Functional Mellin Transforms, the Probability Analogy Remains a Proﬁtable Guide; This Time in the Richer Context of Banach Algebras

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For functional Mellin transforms, the probability analogy remains a profitable guide; this time in the richer context of Banach algebras. Like the functional Fourier transform, it turns out that functional Mellin also encodes a duality — but an algebraic rather than group duality. In particular, functional Mellin provides a means to represent C∗-algebras based on non-abelian topological groups. The purpose of this paper is construct and develop this tool. The functional integral framework we will use to construct functional Mellin transforms is based on topological groups, so we start with a brief exposition of some pertinent results concerning locally compact topological groups and their associated integral operators on Banach algebras. The proposed definition of functional integrals is then briefly reviewed, but we refer to [10] for details.1 The remainder of the paper concentrates on the functional analog of the finite-dimensional Mellin transform. In the functional context, Mellin is not just a simple transformation of Fourier, so developing and investigating the infinite dimensional analog of Mellin transforms is worthwhile. We use the functional Mellin transform to define functional analogs of trace, log and determinant. Contained in a key theorem based on these definitions is the functional generalization of exp tr M = det exp M that, roughly stated, says the Mellin transform and exponential map commute under appropriate conditions. There are many reasons — beyond representing C∗-algebras — to expect that functional Mellin transforms will be useful in applied mathematics. To give just a few: Crossed products, which are useful tools for C∗-algebraic quantization [13]–[15], are closely related to functional Mellin transforms. (Appendix C looks closer at this relationship.) Spectral properties of operators associated with a C∗-algebra are linked to Mellin transforms in the context of functional calculus. Properties of the Mellin transform allow efficient analytic treatment of harmonic integrals, asymptotic analysis of harmonic sums, and Fuchsian type partial differential equations [16, 17]. Finally, functional integrals based on the gamma probability distribution show up in the study of constrained function spaces [18, 19], and these are particular classes of functional Mellin transforms. A note of caution: this exposition follows a physics not a mathematics analysis. Technical mathematical issues are not addressed — especially in §5 where details of operator theory and existence/uniqueness are largely ignored. The goal is to develop tools for physics applications to be mathematically scrutinized later if they prove their worth.

2 Topological groups

The scheme for functional integration proposed in [10] is based on topological groups. So this entire section comprises a selection of particularly pertinent deﬁnitions and theorems (which we state without proof) all of which can be found in [20, 21].

1Because the functional integrals we use are defined in terms of bona fide integrals, appendix A contains a basic review of the finite-dimensional Mellin transform.

2 Definition 2.1 A Hausdorff topological group G is a group endowed with a topology such that; (i) multiplication G × G → G by (g, h) 7→ gh and inversion G → G by g 7→ g−1 are continuous maps, and (ii) {e} is closed. Definition 2.2 G is locally compact if every g ∈ G has a neighborhood basis2 comprised of compact sets. Definition 2.3 G is a Lie group if there exists a neighborhood U of {e} such that, for every subgroup H, if H ⊆ U then H = {e}. Remark that the closure hypothesis on {e} together with the topology and group structure allow the closure property to be ‘transported’ to every element in G. That is, G is Hausdorff iff {e} is closed. Moreover, since G is Hausdorff, it is locally compact iff every g ∈ G possesses a compact neighborhood. The motivation for the following definition comes from analogy with the exponential map for finite Lie groups. Definition 2.4 ([20, ch. 5]) A one-parameter subgroup φ : R → G of a topological group 3 is the unique extension of a continuous homomorphism f ∈ HomC (I ⊆ R,G) such that f(t + s)= f(t)f(s) and f(0) = e ∈ G. Let L(G) denote the set of one-parameter subgroups HomC (R,G) endowed with the uniform convergence topology on compact sets in R. The exponential function is defined by

expG : L(G) → G by expG(g) := φg(1) (2.1) −1 where g ∈ L(G) is identiﬁed with dφg(d/dt) ∈ Te(G). Similarly, its inverse logG := expG is a well-deﬁned function. In particular, if G is an abelian topological group, then L(G) is a topological vector space with the uniform convergence topology on compact sets. Let B be the invertible elements of some (unital) Banach algebra B. Then B is a topological group and there exists a homeomorphism η : B → L(B) such that φη(b)(1) = B expB(η(b)) =: expB(b). Moreover, for any subgroup G ⊆ B, there is an induced homeo- B morphism ηGB : B → L(G ). Consequently, the exponential function extends to the algebra level expB : B → B by b 7→ expB(tb)= φb(t), and it enjoys the standard properties if B is endowed with a Lie bracket.

Deﬁnition 2.5 ([20, def. 5.1]) Let B1(1) be the unit ball about the identity element 1 ∈ B. The exponential, expB : B → B, and logarithm, logB : B1(1) → B, are deﬁned by ∞ 1 exp (b) := bn ∀ b ∈ B, B n! 0 X ∞ (−1)n+1 log (1 + b) := bn for kbk < 1 . (2.2) B n 1 X 2A neighborhood basis at g ∈ G is a family N of neighborhoods such that given any neighborhood U of g there exists an N ∈ N such that N ⊂ U. 3 Denote continuous homomorphisms G → H by HomC (G, H).

3 The two functions are absolutely convergent for the indicated b ∈ B, and they are analytic. Proposition 2.1 ([20, prop. 5.3]) −1 Let N0 be the connected component of the 0-neighborhood of expB B1(1). Then

i) logB(expBb)= b ∀ b ∈ N0 .

ii) expB(logB b)= b ∀ b ∈ B1(1) . iii) exp | : N → B (1) is an analytic homeomorphism with B N0 0 1 analytic inverse logB : B1(1) → N0 . (2.3)

Deﬁnition 2.6 ([20, def. 5.32]) Let BL ⊆ (B, [·, ·]) be a closed Lie subalgebra of some Banach algebra (B, [·, ·]) equipped with a Lie bracket. Let Bl be a subgroup of the multiplica-

tive group of units of B such that expBL is a homeomorphism mapping a neighborhood of {0} ∈ BL into a neighborhood of {e} ∈ Bl. A topological group is a linear Lie group if it is isomorphic to Bl. Proposition 2.2 ([20, th. 5.41]) If G is a linear Lie group, then the set L(G) is a completely

normable topological real Lie algebra and expG is a homeomorphism from a 0-neighborhood of L(G) to an e-neighborhood in G. There are of course many other interesting and useful structures regarding topological group structures (e.g. for subgroups, quotients, product groups, etc.) that will only be recorded as needed. The most important for our application is the following well-known result: Theorem 2.1 If G is locally compact, then there exists a unique (up to positive scalar multiplication) Haar measure. If G is compact, then it is unimodular. If G is a locally compact linear Lie group then it is a dim-L(G) manifold. Evidently, locally compact topological groups can be used as a footing on which to ground functional integration: They supply measure spaces on which to model functional integral domains and their associated integrators. Accordingly, we require Banach-valued integration on locally compact topological groups:

Proposition 2.3 ([21, prop. B.34]) Let Gλ be a locally compact topological group, µ its associated Haar measure, and B a Banach space possibly with an algebraic structure. Then the set 1 of integrable functions L (Gλ, B) ∋ f, consisting of equivalence classes of measurable functions equal almost everywhere with norm kfk1 := kf(gλ)kdµ(gλ) ≤kfk∞ µ(supp f) < ∞, Gλ is a Banach space. Moreover, f 7→ f(gλ)dµ(gλ) is a linear map such that Gλ R R k f(gλ)dµ(gλ)k≤kfk∞ µ(supp f) (2.4) ZGλ 1 for all f ∈ L (Gλ, B),

ϕ f(gλ)dµ(gλ) = ϕ (f(gλ)) dµ(gλ) (2.5) ZGλ ZGλ

4 for all ϕ ∈ B′, and

LB f(gλ)dµ(gλ) = LB (f(gλ)) dµ(gλ) (2.6) ZGλ ZGλ for bounded linear maps LB : B → B2. Moreover, Fubini’s theorem holds for all equivalence 1 classes f ∈ L (G1 × G2, B).

∗ ∗ ∗ Corollary 2.1 Let B be a C -algebra and π : B → LB(H) a representation with LB(H) the algebra of bounded linear operators on Hilbert space H. Then

π f(gλ)dµ(gλ) v|w = hπ (f(gλ)) v|wi dµ(gλ) , (2.7) ZGλ ZGλ ∗ ∗ f(gλ)dµ(gλ) = f(gλ) dµ(gλ) , (2.8) ZGλ ZGλ and

a f(gλ)dµ(gλ)b = af(gλ)bdµ(gλ) (2.9) ZGλ ZGλ where v,w ∈ H and a, b ∈ M(B∗) with M(B∗) the multiplier algebra4 of B∗.

1 ∗ It can be shown ([21, appd. B]) that L (Gλ, B ) is a Banach ∗-algebra when equipped with the k·k1 norm, the convolution

−1 f1 ∗ f2(gλ) := f1(hλ)f2(hλ gλ)dµ(hλ) , (2.10) ZGλ and the involution ∗ −1 ∗ −1 f (gλ) := f(gλ ) ∆(gλ ) (2.11) where ∆ is the modular function on Gλ. Let Gλ be locally compact and U : Gλ → U(H) be a unitary representation furnished by some Hilbert space H. Consider integrals of the type π(f(gλ))U(gλ)dν(gλ). As it Gλ stands, this integral is not well-deﬁned because π(f(g ))U(g ) is not a continuous function λ Rλ in general. To ﬁx the problem, it is enough to equip the multiplier algebra of B∗ with the ∗ ∗ strict topology since then π(f(gλ))U(gλ) is continuous for f(gλ) ∈ Ms(B ) where Ms(B ) denotes M(B∗) endowed with the strict topology ([21, §1.5]):5

4 ∗ Recall the multiplier algebra can be characterized as the set of adjointable linear operators L∗(BB) on B∗ viewed as a right Hilbert module over itself. If B∗ is unital then M(B∗)= B∗. 5 1 Prop. 2.3 and consequently Def. 3.1 were stated for f ∈ L (Gλ, B). In order to quote [21] precisely ∗ ∗ and thereby avoid introducing technical diﬃculties, we restrict here to f ∈ CC (Gλ, B ) where CC (Gλ, B ) ∗ ∗ ∗ denotes the set of continuous, compactly-supported functions f : Gλ → B and B is a C -algebra. But we ∗ 1 ∗ point out that CC (Gλ, B ) is dense in L (Gλ, B ) since Gλ is locally compact [6].

5 ∗ ∗ Proposition 2.4 ([21, lemma 1.101]) For f ∈ CC (Gλ, Ms(B )) (i.e. f : Gλ → Ms(B ) is ∗ continuous and compactly supported), and π : M(B ) → LB (H), there exists a linear map ∗ ∗ f 7→ f(gλ)dν(gλ) from CC (Gλ, Ms(B )) to M(B ) such that Gλ R π f(gλ)dν(gλ) v|w = hπ (f(gλ)) v|wi dν(g) , (2.12) ZGλ ZGλ and

l f(gλ)dν(gλ) = l (f(gλ)) dν(gλ) (2.13) ZGλ ZGλ ∗ ∗ for l : M(B ) → M(B2) a non-degenerate homomorphism. With the understanding that the target Banach algebra gets replaced by its multiplier algebra endowed with the strict topology, the integral of interest π(f(gλ))U(gλ)dν(gλ) Gλ has meaning and R Proposition 2.5 ([21, ch. 2.3])

π ⋊ U(f) := π (f(gλ)) U(gλ)dν(gλ) (2.14) ZGλ ∗ deﬁnes a ∗-representation of CC (Gλ, B ) on H. This proposition embodies the crossed-product approach to algebraic quantization [15].

3 Functional integration scheme

Our functional integrals are based on the data (G, B,GΛ) where G is a Hausdorff topological group, B is a Banach space that may have additional associative algebraic structure, and GΛ := {Gλ,λ ∈ Λ} is a countable family of locally compact topological groups indexed by homomorphisms λ : G → Gλ (recall Gλ is locally compact). Pertinent details regarding the definition of functional integrals contained in [10] are briefly recalled here for convenience.

