Algebras Defined by First Order Differential Seminorms

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 1, February 2016, pp. 125–141. c Indian Academy of Sciences

Smooth Frechet subalgebras of C∗-algebras deﬁned by ﬁrst order differential seminorms

SUBHASH J BHATT

Center for Interdisciplinary Studies in Science and Technology (CISST), Sardar Patel University, Vallabh Vidyanagar 388 120, India Present Address: Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388 120, India E-mail: [email protected]

MS received 15 April 2014; revised 17 January 2015

Abstract. The differential structure in a C∗-algebra defined by a dense Frechet subalgebra whose topology is defined by a sequence of differential seminorms of order 1 is investigated. This includes differential Arens–Michael decomposition, spectral invariance, closure under functional calculi as well as intrinsic spectral description. A large number of examples of such Frechet algebras are exhibited; and the smooth structure defined by an unbounded self-adjoint Hilbert space operator is discussed.

Keywords. Smooth subalgebra of a C∗-algebra; spectral invariance; closure under ∗ functional calculus; Arens–Michael decomposition of a Frechet algebra; Frechet D1 - algebra; unbounded self-adjoint Hilbert space operator.

2000 Mathematics Subject Classiﬁcation. Primary: 46H20; Secondary: 46L57.

1. Introduction Noncommutative differential topology requires understanding differential structure in a C∗-algebra. This is generally deﬁned by derivations [11], differential seminorms [7, 9] or a family of seminorms satisfying appropriate growth conditions inspired by the Leib- ∗ niz rule [15]. Let (U, .0) be a C -algebra. A differential Banach algebra of order 1 in ∗ ∗ (U, .0), also called a Banach D1 -subalgebra of U [15, 16], is a -subalgebra A of U which is a Banach ∗-algebra with a norm ·such that x∗=x, xy≤xy and xy≤x0y+xy0 for all x,y in A. In particular, let T = (T0,T1) be a dif- ∗ ferential norm of order 1 [9] on A, so that T0 =·0 and T1 is a -seminorm satisfying T1(xy) ≤ T0(x)T1(y) + T1(x)T0(y) for all x,y in A.Letx=T0(x) + T1(x)(x ∈ A). ∗ If (A, .) is complete, then it is a Banach D1 -algebra. Kissin and Shulman have investi- ∗ gated Banach D1 -algebras in a series of papers [16–18]. As a noncommutative analogue of the algebra Cp[a,b] of p-times continuously differentiable functions, they have pro- ∗ posed (and investigated in [16] the basic properties of) a Banach Dp-subalgebra of U as ∗ a dense -subalgebra A such that there exists a family of seminorms { · i : 0 ≤ i ≤ p} satisfying the following: ∗ (1) For all x,y ∈ A and for all i,1≤ i ≤ p, xyi ≤xiyi, x i =xi. (2) For each i, 1 ≤ i ≤ p, xyi ≤xiyi−1 +xi−1yi for all x,y in A. ∗ (3) (A, .p) is a Banach -algebra.

125 126 Subhash J Bhatt

In [5], we have initiated the investigation of the following C∞-analogue advancing the above line of investigation.

DEFINITION 1.1

∗ ∗ Let (U, ·0) be a C -algebra. Let A be a dense -subalgebra of U. Then A is called a ∗ Frechet D∞-subalgebra of U if there exists a sequence of seminorms {·i : 0 ≤ i<∞} such that the following hold.

∗ (1) For all i, 1 ≤ i<∞, for all x,y in A, xyi ≤xiyi, x i =xi. (2) For each i,1≤ i<∞, xyi ≤xiyi−1 +xi−1yi holds for all x,y in A. (3) B is a Hausdorff Frechet ∗-algebra with the topology τ deﬁned by the seminorms { · i : 0 ≤ i<∞}.

∗ ∗ Thus a Frechet D∞-subalgebra of a C -algebra is envisaged as a noncommutative analogue of C∞[a,b]. In [5], its regularity properties like spectral invariance, closure under appropriate functional calculi, invariance of domains of smooth homomorphisms are investigated using a basic structure theorem proved therein. In the present paper, we investigate the following important class in which the topology τ is deﬁned by a sequence of ﬁrst order seminorms. It follows from the estimates discussed in Examples 8 and 9 in [15] that this includes the classical case C∞[a,b] as well as the C∞ domain of a closed C∗-derivation that is a generator.

DEFINITION 1.2

∗ ∗ Let (U, ·0) be a C -algebra. Let A be a dense -subalgebra of U. Then A is called a ∗ Frechet D1 -subalgebra of U if there exists a sequence of seminorms { · i : 0 ≤ i<∞} such that the following hold:

∗ (1) For all i, 1 ≤ i<∞, for all x,y in A, xyi ≤xiyi, x i =xi. (2) For each i,1≤ i<∞, xyi ≤xiy0 +x0yi holds for all x,y in A. (3) A is a Hausdorff Frechet ∗-algebra with the topology τ deﬁned by the seminorms { · i : 0 ≤ i<∞}.

∗ This provides a Frechet analogue of Banach D1 -algebras. This models Frechet differential algebras of order 1 which are subalgebras of C∗-algebras; e.g. Frechet algebras of continuous functions on a compact interval [a,b] that are C1 in (a, b). In §2, we discuss a structure theorem (Theorems 2.2 and 2.3 constituting the main contribution of ∗ the paper) showing that A as above is an inverse limit of a sequence of Banach D1 - subalgebras of a C∗-algebra; and discuss appropriate functional calculus. This functional calculus is a Frechet analogue of Theorem 12 in [15]; and improves, in the present case, Corollary 3.3 in [5]. We also prove in §3 an intrinsic characterization of a Frechet ∗ ∗ ∗ ∗ D1 -subalgebra of a C -algebra. A Banach -algebra turns out to be a Banach D1 -sub algebra of a C∗-algebra if and only if there exists a D>0 such that for all x,y it holds that xy≤D{xs(y) +ys(x)},s(x) = r(x∗x)1/2 being the Ptak function. ∗ ∗ This refines Theorem 14 in [15]. It follows that a Frechet D1 -subalgebra of a C -algebra ∗ admits a unique topology; and is necessarily the Frechet D1 -subalgebra of its enveloping C∗-algebra [3, 4]. In §4, we discuss approximation in A giving straight forward generalizations of results in [16] that can be obtained by the typical method of inverse limit Smooth Frechet subalgebras of C∗-algebras 127 decomposition when one passes from Banach to Frechet case. However a new structure ∗ arises at the present Frechet D1 -algebra level, viz., the bounded part b(A) of A defined as ∗ b(A) := {x ∈ A :x:=supn xn < ∞} which is a Banach D1 -algebra with norm .. ∗ In §5, we discuss various classes of examples of Frechet D1 -algebras which also provide a motivation to investigate the present case. In particular, we discuss in some details the differential structure defined by an unbounded self-adjoint operator on a Hilbert space. It should be noted that this class of Frechet algebras has been first suggested in [2]. The ∗ present paper provides Frechetization of Banach D1 -algebras discussed in [15, 16] and contributes to understanding differential structure in a C∗-algebra.

