (1 ∨ ω ) − µ x , we have τ ( hy ) = − lim n →∞ Z n − n e s dτ (cid:0) θ s ( e ) y (cid:1) = lim n →∞ (cid:18)Z n − n e s τ (cid:0) θ s ( e ) y (cid:1) ds − e n τ (cid:0) θ n ( e ) y (cid:1) + e − n τ (cid:0) θ − n ( e ) y (cid:1)(cid:19) = lim n →∞ (cid:18)Z n − n ds e s Z R dt π ( µ + it ) τ (cid:0) θ s ( e ) x ∗ ω it x (cid:1) − τ (cid:0) eθ − n ( y ) (cid:1)(cid:19) = lim n →∞ (cid:18)Z n − n ds Z R dt e ist π ( µ + it ) τ ( ex ∗ ω it x ) − π Z R e int µ + it τ ( ex ∗ ω it x ) dt (cid:19) . By the lemma below, the function τ ( ex ∗ ω it x ) / ( µ + it ) is integrable, whencelim n →∞ Z R e int µ + it τ ( ex ∗ ω it x ) dt = 0 . Lemma 8.2.
We have τ ( ex ∗ ω it x ) = 12 π (1 − it ) ϕ ( x ∗ ω it xϕ − it ) . Proof.
From the expression τ ( ex ∗ ω is x ) = ( xeτ / | ω is xeτ / ) with ω is xeτ / = 12 π I R dt i ( t − s ) + 1 / ω is xϕ − is ϕ it +1 / ,τ ( ex ∗ ω is x ) = 1(2 π ) Z R dt − it + 1 / i ( t − s ) + 1 / xϕ it +1 / | ω is xϕ − is ϕ it +1 / )= 1(2 π ) Z R dt − it + 1 / i ( t − s ) + 1 / ϕ ( x ∗ ω is xϕ − is )= 12 π − is ϕ ( x ∗ ω is xϕ − is ) , (cid:3) To deal with the first term in the last expression of τ ( hy ), we use the relation2 π ( µ + it ) − = g ∗ ∗ g for g ( t ) = 1 / ( it + µ/
2) to see that Z R e ist µ + it ω it dt = Z R dt ′ e − ist ′ g ( t ′ ) ω − it ′ Z R dt e ist g ( t ) ω it and hence2 π Z R e ist µ + it ( xξ | ω it xξ ) dt = ( Z R dt ′ e ist ′ g ( t ′ ) ω it ′ xξ | Z R dt e ist g ( t ) ω it xξ )= X j ( Z R dt ′ e ist ′ g ( t ′ ) ω it ′ xξ | δ j )( δ j | Z R dt e ist g ( t ) ω it xξ )= X j Z R dt e ist ( F ∗ j ∗ F j )( t ) = X j | c F j ( s ) | , where { δ j } is an orthonormal system in H supporting vectors { ω it xξ } t ∈ R and F j ( t ) = g ( t )( δ j | ω it xξ ) together with their Fourier transforms c F j ( s ) = R R e ist F j ( t ) dt belong to L ( R ).The Plancherel formula is then applied to each F j to get(2 π ) τ ( hy ) = Z ∞−∞ X j | c F j ( s ) | ds = X j Z ∞−∞ | c F j ( s ) | ds = 2 π X j Z R | F j ( t ) | dt = 2 π Z R X j | F j ( t ) | dt = 2 πµ ( xξ | xξ ) = 2 πµ τ ( ex ∗ x ) . Similarly and more easily, the side identity follows from2 π Z R ( ξ | θ s ( y ) ξ ) = Z R ds Z R dt µ + it ( xξ | θ s ( ω it ) xξ )= Z R ds Z R dt e − ist µ + it ( xξ | ω it xξ ) = 2 πµ ( ξ | x ∗ xξ )for each ξ ∈ L ( N ). Theorem 8.3 (Haagerup correspondence) . There is a linear isomorphism between M ∗ and the linear space of τ -measurable operators h on L ( N ) satisfying θ s ( h ) = e − s h and so that ϕ ∈ M + ∗ corresponds to the analytic generator h ϕ of the one-parameter group { ϕ it } of partial isometries in N .Moreover the correspondence preserves N *-bimodule structures as well as posi-tivity. Proof.
The correspondence is already established for positive parts and Lemma 8.1is used to get the additivity by12 πµ φ ( x ∗ x ) = h ( h ϕ + h ψ ) x ∗ (1 ∨ ω ) − µ x i = 12 πµ ( ϕ ( x ∗ x ) + ψ ( x ∗ x )) . Here ϕ, ψ ∈ M + ∗ and φ ∈ M + ∗ is specified by h φ = h ϕ + h ψ .Once the semilinearity is obtained, the other part is almost automatic. The linearextension is well-defined by h ϕ = h ϕ − h ϕ + ih ϕ − ih ϕ for ϕ = ϕ − ϕ + iϕ − iϕ ∈ RACE FORMULAS 25 M ∗ with ϕ j ∈ M + ∗ . The identity ah ϕ a ∗ = h aϕa ∗ for a ∈ M follows again fromLemma 8.1 as τ ( aha ∗ x ∗ (1 ∨ ω ) − µ x ) = τ ( ha ∗ x ∗ (1 ∨ ω ) − µ xa ) = 2 πµ ϕ ( a ∗ x ∗ xa ) = 2 πµ ( aϕa ∗ )( x ∗ x )and then ah ϕ b ∗ = h aϕb ∗ by polarization. (cid:3) References [1] O. Bratteli and D.W. Robinson, Operator algebras and quantum statistical mechanics, Vol.1,Springer, 1979.[2] U. Haagerup, On the dual weights for crossed products of von Neumann algebras I,Math. Scand., 43(1978), 99–118.[3] U. Haagerup, On the dual weights for crossed products of von Neumann algebras II,Math. Scand., 43(1978), 119–140.[4] U. Haagerup, Operator valued weights in von Neumann algebras I, J. Funct. Anal., 32(1979),175–206.[5] U. Haagerup, Operator valued weights in von Neumann algebras II, J. Funct. Anal., 33(1979),339–361.[6] U. Haagerup, L p -spaces associated with an arbitrary von Neumann algebra, Colloques Inter-nationaux CNRS, No. 274, 175–184, 1979.[7] I.E. Segal, A non-commutative extension of abstract integration, Annal Math., 57(1953),401–457.[8] M. Takesaki, Theory of operator algebras, Vol.2, Springer, 2003.[9] M. Terp, L pp