Deﬁnition 3.1 Let F(G) represent a space of functionals F: G → B, and denote the 6 restriction of F to Gλ by f := F|Gλ . Let ν be a left Haar measure on Gλ. A family of integral operators intΛ : F(G) → B is deﬁned by

intλ(F) = F(g)Dλg := f(gλ) dν(gλ) (3.1) ZG ZGλ 1 such that f ∈ L (Gλ, B) for all λ ∈ Λ. We say that F is integrable with respect to the integrator family DΛg, and F(G) ⊆ F(G) is the space of integrable functionals (with respect to Λ).

6The Haar measure ν does not necessarily have unit normalization. Also, recall that if ν and µ are left −1 −1 −1 and right Haar measures respectively, then ν(Gλ) = µ(Gλ ) and dν(gλ) = ∆(gλ )dµ(gλ ) where ∆ is the modular function on Gλ.

6 Further, if B is an algebra, deﬁne the functional ∗-convolution and ⋆-convolution by

−1 (F1 ∗ F2)λ (g) := F1(˜g)F2(˜g g)Dλg˜ (3.2) ZG and

(F1 ⋆ F2)λ (g) := F1(˜gg)F2(˜gg˜)Dλg˜ (3.3) ZG for each λ ∈ Λ.7

For any given λ, the integral operator is linear and bounded according to

kintλ(F)k ≤ kf(gλ)k dν(gλ)= kfk1,λ < ∞ . (3.4) ZGλ Linearity is obvious: to see that it is bounded, use the Cauchy-Schwarz inequality along with 1 Prop. 2.3. Because kfk1,λ is a norm on L (Gλ, B), it follows that kFkF := supλkintλFk is a norm on F(G). Since F(G) is a normed space, its completion (which will be denoted by the same symbol) is a Banach space. The ∗-convolution then implies

intλ(F1 ∗ F2) = (F1 ∗ F2)(g)Dλg ZG −1 = f1(˜gλ)f2(˜gλ gλ) dν(˜gλ,gλ) ZGλ ZGλ

= f1(˜gλ)f2(gλ) dν(˜gλ)dν(gλ) ZGλ ZGλ

= f1(˜gλ)f2(gλ) dν(˜gλ)dν(gλ) ZGλ ZGλ = intλ(F1) intλ(F2) (3.5) where the second line follows from left-invariance of the Haar measure and the last line follows from Fubini. A similar computation (using left-invariance and Fubini) establishes associativity (F1 ∗ F2) ∗ F3 =F1 ∗ (F2 ∗ F3). Finally, given that B is Banach, eq. (3.5) implies kF1 ∗ F2kF ≤ kF1kF kF2kF. Consequently, F(G) inherits the algebraic structure of B and we have shown:

Proposition 3.1 F(G) equipped with the ∗-convolution is a Banach algebra when completed with respect to the norm kFkF := supλkintλ(F)k.

Adding an involutive structure to B leads to

7 Note the extra factor ofg ˜ in the argument of F2. It is not what one would get from a straight analogy with the usual ⋆-convolution, but it enables a comparison of the two functional convolutions using the Mellin integrator as we will see later.

7 Proposition 3.2 ([10, prop. 2.5]) If B is a Banach ∗-algebra. Then intλ is a ∗-homomorphism, and F(G) — endowed with a suitable topology and involution F∗(g) := F(g−1)∗∆(g−1) and completed with respect to the norm k·kF — is a Banach ∗-algebra.

Corollary 3.1 If B is a C∗-algebra, then F(G) is C∗-algebra when completed w.r.t. the norm k·kF.

Proof : First,

∗ ∗ intλ(F ) = F (g)Dλg ZG ∗ = f (gλ) dν(gλ) ZGλ −1 ∗ −1 = f(gλ ) ∆(gλ ) dν(gλ) ZGλ ∗ = f(gλ) dν(gλ) Gλ Z ∗

= f(gλ) dν(gλ) Gλ Z ∗ = intλ(F) , (3.6) where the fourth line follows by virtue of the Haar measure. Together with (3.5), this shows that the integral operators are ∗-homomorphisms. In particular, Id∗ = Id. It remains to verify the ∗-algebra axioms. The ∗-operation is continuous for a suitable choice of topology, and linearity is obvious. Next,

(F∗)∗(g) := F∗(g−1)∗∆(g−1) = (F(g)∗)∗∆(g)∆(g−1)=F(g) (3.7) and

∗ ∗ ∗ ∗ −1 (F1 ∗ F2)λ (g) := f1 (˜gλ)f2 (˜gλ gλ) dν(˜gλ) ZGλ −1 −1 −1 −1 ∗ = f2(gλ g˜λ)∆(gλ g˜λ)f1(˜gλ )∆(˜gλ ) dν(˜gλ) ZGλ ∗ −1 −1 −1 = f2(gλ g˜λ)f1(˜gλ )∆(gλ ) dν(˜gλ) ZGλ −1 ∗ −1 = (F2 ∗ F1)λ(g ) ∆(g ) ∗ = (F2 ∗ F1)λ (g) (3.8) where we used the deﬁnition of involution, left-invariance of the Haar measure, and the fact that the modular function ∆ is a homomorphism. For the norm, B a ∗-algebra and eq. (3.6) ∗ ∗ ∗ imply kintλ(F )k = kintλ(F) k = kintλ(F)k which implies kF kF = kFkF. Conclude that F(G) is a ∗-algebra.

8 If B is a C∗-algebra, the corollary follows from (3.5) and (3.6) since now

∗ ∗ ∗ 2 kintλ(F ∗ F )k = kintλ(F) intλ(F) k = kintλ(F)kkintλ(F) k = kintλ(F)k (3.9)

∗ 2 implies kF ∗ F kF = kFkF.

4 Functional Mellin transform

The close relationship between Fourier and Mellin transforms suggests introducing the functional analog of the Mellin transform. However, contrary to the finite-dimensional case, the functional analog of the Mellin transform does not seem to be directly related to the functional Fourier transform: it is not clear how the change of variable in the finite-dimensional case carries over into the duality structure. Instead, we propose to interpret the functional Mellin transform in terms of one-parameter subgroups of a topological group. To define the functional Mellin transform, we restrict the functional integral data to the case of B ≡ C∗ a unital C∗-algebra and G a topological linear Lie group isomorphic to the group (or subgroup) of units Al of some Banach algebra AL with a Lie bracket. In order to utilize results from [20], we will analytically continue the one-parameter subgroups of a topological linear Lie group that were outlined in §2.

4.1 The definition Recall the definition of the exponential function on G: it associates an element g1 ∈ G 1 with some φg ∈ L(G) by φg(1) = expG(g) =: g . Then, by the definition of one-parameter t R t subgroups, φg(t) = expG(tg) =: g with t ∈ , and the short-hand notation g can be formally t 1 interpreted as the t-th power of g in the sense that g = expG(t logGg ). Evidently the t-th power is characterized by two fiducial points φg(0) = e and φg(1) = g. Now let γ0 : R → C by t 7→ z ∈ C be a continuous injective homomorphism such that −1 C γ0(0) = 0. Denote the set of one-parameter complex subgroups φg := φg ◦ γ0 by L(G) := C 1 z HomC ( ,G) such that g = expG(g) = φg(γ0(1)). Then φg(z) = expG(zg) =: g is the complex analytic extension of a one-parameter subgroup of G parametrizede by the complex parameter z and subject to the conditionseφg(γ0(1)) = expG(eg)= g and φg(γ0(0)) = e. We thus have an analytic exponential map which allows to define the complex group C C z G := expG(L(G) ). Formally interpret g eas a complex power of g. Thise is what is needed C ∗ C for functional Mellin. So the functional integral data is now (G , C ,GΛ) in this entire section.

Deﬁnition 4.1 Consider the subspace of integrable, equivariant functionals F(GC) ⊆ F(GC) C ∗ C such that F ∈ MorC (G , Ms(C )) is equivariant under right-translations by G according to F(gh) = F(g)ρ(h).8 Let C∗ be a unital C∗-algebra whose involution (extendede to GC) is

8This prescription is for left-invariant Haar measures. For right-invariant Haar measures impose equivariance under left-translations.

9 given by F∗(ggα) := ρ(g−α)∗F(g−1)∗∆(g−1). Let ρ : GC → C∗ be a continuous, injective ∗ 9 homomorphism, and π : C → LB(H) be a non-degenerate ∗-homomorphism. Then the C ∗ functional Mellin transform Mλ : F(G ) → C is deﬁned by

α α Mλ [F; α] := e F(gg ) Dλg = F(g)ρ(g ) Dλg (4.1) C C ZG ZG S C α α with α ∈ ⊂ , g := expG(α logG g) and π(F(g)ρ(g )) ∈ LB(H) where the space of bounded linear operators LB(H) is given the strict topology. Denote the space of Mellin C 10 integrable functionals by FS(G ).

Since we don’t have a definition of Mellin transform for generic locally compact topological groups, we have first defined the functional Mellin transform. Then, according to Def. 3.1, a sufficient condition for the functional Mellin transform to exist is f(g)ρ(gα) integrable precisely when α ∈ S. This will supply us with a Mellin transform that extends the usual definition to the case of Banach-valued integrals over locally compact topological groups:

C ∗ Deﬁnition 4.2 Let the map ρ : Gλ → C be a continuous injective homomorphism, and C ∗ 1+α α 1 C ∗ consider equivariant f ∈ MorC (Gλ , C ) by g 7→ f(g)ρ(g ) such that f ∈ L (Gλ , C ) for all α ∈ S ⊂ C. Then F is Mellin integrable since

α S |Mλ [F; α]| ≤ |f(gλ)ρ(gλ )| dν(gλ) < ∞ , α ∈ . (4.2) GC Z λ

We say the Mellin transform Mλ [F; α] exists in the fundamental region S. Identifying the Lie algebra GC of GC at the identity element with L(GC), the Mellin integral can be explicitly formulated as

α f(gλ)ρ(gλ ) dν(gλ) GC Z λ

= f(exp C (g))ρ(exp C (αg)) | det dg exp C (g)| dg . (4.3) Gλ Gλ Gλ L(GC) Z λ Roughly speaking, the functional Mellin transform is a family of integrals represented by the right-hand side of (4.3) which can be interpreted as a generalized two-sided Laplace transform provided α ∈ S. To emphasize that the fundamental region depends on λ, we will sometimes write Sλ. Stating the deﬁnitions is relatively easy: the hard work involves determining S given f, ∗ C ρ and λ. For example, let C = C and λ : G → R+ the strictly positive reals. Choose the

9Non-degenerate here means π(C∗)(H) is a dense subset in H. 10The class of functional Mellin transforms defined here includes the integrated form of a covariant representation of a dynamical system[21] as a special case. To relate the integrated form to functional Mellin C transforms, require π ◦ ρ to be a strongly continuous unitary representation U : Gλ → LB(H). Then π(f(gλhλ)) = π(f(gλ)ρ(hλ)) = π(f(gλ))U(hλ) and the integrated form π⋊U(f) is equivalent to π(Mλ [F; 1]) — although our definitions of ∗-convolution and involution are certainly different.

10 standard normalization for the Haar measure ν(gλ) = log(gλ). Then Mλ [F; α] reduces to the usual ﬁnite-dimensional Mellin transform for suitable f (see appendix A);

∞ dx ∞ M [F; α]= f(x)xα = f(x)xα−1 dx α ∈ha, bi ≡ S (4.4) H x H H Z0 Z0 where the subscript H indicates the normalized Haar measure.