2. Structure theorem

Let (.n) be a defining sequence of seminorms defining the topology τ of A satisfying ∗ for all x,y in A, x n =xn, xyn ≤xnyn, xyn ≤x0yn +xny0. Further (.n) can be assumed to be increasing (replacing, if necessary, .n by the inductively defined |.|n := max{|.|n−1, .n}). Notice that we can take .0 ≤.n for all n. Indeed by a standard automatic continuity argument, the inclusion id : (A,τ) → (U, .0) is continuous; hence there is a scalar K>0 and an n such that x0 ≤ Kxn 2 ∗ ∗ 2 1/2m for all x ∈ A. Then x0 =x x0 ≤ Kx xn ≤ Kxn; hence x0 ≤ K xn for all x and for all m, and so x0 ≤xn for all x ∈ A. Now we can delete (.k : k = 1, 2, 3,...,n− 1), {.n} being increasing; and assume .0 ≤.n for all n. By following the standard technique of Arens–Michael decomposition [13], we express A as an inverse limit of a sequence of Banach ∗-algebras as follows. Let the Banach ∗- algebra An be the completion of A in the norm .n. The inclusion id : A → U extends ∗ ∗ as a continuous -homomorphism φn : An → U making An to be a C -seminormed ∗ ∗ algebra with the C -seminorm .0 induced by the C -norm on U and extending it from A. It is easily seen that the norm .n on An satisfies the first order differential inequality xyn ≤x0yn +xny0 for all x,y in An. Since .n ≤.n+1 on A, there ∗ exists a -homomorphism πn : An+1 → An for each n such that the chain

···←−An−1 ←− An ←− An+1 ←−··· ∗ forms an inverse system of Banach -algebras, and A = lim An. Recall that A is a Q- ←− algebra [13] if the quasi-invertible elements of A (invertible elements of A in case A is unital) form an open set in the topology of A.

Lemma 2.1. The algebra A is spectrally invariant in U, is hermitian and is closed under holomorphic functional calculus of U. Further the Frechet algebra A, each of the Banach ∗ ∗ -algebra An and the C -algebra U have the same K-theory.

2 2k k 2k−1 Proof. Fix n. For any x ∈ An, x n ≤ 2x0xn; and so x n ≤ 2 x0 xn for k 1/k any k = 1, 2, 3,.... Thus the spectral radius of x in An is rn(x) := limk→∞ x n = k 2k 1/2 A limk→∞ x n ≤x0. By [8], for any x ∈ n, the spectrum SpAn (x) = A ∞ SpU (φn(x)). Then for any x ∈ ,SpA(x) =∪n=0SpAn (x); and so the spectral radius r(x) = supn rn(x) ≤x0, showing that A is spectrally invariant in U; and (A, .0) is a Q-algebra (pre-C∗-algebra [12]). The remaining assertions follow from [8], [12]. To be able to get closure under functional calculus by functions of regularity classes of appropriate generality, one needs a reﬁned version of inverse limit decomposition 128 Subhash J Bhatt

∗ discussed below. It is shown in Theorem 3.1 of [5] that a Frechet (D∞)-subalgebra of a C∗-algebra decomposes as an inverse limit, over n, of a sequence of Banach ∗ ∗ (Dn)-subalgebras of C -algebras. The following is an analogous result in the present case.

∗ ∗ Theorem 2.2. Let A be a Frechet D1 -subalgebra of a C -algebra U. Then there exists an inverse limit sequence

···←−Añ−1 ←− Añ ←− Añ+1 ←−··· such that the following hold: ˜ ∗ (1) Each An is a Banach D1 -subalgebra of U. (2) A = lim Añ. ←− (3) If (xn), xn ∈ Añ, is a coherent sequence such that xn = x for all n, then x ∈ A. ∗ (4) Further if the norms .n on A are closable with respect to the C -norm .0, then ˜ ˜ ∞ ˜ An+1 ⊂ An for all n; and A =∩n=0An.

Proof. Let In := ker φn ={x ∈ An : φn(x) = 0} a closed ideal in An. Then In ∩ A = (0); and An ⊂ U iff .n is closable with respect to .0.LetAñ := A/In the completion of A/In with the quotient norm (denoted by .n itself) from the norm .n on An. Since In is closed in An, Añ = An/In.Letjn : An → Añ be the natural quotient map jn(x) = x + In =˜x say. Notice that ˜xn = inf{x + zn : z ∈ In}.Themap ∗ φn˜ : Añ → U, φñ(x)˜ = φn(x) is an injective -homomorphism; and |˜x|:=φn(x)0 defines a norm on Añ such that the completion of (Añ, |.|) is isomorphic to U.Weshow that for x,˜ y˜ in Añ, it holds that

˜xyñ ≤|˜x|y ñ +|˜y|x ñ.

Let x,y be in A. Then for any u, v in In,

˜xy˜n =(xy)˜ n = inf{xy + zn : z ∈ In} ≤ inf{(x + u)(y + v) : u ∈ In,v ∈ In} ≤{x + u0y + vn +y + v0x + un} ≤ (x0 +u0)y + vn + (y0 +v0)x + un.

Taking inf over v and v in In, we get ˜xyñ ≤x0˜yn +y0˜xn =|˜x|y ñ + |˜y|x ñ since φn|A = id and x,y are in A.Givenx ∈ An, there exist a sequence (yk) in A such that yk → x in .n. By continuity, φn(yk) = yk → φn(x) in (U, .0). Thus |˜x|=φn(x)0 = lim yk0. Using this in the above, for each x,y in An,we have ˜xyñ ≤|˜x|y ñ +|˜y|x ñ; and by continuity, this also holds for all x,˜ y˜ in Añ. ˜ ∗ ∗ This establishes the claim. Thus (An, .n) is a Banach D1 -subalgebra of the C -algebra (U, .0). Notice that A = lim An. We show that A = lim Añ. ←− ←− Consider the maps πñ : Añ+1 → Añ, πñ(x+In+1) = πn(x)+In. Since .n ≤.n+1 on A, πn(x))n ≤xn+1 for all x ∈ An+1. Hence πn(In+1) ⊂ In; and so πñ is a well ∗ defined -homomorphism. Now it is easily seen that (Añ, πñ) is an inverse limit sequence and A = lim Añ. Also, if σn : A → Añ is the defining projection in the construction of ←− this inverse limit, then σn(x) = jn(x) = x + In; i.e. σn = jn|A and σn is one-one on A with the result that A is a subalgebra of each of Añ. Smooth Frechet subalgebras of C∗-algebras 129

∗ Finally assume that each of the norms .n is closable with respect to the C -norm .0. Since .0 ≤.n ≤.n+1 for all n, it follows that .n+1 is closable with respect to .n, with the result Añ+1 ⊂ Añ ⊂ U, and it follows that A =∩Añ. The following is a converse.