Example 4.1 For a less trivial example, consider the Mellin transform of the heat kernel of a free particle on Rn with point-to-point boundary conditions. The degrees of freedom n associated with a free particle are encoded by a continuous map x : R+ → R which dictates C ∗ −S(g) λ : G → R+. In this context, C = R and F is the heat kernel F(g) ≡ e with the ‘eﬀective action’ functional S : L(R+) → R given by

2 n S(g)= π|x ′ − x | /g + log g . (4.5) a a 2

−S With the choice of the usual Haar normalization, MH E ;1 is the elementary kernel of the Laplacian ∆ on Rn associated with point-to-point boundary conditions. Consequently, the point-to-point elementary kernel of the Laplacian on Rn is given by

−1 −∆ K (xa, xa′ ) := hxa′ |∆ |xai = hxa′ |MH E ;1 |xai −∆ = MH hxa′|E |xai;1 −S = MH E ;1 −π|x ′ − x |2 = exp a a g−n/2 dg R g Z + −2 log |x ′ − x | n =2 = a a . (4.6) π1−n/2Γ(n/2 − 1)|x ′ − x |2−n n =26 a a Evidently functional Mellin has application to partial diﬀerential equations and quantum mechanics.

4.2 Algebraic properties

The functional Mellin transform inherits important properties from intΛ that follow from Prop. 2.3, Def. 3.1, and equivariance. First note that if α = 0 ∈ S then functional Mellin reduces Mλ → intλ, so we will not consider this case any longer. C S C Deﬁne a norm on FS(G ) by kFkS := supα,λkMλ [F; α] k with α ∈ λ. Complete FS(G ) C with respect to k·kS (or some other suitably deﬁned norm). Then FS(G ) is Banach because C∗ is Banach.

C Lemma 4.1 If F ∈ FS(G ), then

π Mλ [F; α] := π(Mλ [F; α]) = M [π ◦ F; α] . (4.7)

11 C Proof : By deﬁnition, F ∈ FS(G ) implies f is Mellin integrable for some Sλ. So

α π(Mλ [F; α]) = π F(gg )Dλg C ZG α S = π f(gλgλ ) dν(gλ) , α ∈ λ GC Z λ ! α S = π (f(gλgλ )) dν(gλ) , α ∈ λ (4.8) GC Z λ and the third line follows from Prop. 2.3.

Crucially, under certain conditions, Mλ is a ∗-homomorphism:

Lemma 4.2 If C∗ is commutative, then

Mλ [(F1 ∗ F2); α]= Mλ [F1; α] Mλ [F2; α] (4.9)

and Mλ [(F1 ⋆ F2); α]= Mλ [F1; α] Mλ [F2;1 − α] . (4.10)

Proof : For C∗ commutative,

′ ′ ′ ρ((gh)α)= eαρ (logG g+logG h) = eα ρ (g)+αρ (h) ′ ′ = eα ρ (g)eα ρ (h) = ρ(gα)ρ(hα) = ρ(hα)ρ(gα) . (4.11)

Put g → gg˜ in the functional Mellin transform of the ∗-convolution. The functional integral is invariant under this transformation by virtue of the left-invariant Haar measure,

α Mλ [(F1 ∗ F2); α] = F1(˜g)F2(g(˜gg) )Dλg˜Dλg C C ZG ×G α α = f1(˜gλ)ρ(˜gλ )f2(gλ)ρ(gλ ) dνλ(˜gλ)dνλ(gλ) GC×GC Z λ λ α α = F1(gg )F2(gg ) DλgDλg C C ZG ×G = M [F ; α] M [F ; α] (4.12) λ 1 ee λ 2 e where the second line uses the commutativity of C∗ and the last equality follows from functional Fubini which follows from Prop. 2.3 and Def. 3.1. The proof for the ⋆-convolution follows similarly.

12 Lemma 4.3 If C∗ is non-commutative, but GC is abelian and ρ is unitary, then

Mλ [(F1 ∗ F2); αℜ]= Mλ [F1; αℜ] Mλ [F2; αℜ] (4.13)

and Mλ [(F1 ⋆ F2); αℜ]= Mλ [F1; αℜ] Mλ [F2;1 − αℜ] (4.14) ∗ R S where αℜ = αℜ, i.e. αℜ ∈ ∩ .

C Proof : If Gλ is abelian,

α ρ((gh)α)= ρ(eα log gh)= ρ(elog(hg) )= ρ(hαgα)= ρ(hα)ρ(gα) . (4.15)

∗ ′ ∗ Since ρ is unitary, ρ′(g)= ρ′(log g) is anti-Hermitian and therefore ρ(g−αℜ )∗ = e−αℜρ (g) = ∗ ′ G αℜρ (g) αℜ e = ρ(g ). If instead ρ is assumed real then αℜ gets replaced by αℑ ∈ iR ∩ S. Hence,

∗ ∗ α F1(˜g)(F2) (g(˜gg) )Dλg˜Dλg C C ZG ×G α −α ∗ ∗ −1 ∗ −1 = f1(˜gλ)ρ(˜gλ )ρ(gλ ) (f2 )(gλ ) ∆(gλ ) dνλ(˜gλ)dνλ(gλ) GC×GC Z λ λ α ∗ ∗ α = F1(gg )(F2) (gg ) DλgDλg C C ZG ×G α α = F1(gege )F2(gg ) DλgDeλg (4.16) C C ZG ×G where the ﬁrst equality line follows fromee ρ(˜g−αℜ )∗ = ρe(˜gαℜ ) and the deﬁnition of involution. The ⋆-convolution result follows similarly. Notice if ρ(g) is in the center of C∗ then there is no restriction on α ∈ S. For complex α we don’t get representations, but with ρ unitary we have (for abelian GC)

∗ Mλ [(F1 ∗ F2); α]= Mλ [F1; α ] Mλ [F2; α] , (4.17)

and for ρ real ∗ Mλ [(F1 ∗ F2); α]= Mλ [F1; −α ] Mλ [F2; α] . (4.18) Finally, for the most general case of non-commutative C∗ and non-abelian GC, we must restrict to α = 1 for unitary ρ (or ±i for real ρ) to get an algebra representation: Lemma 4.4 If C∗ is non-commutative and GC is non-abelian, but ρ is unitary, then

Mλ [(F1 ∗ F2);1]= Mλ [F1; 1] Mλ [F2; 1] (4.19) and Mλ [(F1 ⋆ F2);1] = Mλ [F1; 1] Mλ [F2; 0] . (4.20) Proof : Use ρ((gh)−1)∗ = ρ(gh)= ρ(g)ρ(h)= ρ(g)ρ(h−1)∗ in the previous argument.

13 Corollary 4.1 If α =1/2 and C∗ is commutative or GC is abelian with ρ unitary,

∗ 2 ∗ Mλ [(F ∗ F );1/2] = |Mλ [F; 1/2]| = Mλ [(F ⋆ F );1/2] . (4.21)

Proposition 4.1 With α ∈ S suitably restricted according to the previous lemmas, Mλ is a ∗-homomorphism. Proof : Given the preceding lemmas, we only need to show ∗ ∗ ∗ α Mλ [F; α] := (Mλ [F; α]) = (F(gg ))Dλg C ZG α ∗ = (f(gλgλ )) dν(gλ) GC Z λ α ∗ ∗ = ρ(gλ ) (f(gλ)) dν(gλ) GC Z λ −α ∗ −1 ∗ −1 = ρ(gλ ) f(gλ ) ∆(gλ ) dν(gλ) GC Z λ ∗ α = F (gg )Dλg GC Z ∗ = Mλ [F ; α] (4.22)

∗ α −α ∗ −1 ∗ −1 −α ∗ ∗ ∗ where we used f (gλgλ )= ρ(gλ ) f(gλ ) ∆(gλ )= ρ(gλ ) f (gλ). In particular Id = Id if B is a ∗-algebra.

Theorem 4.1 Let SR denote the fundamental region with α ∈ S suﬃciently restricted to C ∗ render Mλ a ∗-homomorphism. Then FSR (G ) is a Banach C -algebra — when endowed with an involution deﬁned by F∗(g1+α) := F(g−1−α)∗∆(g−1) and suitable topology. Proof : Linearity and (F∗)∗ = F are obvious. Next,

∗ ∗ 1+α ∗ ∗ −1 1+α (F1 ∗ F2)λ (g ) := f1 (˜gλ)f2 (˜gλ gλ ) dν(˜gλ) ZGλ −1−α −1 −1 −1 ∗ = f2(gλ g˜λ)∆(gλ g˜λ)f1(˜gλ )∆(˜gλ ) dν(˜gλ) ZGλ ∗ −1−α −1 −1 = f2(gλ g˜λ)f1(˜gλ ) dν(˜gλ) ∆(gλ ) ZGλ −1−α ∗ −1 = (F2 ∗ F1)λ(g ) ∆(g ) ∗ 1+α = (F2 ∗ F1)λ (g ) (4.23) 1+α using left-invariance of the Haar measure to putg ˜λ → gλ g˜λ in the fourth line. This gives ∗ ∗ ∗ ∗ ∗ Mλ [(F1 ∗ F2); α] = Mλ [(F2 ∗ F1) ; α]. Lastly, since C is a C -algebra in any case, it fol- ∗ ∗ 2 lows that kFkS = kF kS and the lemmas imply kF ∗ F kS = kFkS.

14 (α) It is convenient to denote by Πλ := π(Mλ) the π-representation of functional Mellin (α) C and Π (FSR (G )) its corresponding image in LB(H) under the various conditions that render Mλ a ∗-homomorphism.

(α) Corollary 4.2 Πλ is a ∗-representation on Hilbert H.

∗ (αℜ) For example, if C is non-commutative and unital, then Πλ denotes a functional Mellin (1) ∗-representation for an abelian group and unitary ρ. Similarly, Πλ is a ∗-representation for ∗ C (1) C non-commutative C and non-abelian G . In fact, as already mentioned, Π (FS(G )) is closely related to a crossed product[21]. These inherited properties of the functional Mellin transform (at least partially) explain the utility of the resulting integrals in probing the local structure of linear Lie groups GC and C ∗-algebras FS(G ). More importantly, (and this is the main theme of this paper) since it is not merely a restatement of Fourier transform it provides an independent tool to investigate these spaces. The following subsection is exemplary.

5 Mellin functional tools

Before extracting useful tools from the functional Mellin transform, it is a good idea to gain some experience and insight by analyzing its reduction to ﬁnite-dimensional integrals under various conditions. Appendix B contains several examples. They suggest how to deﬁne Mellin functional counterparts of resolvents, traces, logarithms, and determinants. For the most part, these are familiar objects and many have been constructed and extensively analyzed using a variety of approaches in the literature — in particular, resolvents, fractional powers of operators, and zeta functions. Our purpose is to establish them in the functional integral context and to show consistency. From now on to clean up notation, no distinction will be made between g, gλ, and ρ(gλ) when it will not cause confusion. Instead of detailing integrability conditions, we will generally assume Mellin integrable functions from the beginning. And as mentioned in the introduction, we will ignore operator theory issues in order to concentrate on tool building.

5.1 Functional resolvent Analysis of the functional exponential in appendix B suggests a resolvent operator can be represented by a functional Mellin transform:

C ∗ −A C C ∗ Deﬁnition 5.1 Let A ∈ MorC (G , C ) such that E ∈ FS(G ). Let Z ∈ MorC (G ,Z(C )) with Z(C∗) the center of C∗. The functional resolvent of A is deﬁned by e −(A−Z) Rα,Aλ (Zλ) := Mλ E ; α . (5.1)

∗ 2 n ∗ C Example 5.1 For a quick example, let C = LB(L (C )) and take AL ≡ C so that G is the group of units of C∗. Suppose A(g)= ρ(ag) = ρ(a)ρ(g) =: Ag with a ∈ GC and A ∈ C∗

15 C such that A is self-adjoint. Let λ : G → φlog a(i R+); the one-parameter unitary semi-group generated by log a. The functional resolvent in this case reduces to a Laplace transform

−(A−zId)g α −α S Rα,AΓ (ZΓ)= e g dν(gΓ)=(A − zId)Γ α ∈ . (5.2) R Zφlog a(i +) C To see this, note that log a ∈ Gλ . For z∈ / σ(A) a regular value, A − zId is invertible and commutes with all g ∈ φlog a(iR+), and so it can be extracted from the integral using the invariance of the Haar measure. The remaining integral is a normalization we absorb into the measure (see appendix B). On the other hand, if z ∈ σ(A), the right-hand side formally corresponds to a derivative of a delta functional according to [10].11

C C This well-known example is a significant simplification because A is linear on G and Gλ is a rather drastic reduction to a one-dimensional abelian subgroup. A still-seemingly tractable C case is for Gλ any locally compact topological abelian group (which is finite dimensional) for which the Mellin distributions of appendix A.4 become relevant. In the generic case, it

is much harder (if not impossible) to exhibit a closed form for Rα,Aλ (Zλ). Specializing functional resolvents defines the complex power of elements in C∗: Definition 5.2 The functional inverse power of A is defined by

−α −α −A Aλ := (A )λ = Rα,Aλ (0) = Mλ E ; α . (5.4) Note that, for the set-up from Example 5.1,

−A −B 1 −Ag˜ −Bg˜−1g α 2 MΓ E ∗ E ; α = 2 e e g d(logg ˜) d(log g) Γ(α) R R Zφlog a(i +) Zφlog a(i +) −α −α = AΓ BΓ . (5.5) Keep in mind that complex powers are typically only valid for ℜ(α) > 0. However, one is α × C ∗ ∗ often interested in positive powers Aλ. To that end, let Z ∈ MorC (G , R(C )) where R(C ) is the multiplier ring of C∗. Then: Deﬁnition 5.3 The functional power of invertible12 A can be deﬁned by

α α × −1 Aλ := (A )λ := Mλ Rβ,Id(Z A ); α (5.6)

× −1 (−1)αΓ(β) × −1 −β where Rβ,Id(Z A ) := Γ(β−α) (Id − Z A ) .

11When A is linear on GC it is useful to generalize for any C∗ the delta functional deﬁned in [10] and to formally write (under appropriate conditions on A)

′ −α (α−1) Rα,A(Z) = p.v.(A − Z) − πiδ (A − Z) . (5.3) Consequently, at α = 1, p.v. (A − Z)−1 and δ(A − Z) correspond to the functional resolvent set and functional spectrum respectively. The appearance of distributions here motivates extending the theory of Mellin transforms of distributions (brieﬂy outlined in the appendix) to the functional context, but this will be left to future work. 12 α R If A is not invertible we must restrict to β = 1 and deﬁne Aλ in the usual way and then αℜ ∈ ∩ h1, 2i.

16 Returning to Example 5.1 (still with self-adjoint A), this yields (for unitary ρ)

α α (−1) Γ(β) β −β −1 α − R AΓ = (−1) g Rβ,A 1 (g ) g dν(gΓ) αℜ ∈ ∩h0, βi Γ(α)Γ(β − α) R Zφlog a(i +) α (−1) Γ(β) −1 −β α R = (Id − gA )Γ g dν(gΓ) αℜ ∈ ∩h0, βi (5.7) Γ(α)Γ(β − α) R Zφlog a(i +) which (for β = 1 and under suitable restrictions on A) is the deﬁnition proposed in [11, 12]. × −1 × −1 α α Just like for inverse powers, one ﬁnds Mλ [Rβ,Id(Z A ) ∗ Rβ,Id(Z B ); α] = AΓBΓ . And, using −1 (Id − gA˜ −1)−β e−Ag˜ g gα d(logg ˜) d(log g) , (5.8) R R Zφlog a(i +) Zφlog a(i +) it is straightforward to verify that

× −1 −A α −α MΓ Rβ,Id(Z A ) ∗ E ; α = AΓAΓ = Id αℜ ∈ (0, β) (5.9)

in this example. The left-hand side is actually valid for α ∈h0, ∞i, but it is not expressible as offsetting powers of A outside the open interval αℜ ∈ (0, β) ⊂ R. It is not surprising that the fundamental strip of the functional resolvent of A fixes the range of validity of the (positive) complex power of A. Afterall, it fixes the inverse power as well. Clearly ± complex powers of A are not generally inverse to each other in the sense that α −α Aλ Aλ =6 Id — special cases notwithstanding. Indeed from the previous section we know that M[F1 ∗ F2; α]= M[F1; α]M[F2; α]= Id only for carefully chosen F1 , F2 and restricted α depending on the nature of C∗ and GC. −α We stress that, except in limited cases, the functional form of Aλ will not resemble −α (Aλ) , and computation of the associated integral is generically rather involved. But Def. 5.1 and the characterization of delta functionals in [10], suggest that the functional inverse complex power of A implicitly includes derivatives of Dirac delta functionals if it has a non- trivial kernel, i.e. if z =0 ∈ σ(Aλ). If on the other hand Aλ has no non-trivial kernel, then we can define a functional zeta:

−α Deﬁnition 5.4 If Aλ is trace class with 0 ∈/ σ(Aλ), then the functional zeta is deﬁned by

−α −A ζAλ (α) := tr Aλ = tr Mλ E ; α . (5.10)

Of course this trace is not always a well-defined object for all αin the fundamental region of −α Aλ . Rather, the fundamental region containing α is dictated by λ and is often restricted. An effective strategy to quantify the restriction is to move the trace inside the integral. Presumably the trace of the integrand will be well-defined for certain choices of λ and the properties of the functional Mellin transform will allow the domain of α to be determined. This strategy leads us to the next subsection.

17 5.2 Functional trace, logarithm, and determinant

C ∗ −A C Deﬁnition 5.5 Let A ∈ MorC (G , C ) such that E ∈ FS(G ) for some λ-dependent fundamental region α ∈ Sλ. The functional trace of A is deﬁned by

−α −α −A −A(g) α Tr Aλ := (Tr A )λ := Mλ Tr E ; α := tr e g Dλg . (5.11) C ZG Remark that, as a consequence of Prop. 2.3, the interchange of the ordinary trace and −A C functional integral is valid only for E ∈ FS(G ) and appropriate λ. Then according to the deﬁnition,

−α −A tr(Aλ ) = tr Mλ E ; α

−A(g) α = tr e g Dλg , ∀ α ∈ Sλ C ZG −A(g) α = tr e g Dλg, ∀ α ∈ Sλ C ZG −A S = Mλ Tr E ; α , ∀ α ∈ λ . e (5.12) Evidently, the functional trace and ordinary trace possess the same functional form. But e the fundamental region of the functional trace depends on the chosen normalization, and taking the ordinary trace inside the integral often requires a restriction on the fundamental −A region of Mλ Tr E ; α . The point is we can turn the calculation around and (with appropriate normalization/regularization) give meaning to the object M Tr E−A; α through λ the ordinary trace and an adjustment to S . λ In particular, the functional zeta can be represented as

−A ζAλ (α) ≡Mλ Tr E ; α (5.13) but only for appropriate λ and α. For example, ∗ Example 5.2 Let H be a separable Hilbert space. Suppose M ∈ C = LB (H) is self-adjoint with σ(M) = N+, and let {|ii,εi} with i ∈{1,..., ∞} denote the set of orthonormal eigen- C vectors and associated eigenvalues of M. Choose λ : G → R+ and ρ such that g 7→ g · Id. The Riemann zeta function associated with M can be deﬁned by

−Mg α ζMΓ (α) = tr e ρ(g ) dν(gΓ) R Z + ∞ −εig+α(log g) = e hi|ii dν(gΓ) R i=1 Z + X ∞ −εig+α(log g) = e dν(gΓ) , α ∈h1, ∞iΓ R Z + i=1 ∞X 1 α = g g dν(gΓ) , α ∈h1, ∞iΓ 0 e − 1 Z −M = MΓ Tr E ; α , α ∈h1, ∞iΓ (5.14)

18 where ν(gΓ) := ν(g)/Γ(α). Note that the integral in the ﬁrst line is valid for α ∈ h0, ∞i, so exchanging summation and integration here comes with the price of restricting the fundamental strip. If instead GC → C a smooth contour in C \{0} with the branch cut for log along the positive real axis, this has the well-known representation 1 ζ (α) = gα dν(g ), α ∈h0, ∞i\{1} MΓC g ΓC ΓC C e − 1 Z −M = MΓC Tr E ; α , α ∈h0, ∞i\{1}ΓC (5.15) π csc(πα) where ν(gΓC ) := 2πı ν(g)/Γ(α) and the contour starts at +∞ just above the real axis, passes around the origin counter-clockwise, and then continues back to +∞ just below the real axis.13 This is an explicit illustration of the fact that S depends on λ through both the normalization and the averaging group.

∗ Example 5.3 Again for M ∈ LB(H) with M = M and σ(M) = N+, the Dirichlet eta function associated with M is given by

−M(g+iπ) α ηMΓ (α) = tr −e ρ(g ) dν(gΓ) R Z + ∞ −εi(g+iπ)+α(log g) = − e dν(gΓ) , α ∈h0, ∞iΓ R + i=1 Z X ∞ 1 = gα dν(g ) , α ∈h0, ∞i e g +1 Γ Γ Z0 −Mη = MΓ Tr E ; α , α ∈h0, ∞iΓ −M = MΓη Tr E ; α , α ∈h0, ∞iΓη (5.16) 1−α where Mη(g)= M(g + iπ) and ν(gΓη ) := (1 − 2 )ν(g)/Γ(α). Going from the first to second line uses the fact that e−ǫiggα dν(g) converges for α ∈h0, ∞i. Notice the same functionalP R Tr E−M yields different objects depending on the choice of λ. Continuing with this strategy, define the functional logarithm: C ∗ −A C Definition 5.6 Let A ∈ MorC (G , C ) be invertible and suppose that E ∈ FS(G ) for some fundamental region α ∈ Sλ. The functional logarithm of A is defined by

−1 −1 d −A −A(g) α Log Aλ := (Log A )λ := Mλ E ; α =: e g logλ g Dλg dα + C + α→0 ZG α→0 −A(g) α =: e g Dλg C ZG α→0+ −A =: Mλ E ;0 b (5.17)

d α α if the limit exists. Here log g Dλg := g dν(gλ) =: g dν(gλ) (see appendix B). λ dα c 13Of course π csc(πα)/Γ(α) can be analytically continued to the left-half plane thus obtaining a meromorphic representation of Riemann zeta. b

19 C ∗ Example 5.4 In particular let G → R+ and C = C, and suppose that A(g)= Ag with A invertible, positive-deﬁnite. Choose the gamma normalization to get

−1 −Ag α Log AΓ = lim e g dν(gΓ) α→0+ R Z + −Ag α = lim e g logb ψ g dν(gΓ) α→0+ R Z + −Ag α = lim e g (log g − ψ(α)) dν(gΓ) α→0+ R Z + = − log A (5.18)

Γ′(α) where ψ(α)= Γ(α) . −B −B −1 −B Notice that, for A = E , we have formally (Log(E ) )λ = (−Log (E ))λ = Bλ since e−B(g)= e−Bg. Also, when the limit exists

−1 −A(g) α −A Tr(Log Aλ )= tr e g logλ g Dλg = Mλ Tr E ;0 C ZG α→0+ ′ = ζcA(α)|α→0+ (5.19)

Functional Log can be trivially extended to other α ∈ C by choosing the normalization

ν(gΓα ) := Γ(α + 1)ν(gΓ)/Γ(α) to get −α −1 Log AΓ := Log AΓα

Γ(α + 1) −A(g) α = e g dν(gΓ) , α ∈ h−∞, ∞iΓα . Γ(α) R Z + α→0+

(5.20) b This yields the expected behavior for exponents when A(g)= Ag. Functional Log does not seem as useful as the functional Tr, because its complex extension is simply determined by a normalization. However, to the extent this definition can be made mathematically rigorous, it can be used to justify the replica trick often used in statistical mechanics. The final component of our triad is the functional determinant of a complex power which is defined in terms of the functional trace: C ∗ Definition 5.7 Let A ∈ MorC (G , C ) with A(g) trace class and 0 ∈/ σ(Aλ). Suppose that −Tr A C E ∈ FS(G ) for α ∈ Sλ. Define the functional determinant of A by

−α −α −A −A(g) α Det Aλ := (Det A )λ := Mλ Det E ; α := det e g Dλg C ZG −tr(A(g)) α = e det(g ) Dλg GC Z −Tr A =: Mλ E ; α (5.21) where ρ : GC → C by g 7→ det(gα) := e α tr(g) and the trace tr is with respect to C∗.