Theorem 2.3. Let ··· ←− An−1 ←− An ←− An+1 ←−··· be an inverse limit ∗ ∗ sequence of Banach -algebras (An, .n) such that each An is a Banach D1 -subalgebra ∗ ∗ of a C -algebra (U, .0). Let A = lim An. Then A is a Frechet D -subalgebra of U. ←− 1

Proof. Let πn : An+1 → An be the deﬁning linking homomorphisms in the given inverse limit sequence. By construction, A ={x = (xn) : xn ∈ An,π(xn+1) = xn for all n} with the relative product topology. Let pn : A → An be the projection maps pn(x) = xn.The topology on A is deﬁned by the sequence of semi norms {|.|n}, |x|n =|(xn)|n =xnn. ∗ ∗ ∗ Now each An is -semi simple, being a subalgebra of a C -algebra. Hence A is -semi ∗ ∗ ∗ simple having enveloping C -algebra C (A) = lim C (An) = U the last equality being a ←− ∗ consequence of the fact that each An is a Banach D1 -subalgebra of U. Thus A is a dense ∗ -subalgebra of U.Nowforanyx = (xn), y = (yn) in A,

|xy|n =xnynn ≤xn0ynn +xnnyn0 =xn0|y|n +|x|nyn0 ≤x0|y|n +|x|ny0 ∗ showing that A is a Frechet D1 -subalgebra of U. The last inequality follows thus. Since ∗ An is a Banach D1 -subalgebra of U, it is spectrally invariant in U and is a Q-normed algebra in the norm .0. Hence A is also a Q-normed algebra in the norm .0. Hence 2 ∗ ∗ ∗ ∗ ∗ ∗ xn0 =xnxn0 = rU (xnxn) = rAn (xnxn) = rAn (pn(x x) ≤ rA(x x) ≤x x0 = 2 x0. ∗ ∗ Easy arguments, as in [5], show that if A is a Frechet D1 -subalgebra of a C -algebra U, then the unitification Ae, the matrix algebra Mn(A) over A, the direct sum as well as ∗ ∗ appropriate quotients are all Frechet D1 -subalgebras of C -algebras. Following Blackadar and Cuntz [9], a ∗-subalgebra A of a C∗-algebra U is smooth if it is complete in the topology defined by the family of all closable derived norms. (A derived norm is the quotient norm of the total norm of a differential norm of finite total order, see also [7].)

COROLLARY 2.4

∞ ∗ ∗ Let (A, {.n}0 ) be a Frechet D1 -subalgebra of a C -algebra (U, .0). Then A is a smooth subalgebra of U.

∗ Proof. By Theorem 2.2, A = lim Añ, each Añ with norm .n being a Banach D - ←− 1 (n) subalgebra of U. Thus, for each n, T ={.0, .n} is a differential norm of order 1 on Añ, and so on A. Hence each Añ is a differential Banach algebra of order 1 with norm n Ttot =.0 +.n. Thus A is a Frechet differential algebra of order 1 in the topology n defined by the norms {Ttot(.)}; and this topology is same as the topology τ of A defined ∞ by {.n}0 . Now the total norm of a differential norm (of order 1) is necessarily a derived norm; and so the Open Mapping theorem implies that τ is also the topology defined by the family of all closable derived norms on A. 130 Subhash J Bhatt

∞ ∗ The C -functional calculus for Frechet D∞-algebras discussed in Corollary 3.3 in ∗ ∗ [5] can be considerably reﬁned for Frechet D1 -algebras along the lines of Banach D1 - ∗ algebras discussed in Theorem 12 in [15]. Let A be a Frechet D1 -subalgebra of U as above. Assume that U is unital. Then by Theorem 5 in [15], the identity 1 ∈ A˜n for all n; and the inverse limit decomposition shows that 1 ∈ A. The following can be easily proved by applying the arguments in Theorem 12 in [15] to each of the factor Banach ∗ D1 -algebras in the inverse limit decomposition in Theorem 2.2.

COROLLARY 2.5

Let f be a continuous function on the real line R supported on a closed and bounded ∗ interval [a,b]. Let x ∈ A be such that x = x and SpA(x) ⊂[a,b]. Assume 2 ∞ ˆ A that either f is a C -function or −∞ |s||f(s)| ds < ∞. Then f(x) ∈ and f(x) = 1/2 ∞ isx ˆ (1/(2π)) −∞ e f(s)ds.

When x0 <π, the above can be further strengthened using ideas leading to Theorem 6.4 in [9]. Let 0 <ǫ<1/2. Let C(1+ǫ)+(S1) consist of functions f contin- 1 uous on the circle S such that the Fourier coefficients {f(s)ˆ } satisfies f (1+ǫ)+ := ˆ 1+ǫ 2 1 s∈Z |f(s)||s| < ∞, a Banach algebra with norm .(1+ǫ)+ satisfying C (S ) ⊂ (1+ǫ)+ 1 1 1 C (S ) ⊂ C (S ).NowTn := {.0, .n} is a differential norm of order 1 and logarithmic order 1 on A. By Proposition 3.10 in [9], the total order of Tn is ≤ 1. Thus the total norm αn := .0 +.n of Tn is a derived norm of order ≤ 1 and is equiv- (1+ǫ)+ 1 ˆ ist ∗ alent to .n.Nowletf ∈ C (S ), f (t) = s∈Z f(s)e .BytheC -algebra ˆ isx U A isx A functional calculus, f(x) = s∈Z f(s)e ∈ . Since x ∈ ,e ∈ ; and so, for ˆ isx A each n ∈ N, fn(x) := |s|≤n f(s)e ∈ . Also, each α is of total order ≤ 1; and hence by [9], for sufficiently large s ∈ N, α(eisx) ≤ s1+ǫ for all ǫ>0. For s with (−s) ∈ N,α(eisx) = α(ei(−s)(−x)) ≤|s|1+ǫ. Thus for s ∈ Z with sufficiently large |s|, α(eisx) ≤|s|1+ǫ. Therefore, for m>n, m m ˆ isx ˆ 1+ǫ α(fm(x) − fn(x)) ≤ |f(s)|α(e ) ≤ |f(s)||s| → 0 n+1 n+1 as n, m →∞. Thus {fn(x)} is a Cauchy sequence in the Frechet topology of A. It follows that f(x)∈ A. Thus the following is proved.