20 As with the functional trace and logarithm, the determinant essentially can be ‘taken outside the integral’ only for special choices of λ for which the trace in the integrand is well- deﬁned given a particular representation. It is important to realize that, at the functional level,

Proposition 5.1 If A1(g) and A2(g) are trace class, then

−α −α −α Det(A1 ∗ A2)λ = DetA1λ DetA2λ . (5.22)

Consequently, the functional determinant possesses the multiplicative property if the pertinent functional determinants have overlapping critical strips.

Proof : Since in this case C∗ = C is commutative, the result follows from the deﬁnition of the functional determinant and Th. 4.1.

However, at the function level the determinant may not have the multiplicative property ∗ with respect to α. For starters, if A is not in the multiplier algebra Ms(C ), the functional −α −α determinant DetAλ doesn’t have the same form as detA . But even if A1 and A2 act linearly, there may not exist a consistent choice of λ that renders both A1λ and A2λ simultaneously analytic at a common value of α; even if their convolution is analytic there. Moreover, if such a λ does exist, the localization implicit in λ that achieves the reduction Det → det may introduce a “multiplicative anomaly”14:

Proposition 5.2 Suppose A(g) = Ag (explicitly, A · ρ(g)) with A invertible and Ag trace class. Then, −α −α iϕA(α) S DetAλ = Nλ(α) | det Aλ | e , α ∈ λ (5.23)

where Nλ(α) is a λ-dependent normalization. Proof : Using the invariance of the Haar measure with g → A−1g yields

−α −A −tr(g) −1 α DetAλ = Mλ Det E ; α = e det[(A g) ] dν(gλ) GC Z λ −tr g −1 α iϕA,g (α) = e (det(A g)) e dν(gλ) GC Z λ

−α iϕA(α) −tr g α = | det Aλ | e e detg dν(gλ) GC Z λ −α iϕA(α) S =: | det Aλ | e Nλ(α) , α ∈ λ (5.24)

−1 where ϕA(α)= α(arg A +2nπ).

14For example, if one chooses zeta function regularization to eﬀect the reduction Det → det, it is well- known that a non-vanishing Wodzicki residue at α = 0 leads to a multiplicative anomaly.

21 −α The normalization Nλ(α) := Det Idλ requires scrutiny. The definition of functional determinant assumes A(g) is trace class. However, Id(g) = g will not be trace class for C generic G . If it is not, then we regulate Nλ(α) with a positive-definite invertible fixed C −tr(Rg) C element R ∈ Gλ such that Rg is trace class and e ∈ FS(G ). Let HA furnish a ∗ representation π of C . Pick a basis in HA for which R is diagonal. Then

d −trπ(Rg) α −α S Nλ(R; α) := e detπg dν(gλ)= ri , α ∈ λ . (5.25) C Gλ i=1 Z Y where dim(HA)= d. Even if d = ∞, this product can be rendered finite and well-defined if a suitable regulator R exists. Alternatively, we can simply choose the normalization/regularization associated with the choice of λ to set Nλ(R; α) = 1 — which amounts to formally dividing out this factor from the functional determinant (similarly to what is done with Γ(α)). The corresponding regularized functional determinant of an operator π(O) can then be defined by

−α −α Detπ Oλ,R Detπ Oλ := Nλ(R; α)

−1 1 −trπ(ROR g) α := e detπg dν(gλ) Nλ(R; α) GC Z λ −α 1 detπO −trπ(Rg) α = −α e detπg dν(gλ) Nλ(R; α) detπR GC Z λ −1 α detπO = eiϕO,R(α) det R−1 π −α =: detπ(O ) . (5.26)

In essence, the functional determinant is formally regularized by dividing out the possibly inﬁnite factor Nλ(α). This is common practice: One knows how π(R) acts on HA ′ and then deﬁnes the regularized determinant of π(A) by DetπAΓ := DetπAΓ/DetπR Γ = detπ(A)/ detπ(R). −1 So in particular, if arg A =0 mod2π and α =1 ∈ Sλ, then

−1 −1 −2 −2 DetπAλ DetπAλ = DetπAλ = (detπAλ) (5.27) and the regularized determinant enjoys the usual multiplicative property. However, if instead arg A−1 =06 mod2π, a multiplicative anomaly obtains.

6 Commuting Mellin and the exponential

The fundamental relationship between exponentials, determinants, and traces in ﬁnite di- mensions, i.e. exp trM = det exp M, also characterizes the functional analogs. In order to re- C late functional trace and determinant, consider F = −LogA. Then formally, for F ∈ FS(G )

22 and appropriate choice of λ,

−1 −A(g) Tr Log Aλ RGC tr e logλ g Dλg −tr A(g) −1 e ∼ e ∼ e det g Dλg = Det Aλ . (6.1) C ZG This important relation represents a deep connection between Poisson processes and functional Mellin transforms/gamma integrators (as indicated for example in [10]). It is a particular case of the following theorem: C ∗ Theorem 6.1 Suppose A ∈ MorC (G , C ) by g 7→ A(g) is trace class such that 0 ∈/ σ(A), −A C ∗ −A −A ∗ −A −Tr A and let E ∈ MorC (G , C ) by g 7→ e (g) with e ∈ C . Assume that Tr E and E are Mellin integrable for a common domain α ∈ Sλ. Then −α −Tr Aλ −A −α S e = Det(E )λ , α ∈ λ . (6.2) Proof : First, recall that an immediate consequence of the deﬁnitions and the relationship between (exp tr) and (det exp) is

−A −A(g) α Mλ Det E ; α = det e g Dλg C ZG −trA(g) α = e detg Dλg GC Z −Tr A = Mλ E ; α (6.3) where the second line follows as soon as A(g) is trace class. C Lemma 6.1 Let Tr F ∈ MorC (G , C). Suppose the functional Mellin transforms of Tr F −Tr F and E exist for common α ∈ Sλ for a given λ. Then

−Tr F −Mλ[ Tr F;α] Mλ E ; α = e , α ∈ Sλ . (6.4) proof : The Mellin transform of Tr F exists by assumption so e−Mλ[Tr F;α] represents an absolutely convergent series for α ∈ Sλ. Hence, ∞ (−1)n ∞ (−1)n e−Mλ[Tr F;α] = M [Tr F; α]n = M [(Tr F)n; α] n! λ n! λ n=0 n=0 X X ∞ (−1)n = M (Tr F)n; α λ n! "n=0 # X−Tr F = Mλ E ; α (6.5) where moving the power of n into the functional Mellin transform in the ﬁrst line follows from induction on Th. 4.1 because the multiplication represented by (Tr F)n is the ∗-convolution and TrF(g) ∈ C, i.e. C∗ is commutative. Equality between the ﬁrst and second lines follows −Mλ[Tr F;α] −Tr F from the absolute convergence of e , the existence of the integral Mλ E ; α for the common domain α ∈ S , and the fact that e−tr F(g) is analytic. λ It should be stressed that the equality in the lemma holds only for α ∈ Sλ properly restricted. ⊟

23 C ∗ Corollary 6.1 Under the same conditions of Lemma 6.1, replace Tr F with V ∈ MorC (G , C ) where now GC is abelian and V(g) is self-adjoint, then the lemma together with Th. 4.1 imply −V −Mλ[V;α] Mλ E ; α = e , α ∈ R ∩ Sλ . (6.6)

To ﬁnish the proof, put F≡ E−A in the lemma and note F(g)= e−A(g) and A(g)= A(g) so −Tr E−A −tr(e−A(g)) α −A −α Mλ E ; α = e det g Dλg = Det(E )λ (6.7) GC h i Z and −A −A(g) α −α Mλ Tr E ; α = tr e ρ(g ) Dλg = Tr Aλ . (6.8) C ZG Hence −α −A −α −Tr Aλ Det(E )λ = e . (6.9)

If E−A is self-adjoint and GC abelian, then the corollary and Prop. 5.2 imply

−A −A −α −A −α −E −Mλ[E ;α] −Aλ (E )λ = Mλ E ; α = e = e (6.10) h i which leads to the remarkable property (in this case)

− − −Tr A α −A α e λ = Det e λ , α ∈ R ∩ Sλ . (6.11) In particular, consider the special case of a self-adjoint linear operator A(g) = Ag and let −1 Mλ = −Aλ , then

− ζ − (1) M 1 M M 1 Tr M tr M [E ,1] M i arg(e λ ) M M e λ = e λ = e λ = Det e λ = e |det e λ | = det e λ . (6.12)

−1 −1 Similarly, take LogMλ = Aλ and use (5.19) to get (if the limit exists)

′ −1 −M −1 −ζM (0) −Tr LogM −Mλ[Tr E ;0] −LogM logMλ e λ = e λ = e c = Det e λ = Det e = det Mλ . (6.13)

Evidently the theorem reproduces the standard expressions for this special case. It should perhaps be emphasized that, for appropriate choice of λ, Th. 6.1 and the above properties are derived from well-deﬁned Haar integrals, and they should be interpreted in the spirit of Def. 3.1: That is, they are a family of statements at the functional level that may be explicitly realized for appropriate choices of λ.

24 7 A comment on ∗ v.s. ⋆

Restrict to abelian G and recall lemmas (4.2) and (4.3). Allowing for complex α and following the proof in the latter lemma implies

∗ Mλ [(F1 ∗ F2); α] = Mλ [F1; α ] Mλ [F2; α] (7.1) ∗ ∗ Mλ [(F1 ⋆ F2); α] = Mλ [F1; α] Mλ [F2;1 − α ] . (7.2) Accordingly, at α = 1/2 we get simultaneous representations for both commutative and ∗ non-commutative C and norm equality kMλ [(ψ ∗ ψ);1/2] k = kMλ [(ψ ⋆ ψ);1/2] k. But generically we get neither a simultaneous representation nor norm equality. ∗ However, if there exists a class of holomorphic Ψλ(α) := Mλ [ψ; α] ∈ C that satisfy Ψ(α)∗ = Ψ(α∗) and Ψ(α)Ψ(α)∗ = Ψ(α)∗Ψ(α). And if Ψ(α) further satisﬁes a functional equation of the form Ψ(α)= ε(α,λ)Ψ(1 − α) with ε(α,λ) ∈ C such that |ε(α,λ)| = 1, then (for this class) the ∗-convolution and ⋆-convolution yield simultaneous representations up to a phase and norm equivalence for all α ∈ S:

∗ Lemma 7.1 Suppose G is abelian and there exist holomorphic Ψλ(α) := Mλ [ψ; α] ∈ C such that; i) Ψ(α)∗ = Ψ(α∗), ii) Ψ(α)Ψ(α)∗ = Ψ(α)∗Ψ(α), and iii) Ψ(α)= ε(α,λ)Ψ(1 − α) with ε(α,λ) ∈ C and |ε(α,λ)| =1. Then

2 kMλ [(ψ ∗ ψ); α] k = kMλ [(ψ ⋆ ψ); α] k = kΨλ(α)k ∀α ∈ S . (7.3)

To gain some insight into this observation, recall the nature of the ∗ and ⋆ convolutions in the context of standard Mellin transforms: The former is integrated along R+ and the latter along R>0 with their canonical Haar measures. Now, a particular type of functional Mellin transform is the gamma integrator[10]. It results from considering the space of continuous pointed maps T0 ∋ τ : (T+, ta) → (C+, 0) where T+ ⊆ R+ and C+ = R+ × iR. For a suitable topology, this space is an abelian topological group under point-wise multiplication in the ﬁrst component and point-wise addition in the second component. One can attach a counting interpretation to functional Mellin for τ(t) along the R+ direction and an evolution ∗ interpretation along the ±iR>0 directions.[10] To apply functional Mellin, take C = C and C ∼ C impose periodicity on τ so that Gλ = Γ where Γ is some closed contour in +. The gamma functional integral under the linear map L : T0 → Γ for some contour Γ ⊂ C+ reduces:

−ωt α−1 Dλγα,β′,c(τ) → e t dt α ∈ S (7.4) Γ ZT0 Z ′ where ωt = hω, L(τ)i = hL(ω), τi = hβ , τi ∈ C+. Under a P SL(2, Z) transformation t 7→ t/(1 − t) this becomes

e −ωt ∞ 1−t −ωt α−1 e α−1 n (α) α−1 S e t dt = α+1 t dt = t Ln (ω) t dt α ∈ (7.5) Γ Γ (1 − t) Γ n=0 Z Ze Ze X (α) where Ln (ω) are the generalized Laguerre polynomials.

25 (s) C Evidently, Laguerre polynomials are germane in this context, so deﬁne Ln ∈ FSR (G ) (for unitaryρ) by

(s) −tr(g) (s) α s − (α +1)+ n Mλ Ln ; α := e Ln (g) det(g ) Dλg := Γ(α + 1) 1 (7.6) C n ZG (s) C where Ln : G → C are generalized Laguerre polynomials with −1

′ (s) (s) −tr(hβ′,gi) α Lλ [F, β ; α,n,s] := F (g) Dλln,α,β′ (g) := F (g)Ln (g) e det(g ) Dλg (7.9) C C ZG ZG (s) where Dλln,α,β′ (g) defines a family of Laguerre integrators in terms of matrix gamma integrators[10]. Under L : T0 → R+, ∞ (s) ′ (s) (s) −tr(hId′,gi) s+1 Lλ Lm , Id ; s +1,n,s = Lm (g)Ln (g) e det(g ) Dλg Z0 Γ(s + n + 1) := δ 1 = δ (s + 1)(n) 1 (7.10) m,n Γ(s + 1) m,n where (s + 1)(n) is the rising factorial. With these definitions, for suitable F one can series (s) (s) expand F = n an Ln with the help of Laguerre integrators. Now, following [24, 25, 26], define P Definition 7.1 For −1

(s) (s) −tr(g) (s) α+s/2 Ψn (α) := Mλ ψn ; α + s/2 := e Ln (2g) det(g ) Dλg . (7.11) GC Z λ 15Recall example 4.1. Evidently, to calculate the elementary kernel of the Laplacian on Rn, one can just −1 2 as well use the eﬀective action S(g) := S(g )= πg |xa′ − xa| − n/2 log(g).

26 −n Suppose we again take L : T0 → R+ and choose the Haar measure ν(gΨ) := 2 ν(g)/Γ(α). This yields a family of the Ψ(α) of lemma 7.1; 1 ∞ Ψ(s)(α) = e−t L(s)(2t) tα+s/2−1 dt 0 < ℜ(α + s/2). n 2nΓ(α + s/2) n Z0 1 ∞ = e−t/2 L(s)(t) tα+s/2−1 dt 0 < ℜ(α + s/2). 2α+s/2+nΓ(α + s/2) n Z0 1 = L [Id, 1/2Id′; α + s/2,n,s] 0 < ℜ(α + s/2). (7.12) 2α+s/2 Ψ Using the series representation of Laguerre, this can be expressed as[24, 26] (s) −n (n) Ψn (α)=(−2) (s + 1) 2F 1(−n, α + s/2; s + 1;2) , (7.13) (s) ∗ (s) ∗ and thereafter analytically continued to all α ∈ C. Clearly, Ψn (α) =Ψn (α ) and trivially (s) (s) ∗ (s) ∗ (s) Ψn (α)Ψn (α) =Ψn (α) Ψn (α). Moreover, Theorem 7.1 (s) n (s) • Ψn (α)=(−1) Ψn (1 − α) ∀ α ∈ C

(s) (s) n(s+n) (s) • Ψn+1(α)=(α − 1/2)Ψn (α)+ 4 Ψn−1(α) (s) • All zeros of Ψn (α) are simple and lie on the critical line ℜ(α)=1/2. proof : The ﬁrst two points follow readily from (7.13) and the identities (respectively) −a 2F 1(a, b; c; z) = (1 − z) 2F 1(a, c − b; c; z) (2a − c − z(a − b)) a(z − 1) F (a − 1, b; c; z) = F (a, b; c; z)+ F (a +1, b; c; z) . 2 1 (a − c) 2 1 (a − c) 2 1 For proof of the third point we refer to [25, th. 4] (see also [24, th. 4]). Note that, in the (s) course of the proof, it is shown that Ψn (1/2+ iσ) are orthogonal on R with respect to the measure 2s+1|Γ(1/2+ iσ + s/2|2 dσ with σ ∈ R.

(s) We have learned that there exist elements ψn ∈ FSR (T0) such that functional Mellin of both ∗ and ⋆ products act by multiplication up to a phase; also (s) (r) n (s) (r) MΨ ψn ∗ ψm ; α + s/2 =(−1) MΨ ψn ⋆ ψm ; α + s/2 , (7.14) and (s) (s) (s) (s) (s) 2 S kMΨ ψn ∗ ψn ; α + s/2 k = kMΨ ψn ⋆ ψn ; α + s/2 k = kΨn (α)k ∀α ∈ . (7.15) (s) 2 Further, kΨn (α0)k = 0 iﬀ α0 = 1/2+ iσ with σ ∈ R and −1 < s. Evidently, functionals of τ ∈ T0 degrees of freedom can be series expanded along the critical line in terms of the (s) (s) class of functions Ψn (α) = MΨ[ψn ; α + s/2]. In light of this, it is curious and perhaps (s) 16 signiﬁcant that (7.3) holds for Ψn (α). We don’t fully understand it’s implications.

~~16However observe that, in a physical setting, LogΨ corresponds to a complex eﬀective action and the~~

~~27 8 Conclusion~~

Fourier transform has been a central theme in functional integration from the beginning. As is well known, it represents duality between locally compact abelian groups. Our contention is that Mellin transform in the functional context is not just a reformulation of Fourier: It represents duality between Banach ∗-algebras. As such, we expect functional Mellin to be a useful addition to mathematical physics. In particular, we constructed functional analogs of resolvent, trace, log, and determinant that are expected to be applicable to C∗-algebras characterizing quantum systems. In fact we found that, given some relevant representations of a topological group G, functional Mellin deﬁnes a C∗-algebra for which the Mellin integrator acts as a ∗-homomorphism to the algebra of bounded linear operators on the Hilbert space carrying the representations of G. This means that, armed with functional Mellin and a starting topological group, we can construct and represent a non-commutative C∗-algebra — without having to somehow deform a commutative algebra. Consequently, if one is fortunate enough to know G and its relevant representations that characterize a physical system, then one can model its quantum properties without ﬁrst passing through the classical realm. We intend to exploit this idea in future work.

~~A Mellin transforms~~

~~A.1 Basics Analysis of the Mellin transform can be found in many references. Most of the following, which includes some non-standard aspects, can be found in [16] and [17].~~

1 Deﬁnition A.1 Let f : (0, ∞) → C be a function such that f ∈ L (R+) with limits −a −b limx→0+ f(x) → O(x ) and limx→∞ f(x) → O(x ) for a, b ∈ R. Then the Mellin transform f(α) with α ∈ha, bi := (a, b) × iR ⊂ C is deﬁned by

∞ e f(α) := M[f(x); α] := f(x)xα−1dx . (A.1) Z0 counting/evolution along Γ is associatede with entropy/action. The fact that functional Mellin of both ∗ and ⋆ products yield (projective)representations supports the idea that τ(t) represents non-unitary evolution along a general contour in C+ and the critical line ℜ(α)=1/2 seems to indicate an isentropic process characterized by a duality or degeneracy between the notions of real-valued action and information entropy. Then perhaps (s) 2 kΨn (α0)k = 0 only on the critical line induces a phase transition if Γ intersects that line. And, if Γ runs vertically closing at inﬁnity, perhaps evolution (with respect to increasing σ) induces irreversibility in σ — −LogΨ insofar as e develops a pole at every zero α0. If these speculations are correct, it may be fruitful to search for a 1-d quantum scattering model of

Riemann’s xi function. Recalling example 5.2 and thinking of ζMΓ (α) as the functional inverse power of (s) (s) a suitable non-Hermitian evolution operator, something like ξ(α) = n an Ψn (α) being associated with a functional resolvent seems well motivated; especially considering the prime-power counting toy model developed in [10]. P

28 The fundamental strip ha, bi ⊂ C indicates the domain of convergence. Since, by deﬁni- tion, −a −b f(x)|x→0+ = O(x ) and f(x)|x→∞ = O(x ) (A.2) then f(α) exists in ha, bi where it is holomorphic and absolutely convergent. More precisely,

1 ∞ e |f(α)| ≤ |f(x)| xℜα−1 dx+ |f(x)| xℜα−1 dx 0 1 Z 1 Z ∞ e ℜα−1−a ℜα−1−b ≤ M1 x dx+ M2 x dx (A.3) Z0 Z1 for some finite constants M1, M2. From the definition follows some important properties (for suitable fundamental strips); c−αf(α) = M[f(cx); α] c> 0 f(α + d) = M[xdf(x); α] d> 0 1 e α f( ) = M[f(xr); α] r ∈ R −{0} , α ∈hra, rbi e|r| r dn ef(α) = M[(log x)nf(x); α] n ∈ N dαn d −αfe(α) = M x f(x); α dx d −f(α −e 1) = M f(x); α dx 1 x − fe(α +1) = M f(x′) dx′; α α Z0 e (A.4) The last three can be extended by iteration: Γ(α + n) dn (−1)n f(α)= M xn f(x); α (A.5) Γ(α) dx N α+n−m (n−m) ∞ for n ∈ and x f (x)|0 =0 ∀me ∈{1,...,n}, Γ(α) (−1)n f(α − n)= M f (n)(x); α (A.6) Γ(α − n) N α−n−1+m (n−m) ∞ for n ∈ and x f (x)|0 =0 e∀m ∈{1,...,n}, and Γ(α) x n (−1)n f(α + n)= M f(x′) dx′ ; α (A.7) Γ(α + n) Z0 x ′ ′ n e where 0 f(x ) dx defines an iterated integral x n x x R ′ ′ f(x ) dx := ··· f(xn) ...f(x1) dx1 ...dxn . (A.8) Z0 Z0 Z0

29 The last two relations show that (for functions with appropriate asymptotic conditions) the Mellin transforms of derivatives and integrals are symmetrical under n → −n. Indeed, this is the basis of the deﬁnition of fractional derivatives. This suggests an application to n i i pseudo-diﬀerential symbols of the type A(x,d/dx) = i=−∞ ai(x)d /dx . The Mellin transform is directly related to the Fourier and (two-sided) Laplace transforms by P M[f(x); α]= F[f(ex); −iα]= L[f(e−x); α] . (A.9) From these relationships, the inverse Mellin transform can be deduced;

1 c+i∞ f(x) a.= e. x−αf(α) dα (A.10) 2πi Zc−i∞ e where c ∈ (a, b) (provided f(α) is integrable along the path). The almost everywhere (a.e.) designation can be dropped if f(x) is continuous. Moreover, if f(x) is of bounded variation about x0, then e f(x+)+ f(x−) 1 c+iT 0 0 = lim x−αf(α) dα . (A.11) 2 T →∞ 2πi Zc−iT Using the inversion formula, the Parseval relation for thee Mellin transform follows from ∞ 1 c+i∞ g(x)h(x)xα−1 dx= g(α′)h(α − α′) dα′ , (A.12) 2πi Z0 Zc−i∞ assuming the necessary conditions on g(x) and h(x) to allowe e for the interchange of integration order. In particular,

~~∞ 1 c+i∞ M [g(x)h(x);1] = g(x)h(x) dx= g(α′)h(1 − α′) dα′ . (A.13) 2πi Z0 Zc−i∞ Similarly, e e~~

∞ ∞ ′ ′ x α−1 dx M [g(x) ∗ h(x); α] := g(x )h ′ x ′ dx= g(α)h(α) , (A.14) 0 0 x x Z Z and e e ∞ ∞ M [g(x) ⋆ h(x); α] := g(xx′)h(x′) xα−1 dx′dx= g(α)h(1 − α) . (A.15) Z0 Z0 e A.2 Expansions e Deﬁnition A.2 The singular expansion of a meromorphic function f(b) with a ﬁnite set P of poles is a sum of its Laurent expansions to order O(b0) about each pole, i.e.