PROPOSITION 2.6

∗ (1+ǫ)+ 1 Let x = x ∈ A, x0 <π. Let 0 <ǫ<1/2. Let f ∈ C (S ). Then f(x)∈ A.

3. Intrinsic characterization ∗ ∗ We discuss an intrinsic characterization of a Frechet D1 -subalgebra of a C -algebra without any reference to the C∗-algebra in which it is embedded. Let A be a Frechet ∗ -algebra with identity whose topology is deﬁned by a sequence (.n) of submulti- ∗ k 1/k plicative -seminorms. For each n, and each x ∈ A,letrn(x) = limk→∞ x n ; ∗ 1/2 sn(x) = rn(x x) . We shall use the following. Here rB(x) denote the spectral radius of ∗ / ∗ x in B; and sB(.) is the Ptak function deﬁned as sB(x) = rB(x x)1 2. A Banach -algebra Smooth Frechet subalgebras of C∗-algebras 131

A is reduced (also called *-semi simple) if for any x ∈ A,f(x∗x) = 0 for all positive linear functionals f on A implies that x = 0. Theorem 3.1 (Theorem 14 in [15]). Let (B, .) be a Banach ∗-algebra with identity. The following are equivalent: (1) B is a reduced algebra satisfying the property: There exist a D>0 such that for all x = x∗,y = y∗ in B,

xy≤D(xrB(y) +yrB(x)).

∗ ∗ (2) B is a D1 -subalgebra of a C -algebra. ∗ (3) B is a reduced algebra and there exist a D>0 such that x x≤2DsB(x) for all x ∈ B.

This has some interesting implications. ∗ ∗ (a) B is a Banach D1 -subalgebra of a C -algebra if and only if the following hold. (4) B is reduced and there exists D>0 such that for all x,y in B, xy≤D{xsB(y)+ ysB(x)}. ∗ ∗ Indeed let B be a D1 -subalgebra of a C -algebra (U, .0) satisfying xy≤ D{xy0 +yx0} for all x,y in B. Since B is spectrally invariant in U, sB(x) = ∗ 1/2 ∗ 1/2 ∗ 1/2 rB(x x) = rU (x x) =x x0 =x0; and (4) follows. On the other hand, ∗ sB(x) = rB(x) for all x = x ; and so (4) implies (1). This conﬁrms the faith that s is the noncommutative r. ∗ ∗ (b) Let A be a dense -subalgebra of a C -algebra (U, .0).Let.1 and .2 be any ∗ ∗ two norms on A making A a D1 -subalgebra of U. The algebra A is -semi simple, hence semi simple; and Johnson’s uniqueness of norm theorem implies that .1 and .2 are equivalent. ∗ ∗ ∗ (c) Let a Banach -algebra (A, .) be a D1 -subalgebra of each of the C -algebras (U1, .1) and (U2, .2). Then A is a Q-normed algebra with each of .1 and .2 by Theorem 5 in [15], and is spectrally invariant in each of U1 and U2. Hence for any A 2 ∗ ∗ ∗ ∗ ∗ 2 x ∈ , x1 =x x1 = rU1 (x x) = rA(x x) = rU2 (x x) =x x2 =x2. Thus .1 =.2. Since A is dense in each of U1 and U2, we get U1 = U2. Thus a ∗ ∗ ∗ Banach -algebra is a D1 -subalgebra of a unique C -algebra, which must be intrinsi cally determined as follows. ∗ ∗ ∗ (d) Let a Banach -algebra (A, .) be a D1 -subalgebra of a C -algebra. Let m(.) be the Gelfand–Naimark semi norm on A which is the greatest C∗-semi norm on A deﬁned as m(x) := sup{π(x):π ∈ Rep A},RepA denoting the set of all ∗- representations of A on Hilbert spaces. Thus m(x) ≤ s(x) for all x ∈ A; and due to ∗-semi simplicity, m is a norm. The enveloping C∗-algebra C∗(A) of A is the completion of A in m. Since A is a hermitian Banach ∗-algebra being spectrally invariant in U,wehaves = m [10]. Thus xy≤D{xm(y) +ym(x)} for all x,y in A ∗ ∗ showing that (A, .) is a Banach D1 -subalgebra of C (A). This we note as follows. PROPOSITION 3.2

∗ ∗ ∗ If a Banach -algebra A is a D1 -subalgebra of a C -algebra U, then U is unique and U = C∗(A). 132 Subhash J Bhatt

The following Frechet analogue of Theorem 3.1 is essentially a corollary of the inverse limit decomposition discussed in previous section. We do not know whether the hermiticity assumption can be omitted or not.

Theorem 3.3. Let A be a Frechet ∗-algebra. The following are equivalent.

(1) A is a ∗-semi simple hermitian Q-algebra; and there exists a sequence of sub mul- ∗ tiplicative -seminorms (.n)deﬁning the topology of A such that for each n, there exists Dn > 0 such that

xyn ≤ Dn{xnrn(y) +ynrn(x)} for all x = x∗,y = y∗ in A. ∗ ∗ (2) A is a Frechet D1 -subalgebra of a C -algebra. (3) A is a ∗-semi simple hermitian Q-algebra; and there exists a sequence of sub mul- ∗ tiplicative -seminorms (.n)deﬁning the topology of A such that for each n, there exists a Dn > 0 such that ∗ x xn ≤ 2Dnxnsn(x) for all x ∈ A. (4) A is a ∗-semi simple hermitian Q-algebra; and there exists a sequence of sub mul- ∗ tiplicative -seminorms (.n)deﬁning the topology of A such that for each n, there exists a Dn > 0 such that

xyn ≤ Dn{xnsn(y) +ynsn(x)} for all x,y in A.

Proof. ∗ ∗ (2) implies (1). Let A be a Frechet D1 -subalgebra of a C -algebra (U, .). Then A is dense in U; and there exists a defining sequence of semi norms (.n) satisfying xyn ≤ xyn +xny for all x,y in A. In the notations of Theorem 2.1, A = lim Añ, where ←− ˜ ∗ each (An, .n) is a Banach D1 -subalgebra of U. We denote a defining semi norm .n on A and the complete norm on the corresponding Banach algebra Añ by the same symbol ∗ ∗ ∗ .n. Since A is -semi simple being a -subalgebra of a C -algebra, so is each Añ. Hence by Theorem 3.1, for each n, there exist Dn > 0 such that xyn ≤ Dn{xnr (y) + Añ ∗ ∗ ynr (x)} for all x = x ,y = y in Añ. Thus for x ∈ A, the spectral radius of x Añ k 1/k in Añ is r (x) = limk→∞ x = rn(x). In the inverse limit decomposition A = Añ n lim Añ,σn : A → Añ is the natural projection σn(x = (xn)) = xn. Thus for x ∈ ←− ∗ ∗ A, xn =σn(x)n =xnn. Hence for x = x ,y = y in A,

xyn =(xy)nn =xnyn≤Dn{xnnrn(y) +ynnrn(y)} = Dn{xnrn(y) +ynrn(x).

Further by assumption (2), the normed algebra (A, .0) as well as the Frechet algebra ∗ (A,τ)are Q-algebras. Hence A is hermitian; and the C -norm .0 from U is the greatest C∗-semi norm on A with the result C∗(A) = U. (2) implies (3). This is proved by arguments analogous to above. Smooth Frechet subalgebras of C∗-algebras 133

(1) implies (2). Since A is a Frechet Q-algebra, it has a C∗-enveloping algebra [3, 4], say C∗(A) = (U, .) a C∗-algebra. Since A is ∗-semi simple, the star radical srad A = 0; and A is a ∗-subalgebra of U continuously embedded in U as id : (A,τ) → (U, .). Then . is the greatest C∗-semi norm on A automatically continuous in τ, A being Frechet. Further being a Q-algebra, A is spectrally bounded. Now let A = lim An be ←− an inverse limit decomposition as in Lemma 2.1. Then for any z in A being expressed k 1/k as a coherent sequence z = (zn) with each zn ∈ An, rn(z) = limk→∞ z n ≤ k 1/k ∗ 1/2 supn limk→∞ z n = rA(z) ≤ sA(z) = (rA(z z)) =z the last expression rA(.) ≤ sA(.) =. being a consequence of the hermiticity of A [13]. Hence for all n and for all x = x∗,y = y∗ in A,

xyn ≤ Dn{xnrn(y) +ynrn(x)}≤Dn{xny+ynx}. The rest of the arguments are analogous to those on p. 420 in [15]. ∗ (3) implies (2). This can be proved as above. Indeed, for all z ∈ A, for all n, z zn ≤ 2Dnznz, and Lemma 3 in [15] applies as above. (1) iff (4). This follows as in the Banach ∗-algebra case discussed above. A σ −C∗-algebra is a Frechet ∗-algebra whose topology is deﬁned by a sequence of C∗- ∗ semi norms. Let (A, {.n}) be a Frechet -algebra. Let Repn(A) be the collection of all operator representations π of A into bounded operators on Hilbert spaces Hπ satisfying ∗ π(x)≤kxn for all x ∈ A.Let|x|n := sup{π(x):π ∈ Repn(A)} a C -seminorm. ∗ The Hausdorff completion E(A) of (A, {|.|n}) is a σ − C -algebra, called the enveloping σ − C∗-algebra of A, which is universal for operator representations of A [3, 13]. When E(A) is a C∗-algebra, it is denoted by C∗(A), and A is called an algebra with a C∗- enveloping algebra [4,8,13].

COROLLARY 3.4

∗ ∗ Let (A, {.n}) be a Frechet D1 -subalgebra of a C -algebra U. Then A admits a unique ∗ ∗ topology making it a Frechet D1 -subalgebra of a C -algebra, U is unique and is the enveloping C∗-algebra C∗(A) of A.

∗ ∗ Proof. Let τ1 and τ2 be topologies on A making A a Frechet D1 -subalgebra of C - ∗ algebras U1 and U2 respectively. Then the C -norms on A induced by U1 and U2 are spectral norms each making A a Q-normed algebra. Since each of τ1 and τ2 is ﬁner ∗ than any of these C -norms, (A,τ1) and (A,τ2) are Frechet Q-algebras. As the topology of a Frechet Q-algebra is unique [13], τ1 = τ2. That U is unique follows as in the case of Banach algebras discussed above. Since A is a Frechet Q-algebra, it admits a ∗ C -enveloping algebra and x∞ := sup{π(x):π ∈ Rep A} deﬁnes greatest con- ∗ ∗ tinuous C -semi norm on A [4, 13]. In fact, due to -semi simplicity, .∞ is a norm; and it is the greatest C∗-semi norm in view of automatic continuity of ∗-representations on A. Being a hermitian Q-algebra, A is spectrally invarient in C∗(A); and so for all x ∈ A,sA(x) = sC∗(A)(x) =x∞. Then by Theorem 3.3, for all x,y in A, xyn ≤xns(y) +yns(x) =xny∞ +ynx∞ for all n, showing that A is a ∗ ∗ Frechet D1 -subalgebra of C (A). ∗ We note that a Frechet D1 -algebra exhibit a regularity feature not likely to be exhib- ∗ ∗ ited by a Frechet D∞-algebra of [5]. Given a Frechet D1 -subalgebra (A, {.n}) of a 134 Subhash J Bhatt

∗ C -algebra (U, .0), we consider the bounded part b(A) of A defined as b(A) := {x ∈ ∗ A : sup xn < ∞} which is a Banach -algebra with norm x:=sup xn satisfying xy≤x0y+xy0 for all x,y in b(A).LetU0 be the .0-closure of b(A) in the ∗ ∗ ∗ C -algebra U. Then b(A) is a Banach D1 -subalgebra of the C -algebra U0. Thus b(A) is a hermitian Q-algebra in each of the topology τ as well as the norm .0; and for any x ∈ b(A), (x) = (x) = (x) = (x) A SpA SpU SpU0 Spb(A) . Since is closed under the holomorphic functional calculus of U, and since b(A) is closed under the holomorphic functional calculus of U0, it follows that b(A) is closed under the holomorphic functional calculus of A. Thus b(A) is a differential Banach algebra of order 1 defined by the differential norm ∗ T ={.0, .} and is thus a smooth subalgebra of the C -algebra U0.By2,b(A) ∗ is closed under appropriate differential functional calculus. Some Banach D1 -subalgebras of b(A) include c0(A) := {x ∈ A : (pn(x)) ∈ c0},c(A) := {x ∈ A : (pn(x)) ∈ c} as p p well as l (A) := {x ∈ A : (pn(x)) ∈ l }, 1 ≤ p<∞ with appropriate norms; e.g., ∞ p 1/p ∗ in the last case, |x|p := (n=1 |pn(x)| ) . Being smooth subalgebras of C -algebras, they behave like C∗-algebras in a certain respect. As an example, a completely positive (CP) map φ : b(A) → B(H) extends as a CP map φ : A → B(H). Indeed, the identity 1 of U is in A; hence by the definition, 1 ∈ b(A). Now the Stinespring dilation in ∗ a unital Banach -algebra [1] shows that φ extends as a CP map φ : U0 → B(H), ∗ ∗ as U0 = C (b(A)) is universal for -representations. Now Arveson’s famous extension theorem extends φ as a CP map φ : U → B(H).