~~f(b) ≍ Laur[f(b),ε; O(b0)] . (A.16) Xε∈P~~

~~30 Theorem A.1 ([16]) Let f(x) have Mellin transform f(α) in ha, bi. Assume~~

ε k M f(x)|x→0+ = cε,kx (log x) e+ O(x ) (A.17) Xε,k where −M < −ε ≤ a and k ∈ N. Then f(α) can be continued to a meromorphic function in h−M, bi where it has the singular expansion e (−1)kk! f(α) ≍ c . (A.18) ε,k (α + ε)k+1 Xε,k Likewise, if e −ε k −M f(x)|x→∞ = cε,kx (log x) + O(x ) (A.19) Xε,k where b ≤ ε < M and k ∈ N. Then f(α) can be continued to a meromorphic function in ha, Mi where it has the singular expansion e (−1)k+1k! f(α) ≍ c . (A.20) ε,k (α − ε)k+1 Xε,k e More generally, it can be shown that for f(α) meromorphic in h−M, bi (respectively ha, Mi) whose poles lie to the right (respectively left) of M, then e α M f(x)|x→0+ = Res f(α)x ,α = εk + O(x ) ε ∈P Xk h i e −α −M f(x)|x→∞ = − Res f(α)x ,α = εk + O(x ) (A.21) ε ∈P Xk h i if f(x) is at least twice diﬀerentiable. e Conversely, Theorem A.2 ([16]) Let f(x) have Mellin transform f(α) in ha, bi. Assume f(α) is meromorphic in h−M, bi such that

−r e e f(α)||α|→∞ = O(|α| ) , r > 1 (A.22) and e ck,ε f(α) ≍ . (A.23) (α − ε)k+1 Xk,ε Then e (−1)k f(x)| = c x−ε(log x)k + O(xM) . (A.24) x→0+ k! k,ε Xk,ε Likewise, if f(α) is meromorphic in ha, Mi, then (−1)k+1 e f(x)| = c x−ε(log x)k + O(x−M) . (A.25) x→∞ k! k,ε Xk,ε

31 A.3 Holomorphic functions The definition of Mellin transform can be extended to include certain holomorphic functions. The following theorems and definitions are taken from [17] with slight modification.

θ1,θ2 Theorem A.3 Let F (b) be analytic in a sector Sa′ := {b : θ1 < arg(b) < θ2} based at the ′ R R θ1,θ2 point a ∈ with θ1 < 0 < θ2 and (θ1, θ2) ∈ . Assume that, for all b ∈ Sa′ , −r F (b)|b→0,∞ = O(b ) (A.26) where r ∈ (a, b). Then: (i) The complex Mellin transform

M[F (b); α]= F (α) := F (b)bα−1 db (A.27) ZCa′ θ ,θ e ′ ′ 1 2 exists and does not depend on the pointed contour Ca (based at b = a ) inside the sector Sa′ . (ii) The complex Mellin transform equals the real Mellin transform in the fundamental strip. −α θ1−arg(w),θ2−arg(w) θ1,θ2 (iii) M[F (wb); α] = w f(α) in the sector Sa′ for w ∈ Sa′ . (iv) The inverse transform exists and is given by e 1 r+i∞ r+i∞ F (b)= b−αF (α) dα = b−αf(α) dα . (A.28) 2πi Zr−i∞ Zr−i∞ Theorem A.4 If f(α) is the Mellin transforme of f(x) in ha, biesuch that f(ρ + iσ)eθ1σ and f(ρ + iσ)eθ2σ are absolutely integrable, then f(x) can be analytically continued to F (b) in the θ1,θ2 e −r e sector Sa′ such that F (b)b→0,∞ = O(b ) for all r ∈ (a, b). Moreover, if f(α) possesses regulare isolated poles in ha′, b′i where a′ < a < b < b′, then the asymptotic expansion of ′ ′ F (b) agrees with the expansion of f(x) up to order O(b−a ) and O(b−b ) at b =0e and b = ∞ respectively.

A.4 Mellin distributions The relationship between Mellin and Fourier transforms allows the development of Mellin distributions. And the extension to holomorphic functions allows development of Mellin distributions on paracompact C∞ complex manifolds. Following [17]; Rn Rn C Deﬁnition A.3 Let fI : + := {y ∈ : 0 < y < ∞} → be a function with support Rn Rn 1 Rn I := {x ∈ + :0 < x ≤ x0 for some x0 ∈ +. Take fI ∈ L ( +) with limits limx→0+ fI (x) → −a −b n O(x ) and limx→∞ fI (x) → O(x ) for a, b ∈ R . Then the Mellin transform f(α) with α ∈ha, bi := (a, b) × iRn ⊂ Cn is deﬁned by (the analytic extension of)17 e −α−1 f(α) := M[fI (x); α] := fI (x)x dx . (A.29) Rn Z + e Rn α α1 αn The notation (a, b) denotes a poly-interval {y ∈ :a < y < b} and x := x1 · ... · xn . 17The substitution α →−α in the exponent of x conforms with reference [17].

~~32 Now, the close relationship with the Fourier transform motivates the deﬁnition~~

~~Deﬁnition A.4 Let v ∈ Rn and deﬁne~~

~~∞ v+1 λ Mv(I) := {ψ ∈ C (I) : sup x (x∂x) ψ(x) < ∞} (A.30) x∈I~~

~~Nn Nn where λ ∈ 0 and 0 is the set of non-negative multi-indices. Endow Mv(I) with the topology deﬁned by the sequence of seminorms~~

~~v+1 λ ̺v,λ(ψ) = sup x (x∂x) ψ(x) . (A.31) x∈I~~

~~Rn R n Then M(w)(I) for w ∈ ∞ := ( ∪ {∞}) is deﬁned to be the inductive limit of Mv(I), i.e. −−→ ′ M(w)(I) = limv~~

~~′ ′ M (I)= M(w)(I) . (A.32) w∈Rn [∞ ′ Finally, the Mellin transform of a distribution T ∈ M(w)(I) is deﬁned by~~

~~T(α) := M [T; α] := hT, x−α−1i , ℜ(α) < w . (A.33)~~

~~Note the topological inclusionse~~

~~′ ′ D(I) ⊂ Mv(I) ⊂ M(w)(I) ⊂ D (I) , (A.34)~~

~~and T(α) is well-deﬁned on the set~~

~~e ΩT := [ℜ(α) < v] . (A.35) {v:T∈M ′ (I)} [(v) Some of the important properties of the 1-dimensional Mellin transform have their ana- logues for distributions:~~

~~Theorem A.5 The Mellin transform T(α) is holomorphic on ΩT and~~

∂ −α−1 T(α) = hT, xe (− log xi)i ∂αi β T(α e− β) = M x T; α ℜ(α) −ℜ(β) < w γ γ Nn α T(α) = M [(x∂x) T; α] γ ∈ 0 , ℜ(α) < w γ e γ (α + 1)T(α + γ) = M [(∂x) T; α] |γ| =1 , ℜ(α) < w − γ . e (A.36) e

~~33 B Exponential exercises~~

The exponential function plays a prominent role in ordinary Mellin transforms, so we want to develop and characterize the functional counterpart by looking at some specific cases of reduction to finite dimensional groups. 1 n C C S Let E := expFS(G ) := n n! (·) stand for the exponential on F (G ) defined with the product given by the ∗-convolution. Suppose C∗ ≡ C, GC → R , and A(g) = Ag where P + A ∈ C+ := R+ × iR. For the choice of λ that corresponds to the standard Haar measure, this is just the usual exponential Mellin transform

−A −Ag α−1 Γ(α) MH E ; α := e g dg = α , α ∈h0, ∞iH . (B.1) R A Z + In particular, −Id MH E ; α = Γ(α) , α ∈h0, ∞iH . (B.2) −gg˜ −g˜ As a quick exercise, use Th. 4.1 with F1(˜gg) = e and F2(˜g)ρ(˜g) = e g˜ to deduce (for A = Id and a choice of λ corresponding to standard normalization)

~~−g˜ −gg˜ Γ(α)Γ(1 − α) = MH e g˜ e d(logg ˜); α R Z + ∞ tα−1 = dt 1+ t Z0 = π csc(πα) , α ∈h0, 1iH . (B.3)~~

Notice the reduction in Sλ. Simple manipulations yield the standard results πα csc(πα) = Γ(1 + α)Γ(1 − α) and Γ(1+ α)/Γ(α − 1) = α(α − 1). However, the functional Mellin transform provides a mechanism to regularize; and with a suitable choice of λ,

−A −Ag α 1 MΓ E ; α := e g dν(gΓ)= α , α ∈h0, ∞iΓ (B.4) R A Z + for ν(gΓ) := log(g)/Γ(α) = ν(g)/Γ(α) where ν(g) is the normalized Haar measure on R+. To extend the fundamental strip to the left of the imaginary axis, one can use

−A Γ(α) p −Ag α MΓp E ; α := (Ag) e g dν(gΓ) Γ(α + p) R Z + p p (−1) p d −A = M g E ; α , α ∈ h−p, ∞i p , p ≥ 0 . Γ(α + p) H dg Γ (B.5) There are other ways to extend the fundamental strip to the imaginary axis and beyond. For example, deﬁning 6e−Ag := e−Ag − e−g yields

~~−A −Ag α 1 MΓ 6E ; α := 6e g dν(gΓ)= α − 1 , α ∈ h−1, ∞iH . (B.6) R A Z +~~

~~34 For α → 0+ this gives 6e−Ag dν(g)= − log(A) , (B.7) R Z + and therefore (in this case)~~

~~−A d −A MΓ 6E ; α α→0+ = MΓ E ; α (B.8) dα + α→0 which suggests the deﬁnition~~

~~′ d α Mλ [F; 0] := Mλ [F; α] =: F(g)g logλ g Dλg dα + C + α→0 ZG α→0~~

~~α =: F(g)g Dλg (B.9) C ZG α→0+~~

C R b if the limit exists. In particular, for G → +, d D g := log g D g := gαdν(g ) =: gα dν(g ) =: gαlog g dν(g ) (B.10) λ λ λ dα λ λ ψ λ ′ 1 2 Γ (α) where ν(gλ)=b 2 logψ gλ with logψ g := log g − Γ(α) =:b log g − ψ(α). For example, choosing 1 2 ν(gΓ)= 2 logψ g/Γ(α) yields dν(gΓ) = logψ g dν(gΓ). This motivates Def. 5.6 for functional Log. b C C C C b Moving on to the non-abelianb case, suppose that G → GL(n, )+ := SL(n, ) × + ∗ −TrA C −tr (A·g) with C+ := R+ × iR and C ≡ C. Deﬁne the functional E : G → C by g 7→ e C with A ∈ GL(n, C)+, and take ρ : G → C+ by g 7→ det g. Then

−TrA −trAg α S MΓn E ; α = e det (g ) dν(gΓn ) , α ∈ Γn C ZGL(n, )+ −trAg α iϕ(α) S = e (det g) e dν(gΓn ) , α ∈ Γn C ZGL(n, )+ −trg −1 α iϕ(α) S = e det A det g e dν(gΓn ) , α ∈ Γn C ZGL(n, )+ −α S = det(A ) , α ∈ Γn (B.11) where ϕ(α) is a complex phase, ν(gΓn ) := ν(g)/Γn(α) with ν(g) the Haar measure on GL(n, C)+, and Γn(α) a complex multi-variate gamma function deﬁned by