4. Approximation ∗ In this section, we discuss approximation properties in a Frechet D1 -algebra. The con- ∗ ∗ tention is that approximation properties for a Banach D1 -subalgebra of a C algebra ∗ ∗ discussed in [16] can be generalized to a Frechet D1 -subalgebra of a C -algebra by the standard Frechetization technique involving Arens–Michael decomposition [13] and sub- sequently using arguments in the Banach algebra case [16]. Thus we omit the proofs in the present section. The following is as in Theorem 1.2 in [16].

PROPOSITION 4.1

∗ ∗ Let A be a Frechet D1 -subalgebra of a C -algebra. (1) Let B be a ∗-sub algebra of A. Then (B2)− = (Bn)− for n ≥ 2, where − denote closure in the Frechet topology τ of A. (2) Let I beaclosedleft(respectively right) ideal of (A,τ). Let x = x∗ ∈ A.Ifx3 ∈ I, then x2 ∈ I.

We introduce a Frechet analogue of the ad hoc mode of convergence introduced in [16]. Let (N , .) be a normed linear space. Let X be a subspace, not necessarily closed, of N .Let(.n) be a sequence of seminorms on X such that .≤.n for all n.A sequence (xn) in X is said to be (∼)-convergent to x in N if xn − x→0 and for each k, there exist Mk such that xnk ≤ Mk for all k.Itisstrongly (∼)-convergent if further supk supn xnk < ∞. ∗ ∗ Now let A be a Frechet D1 -subalgebra of a C -algebra U.LetU+ be the cone of non ∗ negative elements of U.LetA+ = A ∩ U+ ={x = x ∈ A : Sp(x) ≥ 0}. Since A+ is .-closed, it is (∼)-closed. Notice that the set A+ of non negative elements of A is Smooth Frechet subalgebras of C∗-algebras 135

+ ∗ ∗ A := {x = x ∈ A : x = ﬁnite sum of elements of form y y for some y ∈ A}⊂A+. sq 2 1/2 ∗ Then A+ := {x : x ∈ A+}={y ∈ A+ : y ∈ A+}⊂A+. In case of a C -algebra sq ∗ U, U+ = U+ . For a Frechet D1 -subalgebra A of U, the following approximation gives an analogue of Theorem 2.5 in [16].

PROPOSITION 4.2 sq A+ is the (∼)-closure of A+ .

The following contains a Frechet analogue of Lemma 2.6 in [16] (in a reﬁned form), which can be used to prove a Frechet analogue of Theorem 2.8 in [16]. They are follows: Theorem 4.3. ∗ 2 (1) Let x = x ∈ A. Let h ∈ C [−x0, x0]. ′ ′ (a) If h(x0) = h(−x0) = h (x0) = h (−x0) = 0, then there exists Cn > 0 ′′ such that h(x)n ≤ Cnh ∞.Ifx ∈ b(A), then for some C>0 depending only on ′ x, it holds that supn h(x)n ≤ Ch ∞. ′ ′′ (b) There exists Cn > 0 such that h(x)n ≤ Cn(h∞ +h ∞ +h ∞).If ′ ′′ x ∈ b(A), then there is C>0 such that supn h(x)n ≤ C{h∞ +h ∞ +h ∞}. ∗ (2) Let x = x ∈ A. Then there exists a sequence φn of functions continuous on the real 2 line R and vanishing on a neighborhood of 0 such that for each k, x −φn(x)k → 0 as n →∞.

5. Examples Example 5.1. Frechet function algebras

(1) Let C[a,b] be the supnorm C∗-algebra of continuous functions on [a,b].Let C∞[a,b] be the Frechet subalgebra of all C∞-functions on [a,b], the Frechet topology (i) k f ∞ being defined by the sequence of norms {pk}, where pk(f ) = i=0 i! . The alge- ∞ ∗ bra C [a,b] is a Frechet D∞-subalgebra of C[a,b]. By Theorem 8 in [15], for each ∞ p = 1, 2, 3,..., there exists a D(p) > 0 such that for all f, g in C [a,b]fgp ≤ ∞ ∗ D(p)(f ∞gp +f pg∞), C [a,b] is a Frechet D1 -subalgebra of C[a,b]. ∗ (2) Let C0(R) be the C -algebra of all continuous functions on the real line R vanishing ′ at infinity and with the sup norm .∞.LetA ={f ∈ C0(R) : the derivative f ∈ C(R)} be a Frechet algebra with the topology defined by the sequence of norms f n =f ∞ + ′ ′ ′ f ∞,n, where f ∞,n = sup{|f (t)|:t ∈[−n, n]}. Then for all f, g in A, fgn ≤ ∗ ∗ f ∞gn +f ng∞ shows that A is a Frechet D1 -subalgebra of the C -algebra ′ 1 C0(R).Hereb(A) ={f ∈ C0(R) : f ∈ Cb(R)} and C0 (R) ⊂ b(A) ⊂ A ⊂ C0(R), 1 ′ ′ where C0 (R) ={f ∈ C0(R) : f ∈ C0(R)} with norm f ∞ +f ∞. (3) Nachbin’s weighted function algebras.Letσn : R → R be a sequence of continuous functions satisfying σn ≥ 1. Let A := {f ∈ C0(R) : fσn ∈ C0(R) for all n} be a ∗ Frechet D1 -subalgebra of C0(R) with the topology defined by the sequence of norms f n =fσn∞.Hereb(A) ={f ∈ C0(R) : supn fσn∞ < ∞}. (4) Rolle’s algebra.LetA consist of all continuous functions f on a closed and bounded ∗ interval [a,b] such that f is differentiable on the open interval (a, b). It is a Frechet D1 - ′ subalgebra of C[a,b] with the sequence of norms f n =f ∞ + sup{|f (t)|:t ∈ [a + 1/n,b − 1/n]}.Hereb(A) ={f ∈ C[a,b]:f ′ is bounded on (a, b)}.More 136 Subhash J Bhatt