~~−Tr Id −tr g α Γn(α) := MH E ; α := e det (g ) dν(g), α ∈ SH . C ZGL(n, )+ (B.12) S In particular, if α =1 ∈ Γn , then~~

~~−1 −tr Ag −1 det(A )= e (det g) dν(gΓn ) = (detA) (B.13) C ZGL(n, )+~~

35 Remark that Γn(α) is not a well-deﬁned object unless one restricts to a compact subgroup −α of GL(n, C)+. Otherwise, the price of extracting det(A ) from the integral comes with the price of regularizing this possibly singular normalization. C ∗ n Generalizing further, suppose G → GL(n, C)+ but now C ≡ LB(C ) the space of n −A C n −A·g bounded linear maps on C and E : G → LB(C ) by g 7→ e with A = ρ(a) ∈ n LB(C ) such that a ∈ GL(n, C)+. The Haar normalized functional Mellin transform of the exponential yields

−A −Ag α MH E ; α := e g dν(g) , α ∈ SH C ZGL(n, )+ −g −1 α = e (A g) dν(g) , α ∈ SH C ZGL(n, )+ −α S =: AH , α ∈ H (B.14) −α ∗ S which deﬁnes the element AH ∈ Ms(C ) for α ∈ H . Unless a is in the center of GL(n, C)+ or we restrict to a subgroup of GL(n, C)+, this can’t −α −α be reduced further without explicit computation, i.e. AH =6 (A) in general. However, various restrictions allow for various degrees of simpliﬁcation. For example, if A is self-adjoint and GC is restricted to an imaginary one-parameter subgroup generated by log a ∈ gl(n, C), C then more can be done. So let us take G → φlog a(i R). Then, since a ∈ φlog a(i R),

−A −Ag α MΓ E ; α = e g dν(gΓ) , α ∈ SΓ R Zφlog a(i ) −g −1 α = e (A g) dν(gΓ) , α ∈ SΓ R Zφlog a(i ) −α −g α = A e g dν(gΓ) , α ∈ SΓ R Zφlog a(i ) −α =: A NΓ(α) , α ∈h0, ∞iΓ (B.15)

~~The second line follows from the left-invariance of the Haar measure, and the third line from the fact that a and g commute. Most of the time we will absorb the normalization NΓ(α) into the measure.~~

~~C Relation to crossed products~~

The ingredients necessary to deﬁne crossed products[21] are: i) a “dynamical system” (A,G,ε) where A is a C∗-algebra, G is a locally compact group, and ε : G → Aut(A) is a continuous homomorphism; ii) some Hilbert space H; iii) an algebra representation ̟ : A → LB(H); and iv) a unitary, group representation U : G → U(H). The two representations are required to satisfy the ‘covariance condition’

~~∗ ̟(εg(a)) = Ug̟(a)Ug , g ∈ G , a ∈ A. (C.1)~~

~~36 With these objects, a ∗-representation on H of Cc(G, A) (continuous compact morphisms f : G → A) is supplied by the integral~~

~~̟ ⋊ U(f) := ̟(f(g))Ug dµ(g) (C.2) ZG where f ∈ Cc(G, A) and µ is a Haar measure on G. A product and involution are introduced on Cc(G, A) according to~~

−1 (f1 ∗ f2)(g) := f1(˜g)εg˜(f2(˜g g)) dµ(˜g) (C.3) ZG and ∗ −1 −1 ∗ f (g) := ∆(g )εg(f(g ) ) (C.4) where ∆ is the modular function on G. Completion of Cc(G, A) with respect to the norm deﬁned by kfk := supk̟ ⋊ U(f)k (C.5) ∗ is a C -algebra called the crossed product denoted by A ⋊ε G. The crucial property of this construction is a one-to-one correspondence between non- degenerate covariant representations associated with (̟, U) and non-degenerate representa- ∗ tions of A⋊εG which preserves direct sums, irreducibility, and equivalence. So the C -algebra ∗ A ⋊ε G can be used to model the C -algebra encoded in the system (A,G,ε) endowed with a covariant representation (̟, U). We recognize the covariant condition as an algebra auto- morphism by a group element; which, in particular, for the evolution operator in quantum mechanics becomes the integrated Heisenberg equation. C Let’s compare with functional Mellin. Suppose λ : G → Gλ. Identify π ◦ ρ ≡ U (with suitable restrictions if necessary) and choose Dλg ≡ dµ(g) with g ∈ Gλ, then

π (Mλ[F, 1]) = π(f(g))U(g) dµ(g) (C.6) ZGλ ∗ where π : C → LB(H). As soon as π(f(g)) is Mellin integrable w.r.t. Gλ, this integral and the integral in (C.2) represent the same object in LB(H) iff ̟ ◦ f ≡ π ◦ f. Keep in mind 1 ∗ that the nature of f ∈ Cc(G, A) versus f ∈ L (Gλ, C ) is quite dependent on the nature of A versus C∗: If they are both simultaneously commutative or non-commutative, then f and f at least have the chance of representing the same object if A and C∗ are isomorphic. Otherwise, they are distinctly different. Mathematically, we can always choose A ≡ C∗ and ̟ ≡ π. In this case, the difference between crossed products and functional Mellin is that ∗ A ⋊ε G is the C -algebra of f ∈ Cc(G, A) satisfying the covariance condition (C.1) while C ∗ 1 FS(G ) is the C -algebra of equivariant f ∈ L (Gλ, A). For application to quantum physics, the pivotal point comes down to ε : G → Aut(A) and dynamics. Suppose the algebra is commutative. By Gelfand duality, there is some topological space X such that A ≡ C0(X) (the algebra of complex valued continuous morphisms vanishing at infinity). Non-trivial ε reflects a basic assumption about the dynamical

37 −1 system; that G acts on X and this is accounted for by εh(f(g))(x)=f(g)(h · x) for x ∈ X. But then the covariance condition is required to encode dynamics through the adjoint action on LB(H). Insofar as quantization (virtually always) starts with a classical system (C0(X),G,ε) with covariant representation (̟, U), the crossed product A ⋊ε G supplies a concrete realization of quantization of the commutative algebra C0(X). On the other hand, for functional Mellin the group is contained in A by construction so it acts by inner automorphisms which automatically reproduces the covariance condition. Moreover, by equivariance, f(gh)(x)= f(g)ρ(h)(x)= f(g)(h−1 · x). However, the involution and product in functional C ∗ Mellin do not depend on ε — unlike A ⋊ε G. Evidently, even though FS(G ) is a C -algebra it is not isomorphic to A ⋊ε G and it would be difficult to attach physical interpretation to 1 the functions f ∈ L (Gλ, A) and their relation to the dynamical system. Now suppose A is non-commutative and G is its group of units. Then G acts on A by inner automorphisms which means the covariance condition is automatic and ε is unneeded. Setting ε ≡ Id brings the involution and product of crossed products into agreement with functional Mellin. But then, the only way (it seems) to save the non-commutative C∗-algebra C ∼ structure of A⋊Id G is to insist that f be equivariant. In this situation then FS(G ) = A⋊Id G and representations furnished by π (Mλ[F, 1]) are in one-to-one relation to A ⋊Id G and therefore in one-to-one relation to the system (A,G,Id) with covariant representation (̟, U). But note this dynamical system in not classical — until expectations are taken. Whereas the previous paragraph described the quantization process classical → quantum; this paragraph describes quantum → classical. But for quantum → classical how can we know anything about the functions f ≡ f without C0(X)? Happily, spectral theory allows to represent (π ◦ f)(g) in terms of an operator valued function fˆ(ρ(g)) and Mellin integrators supply the resolvent of ρ(g). If we don’t venture outside of A to find evolution operators and we use functional calculus to represent π(f(g)) ≡ fˆ(ρ(g)), then functional Mellin and a choice of GC fully determine a quantum system. That is, once we settle on GC and find relevant representations and ∗ C their furnishing Hilbert spaces, functional Mellin defines a C -algebra FS(G ) that contains quantum dynamics and Mellin integrators furnish representations of this algebra in LB(H). To highlight the similarity of crossed products and functional Mellin further, extend the integrated form of (̟, U) to GC according to

~~(α) ̟ ⋊ U (f) := ̟(f(g))Ugα dµ(g) (C.7) C ZG ∗ 1+α −1 −(1+α) ∗ Likewise extend the involution f (g ) := ∆(g )εgα (f(g ) ) and product~~

α −1 α (f1 ∗ f2)(g ) := f1(˜g)εg˜(f2(˜g g )) dµ(˜g) . (C.8) C ZG C ∗ Then we conjecture A ⋊ε G (after completion w.r.t. to a suitable norm) is a C -algebra and ̟ ⋊ U (α) is a ∗-homomorphism because: • (f1 ∗ f2) ∗ f3 =f1(∗f2 ∗ f3) (C.9)

~~38 follows immediately from (C.8) using the invariance of the Haar measure.~~

• ∗ 1+α ∗ −1 ∗ −(1+α) 1+α f (g ) = (∆(g ) εgα (f(g )) = f(g ) (C.10) where the ﬁrst equality follows from the covariance conditions and the second from the (1+α) −1 ∗ −(1+α) invariance of the Haar measure which implies f(g )=∆(g ) εgα (f(g )). •

∗ ∗ 1+α kf kα := kf (g )k dµ(g) C ZG −1 −(1+α) ∗ = k∆(g )εgα (f(g ) )k dµ(g) ZG −1 ∗ −(1+α) ∗ ∗ = k∆(g ) εgα (f(g ) ) k dµ(g) C ZG −1 ∗ −(1+α) = k∆(g ) εgα (f(g )k dµ(g) C ZG = kf(g−(1+α))k dµ(g) C ZG = kfkα . (C.11)

•

∗ ∗ 1+α ∗ ∗ −1 1+α (f1 ∗ f2 )(g ) = f1 (˜g)εg˜(f2 (˜g g )) dµ(˜g) C ZG −1 −1 ∗ −1 −(1+α) ∗ = ∆(˜g )εg˜(f1(˜g ) )εg˜(∆(g g˜))εg˜−1g1+α (f2(g g˜) ) dµ(˜g) C ZG −1 −1 ∗ −(1+α) ∗ = ∆(g ) εg˜(f1(˜g ) )εg˜ εg˜−1g1+α (f2(g g˜) ) dµ(˜g) C ZG −1 −1 ∗ −(1+α) ∗ = ∆(g ) εg˜(f1(˜g ) )εg1+α (f2(g g˜) ) dµ(˜g) C ZG −1 −1 −(1+α) ∗ ∗ = ∆(g ) εg1+αg˜(f1(˜g g ) )εg1+α (f2(˜g) ) dµ(˜g) C ZG −1 −1 −(1+α) ∗ = ∆(g )εg1+α f2(˜g)f1(˜g g ) dµ(˜g) C ZG −1 (1+α) ∗ = ∆(g )εg1+α f2 ∗ f1(g ∗ 1+α = (f2 ∗ f1) (g ) (C.12)

~~39 •~~

(α) ∗ ∗ ̟ ⋊ U (f) = (̟(f(g))Ugα ) dµ(g) C ZG ∗ = Ug−α ̟(f(g) ) dµ(g) C ZG −1 ∗ −1 = Ugα ̟(f(g ) )∆(g ) dµ(g) C ZG −1 ∗ −1 = ̟ εg(f(g ) ∆(g )) Ugα dµ(g) C ZG ∗ −1 = ̟ f (g ) Ugα dµ(g) C ZG = ̟ ⋊ U(α)(f∗) (C.13)

•

(α) −1 ̟ ⋊ U (f1 ∗ f2) = ̟ f1(g))εg(f2(g g˜)) Ug˜ dµ(g, g˜) C C ZG ×G −1 = ̟ f1(g))Ug̟(f2(g g˜)) Ug−1g˜ dµ(g, g˜) C C ZG ×G = ̟ (f1(g))Ug ̟(f2(˜g))) Ug˜ dµ(g, g˜) C C ZG ×G (α) (α) = ̟ ⋊ U (f1) · ̟ ⋊ U (f2) (C.14)

~~where the last equality follows from Fubini.~~

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