m generally one can consider Am := {f ∈ C[a,b]:f ∈ C (a, b)} with family of norms m (k) f m;=f ∞ + k=1 sup{|f (t)|/k!:a + 1/n ≤ t ≤ b − 1/n}. ∗ (5) Let Cb(R) be the C -algebra of all bounded continuous functions on R.Let CBVloc(R) := {f ∈ Cb(R) : f is of bounded variation on [−n, n] for each n} be a Frechet algebra with topology defined by norms f n := f ∞ + Var n(f ) where Var n(f ) = the total variation of f on [−n, n]. It is easily seen that for f, g in CBVloc(R), fgn ≤f ngn and fgn ≤f ∞gn +g∞f n show that CBVloc(R) is a ∗ Frechet D1 -subalgebra of Cb(R). Several generalizations (corresponding to various generalizations of functions of bounded variations) as well as multivariate analogue of this p can be considered. For 1 ≤ p ≤∞, the algebra ACloc(R) of all f ∈ Cb(R) which, for each n, are absolutely continuous on [−n, n] having f ′ ∈ Lp[−n, n] is a Frechet ∗ n ′ p 1/p D1 -algebra with norms |f |n := f ∞ + (−n |f (t)| dt) . Example 5.2 (Example 1.5(i) in [5]). Let δ : D(δ) → A be a closed ∗-derivation defined ∗ ∗ n n on a dense -subalgebra D(δ) in a C -algebra (A, ·0).LetC (δ) := D(δ ) be n ∗ the domain of δ which is a dense Banach -subalgebra of A with norm xn := n j ∞ ∞ ∞ n ∗ j=0(δ (x)0/j !).TheC -domain C (δ) := ∩n=0C (δ) is a dense Frechet - subalgebra of A with the topology τ defined by the seminorms { · n : n ∈ N ∪{0}} satisfying, for each n and for x,y, xyn ≤xnyn and for each n ≥ 1, xyn ≤ ∞ ∗ xnyn−1 +xn−1yn. Thus (C (δ), τ) is a Frechet D∞-subalgebra of A. Notice that if δ is a generator, then by Theorem 9, p. 412 of [15], the norms ·n are first order norms in the sense that there exist Dn > 0 such that xyn ≤ Dn{xny0 +x0yn} ∞ ∞ ∗ ∞ for all x,y in C (δ) making C (δ) a Frechet D1 -algebra. Here b(A) ={x ∈ C (δ) : ∞ j 0 δ (x)/j ! < ∞}. Example 5.3 (Example 1.7 in [5]).

(a) Let H be a separable inﬁnite dimensional Hilbert space. Let K(H) be the C∗- algebra of all compact operators; and let Cp(H),1 ≤ p<∞, be the dense Banach ∗-subalgebra consisting of operators of Schatten–von Neumann class with Schat- 1 p q ten norm ·p. Then C ⊂ C ⊂ C ⊂ K(H) whenever 1 ≤ p ≤ q<∞ + p and p, q ∈ R , and ·0 ≤·q ≤·p ≤·1. For all x,y in C , 1 1 xyp = 2 {xyp +xyp}≤ 2 {x0yp +y0xp} shows that the pair 1 p 1+ ∞ 1+ p { · 0, ·p} is a differential norm of order 1 on C (H).LetC =∩p=1C (H). 1+ 1+ For x ∈ C ,let|x|p =x 1 for p = 1, 2, 3,.... Then on C , |·|1 ≤|·|2 ≤ 1+ p 1+ ···| · |n ≤|·|n+1 ≤ ···.Also,|xy|n ≤x0|y|n +|x|ny0, for all x,y ∈ C ; and C1(H) ⊂ C1+(H) ⊂ C2(H) ⊂ K(H) are all continuous embeddings with 1 1+ ∗ ·0 ≤|·|n ≤·1 on C (H). The algebra C is a Frechet D1 -subalgebra of K(H) p+ with norms {| · |n : n = 1, 2, 3,...}. We can analogously consider C for p>1. Here 1+ 1+ 1 b(C ) ={x ∈ C : supp |x|p < ∞} = C (H). ∗ (b) Let (A, y0) be a C -algebra of operators on a Hilbert space H.Let(A, H,D)be a Fredholm module such that (1) D : H → H is a bounded self-adjoint operator; and (2) for all x ∈ A, the commutator [D,x]∈K(H).

Let δ : A → B(H) be the derivation deﬁned by D as δ(x) = i[D,x].For1≤ p<∞,let p ∗ Ap := {x ∈ A : δ(x) ∈ C (H)} be the p-summable part of A. It is a Banach -algebra Smooth Frechet subalgebras of C∗-algebras 137 with norm |x|p := x0 +δ(x)p. Assume that each Ap is dense in A.LetAp+ := ∩p

Example 5.4 (C∗-Segal algebras). A C∗-Segal Banach ∗-algebra [14] is a Banach ∗- ∗ algebra (A, .) such that there exists a C -algebra (U, .0) containing A as a dense ∗ -ideal such that ax≤ax0, xa≤x0a for all x ∈ U,a ∈ A. It is a Banach ∗ ∗ ∗ ∗ D1 -subalgebra of a C -algebra. More generally, a C -Segal Frechet -algebra is a Frechet ∗ D1 -subalgebra of U. There are several examples of such algebras.

∞ (k) Example 5.5. Let A = C0 (R) ={f ∈ C0(R) : the derivatives f ∈ C0(R) for all k} with the topology of uniform convergence of functions as well as all their derivatives; or A = S(R) the Schwartz space of rapidly decreasing C∞-functions on R with the Schwartz topology. Let (pn) denote the sequence of seminorms defining either of these Frechet topologies. The algebra A is a Frechet convolution algebra. It is a dense ∗-subalgebra of the C∗-algebra C∗(L1(R)) the group C∗-algebra of R identified with C0(R) with pointwise multiplication. It is well known that differentiation is a multiplier for convolution in the sense that for f, g in A, (f ∗ g)(k) = f (k) ∗ g = f ∗ g(k); and so each (pn) is a first order semi norm satisfying pn(f ∗ g) ≤{f ∞pn(g) + pn(f )g∞}. ∗ ∗ ∗ 1 Thus A is a Frechet D1 -subalgebra of C -algebra C (L (R)). One can consider a multivariate analogue of this. The next example gives an abstract non commutative analogue of this.

Example 5.6 (C∞-domain of an unbounded multiplier). Let U be a C∗-algebra. Let T be an unbounded multiplier of U in the sense that T : D(T ) → U is a linear map defined on a dense ∗-subalgebra D(T ) of U such that for all x,y in D(T ), T(x∗) = T(x)∗ and T(xy) = T(x)y = xT (y). We assume that T n are closed linear maps. For 1 ≤ n< ∞,letD(T n) ={x ∈ D(T ) : T k ∈ D(T ) for all k,1 ≤ k<∞} be a Banach ∗- n ∞ n algebra with norm pn(x) =x+T (x).LetC (T ) =∩{D(T ) : n = 1, 2, 3,...} ∗ ∞ be a Frechet -algebra with the topology defined by (pn).Nowforallx,y in C (T ), ∞ pn(xy) ≤ pn(x)pn(y), pn(xy) ≤xpn(y) +ypn(x) shows that C (T ) is a Frechet ∗ ∗ ∞ D1 -subalgebra of U. On the Banach D1 -algebra (b(C (T )), p(.)), p(x) = sup pn(x), T defines a multiplier that is bounded in the norm p(.).

Example 5.7 (C∞-structure deﬁned by an unbounded self-adjoint operator). Let S be a closed symmetric operator with a dense domain D(S) in a Hilbert space H.Let B(H) and K(H) be the C∗-algebras consisting respectively of all bounded operators 1 and all compact operators on H.LetAS consist of all operators A in B(H) such that AD(S) ⊂ D(S),A∗D(S) ⊂ D(S) and SA−AS extends by closure to a bounded operator − 1 ∗ on H .LetAS := (SA − AS) be the closure. Then AS is a Banach -algebras with norm ∗ A1 := A+AS, · denoting the operator norm. Let US be the C -algebra obtained 1 by completing AS in ·.LetδS be the ∗-derivation deﬁned by S as δS(A) = iAS with 1 ∞ 1 1 1 domain D(δS) = AS in US.LetKS := AS ∩ K(H); JS := {A ∈ KS : AS ∈ K(H)} 138 Subhash J Bhatt

1 1 and FS be the closure in the norm ·1 of all finite rank operators in AS. The algebra 1 ∗ 1 1 1 AS is a Banach D1 -algebra [16]. The algebras KS, JS , FS are closed subalgebras of 1 1 1 1 1 (AS, ·1); and FS ⊂ JS ⊂ KS ⊂ AS. In a series of papers [16–18], Kissin and Shulman have investigated the structure of these algebras regarding them as non commu- ∗ tative differential algebras defined by the derivation δS.TheC -algebra US represents the continuity structure defined by S. Then a comparison with the Banach function algebra Lip[a,b]:={f ∈ C[a,b]:f ′ ∈ L∞[a,b]} of Lipschitz functions on interval [a,b] with ′ the norm f Lip := f ∞ +f ∞ shows that these algebras represent noncommutative Lipschitz structure defined by S. In [6], basic properties of the second order differential 1 1 1 structure defined by S have been investigated. Let AS := {A ∈ AS : δS(A) ∈ AS} ∗ 2 which is a Banach -algebra with norm A2 =A+δS(A)+(1/2)δS(A); 2 2 2 1 2 1 KS = AS ∩ K(H); JS ={A∈ KS : δS(A) ∈ JS } and let FS be the closure in ·2 of 2 1 finite rank operators in AS. Notice that for A in AS, δS(A) ∈ US; and thus the algebra 2 1 AS corresponds to the algebra of C -functions whose derivative is Lipschitzian. Here we consider the frame work of C∞-structure defined by S. Assume that for each n = 1, 2, 3,...,D(Sn+1) := {x ∈ D(Sn) : Snx ∈ D(S)} is dense in H so that each Sn is closable. Then the domain of the closure D(Sn−) is a n Banach space, with norm xn := x+S x, dense in H, with the result, the Frechet space C∞(S) := ∩{D(Sn−) : n = 1, 2, 3,...} is dense in H. We inductively define n n−1 n−1 ∗ AS := {A ∈ AS : δS(A) ∈ AS }, a Banach -algebra in B(H) with norm An := k=n n k Kn An K H J n k=0( Ck)δS(A), .0 being the operator norm. Let S := S ∩ ( ), S := {A ∈ n−1 n n K(H) : δS(A) ∈ JS }, FS := .n− the closure of finite rank operators in AS. Define ∞ ∞ n ∗ AS := ∩n=0AS a Frechet -algebra in the topology defined by the sequence of norms n n n ∞ n (.n).LetKS = AS ∩ K(H), JS ={A ∈ KS : δS(A) ∈ K(H), n = 1, 2, 3,...}, and ∞ ∞ ∞ ∞ FS = the closure in JS of all finite rank operators in AS . Then AS is a differential Frechet algebra in the sense of being defined by a differential norm [7], hence is a Frechet ∗ ∗ ∞ (D∞)-sub algebra of the C -algebra US; and there is a chain of Frechet algebras FS ⊂ ∞ ∞ ∞ JS ⊂ KS ⊂ AS . The smooth structure defined by S is exhibited below: ∞ n+1 n 1 AS ⊂··· ⊂AS ⊂ AS ⊂··· AS ⊂ US ∞ n+1 n 1 KS ⊂··· ⊂KS ⊂ KS ⊂··· KS ∞ n+1 n 1 JS ⊂··· ⊂JS ⊂ JS ⊂··· JS ∞ n+1 n 1 FS ⊂··· ⊂FS ⊂ FS ⊂··· FS . This matrix of Frechet and Banach algebras represents in totality the smooth structure 1 defined by the symmetric operator S. When S is a bounded operator, AS = US = B(H) all 1 1 bounded operators on H; each raw collapses to a single Banach algebra; and JS = FS = 1 KS = K(H) all compact operators on H. Now assume that S is self-adjoint (and unbounded). Then the following hold: (a) The derivation δS is an infinitesimal generator of a 1-parameter group of auto- n n n n morphisms of US. Then as in Example 4 above, each of AS, KS, JS , FS is a Banach ∗ ∞ ∗ D1 -algebra, with the result, the Frechet algebra AS is a Frechet D1 - subalgebra of a C∗-algebra. Smooth Frechet subalgebras of C∗-algebras 139

∞ Z (b) Let S = −∞ λdEλ be the spectral resolution of S.Forn ∈ ,letE(n) = E(−∞,n) be the spectral projections. Let P(n) = E(n + 1) − E(n).Let[S]:= ∞ ∞ −∞ nP (n) = f(S), where f = −∞ nχ[n,n+1] deﬁned by the functional calculus, χ being the characteristic function. The operator S −[S] is bounded with S −[S] ≤ 1; and 1 for any A∈ AS, A[S] := [S]A − A[S] is bounded satisfying δS(A)≤A[S]+2A as 1 1 1 1 1 shown on p. 19 in [16]. As shown in [16], this implies that AS = A[S], KS = K[S], JS = 1 1 1 J[S], FS = F[S]. ∞ 2 (c) By §2, the algebra AS is closed under C - functional calculus; and it is spectrally ∞ 1+ǫ invariant in US. In fact, AS is closed under C -functional calculus for self-adjoint elements. ∞ ∞ (d) We say that a sequence An in the Frechet algebra AS (∼)-converges to A in AS if An − A0 → 0; and for each k, there exists a constant Mk > 0 such that Ank 0, let Bǫ = (A + ǫ1) − ǫ 1. Then by the n 2 holomorphic functional calculus, Bǫ∈ AS and Sp(Bǫ) ≥ 0. Also A − Bǫ 0 → 0as ǫ → 0. Also

2 1/2 1/2 Bǫ n =A + 2ǫ1 − 2ǫ (A + ǫ1) n 1/2 1/2 ≤An + 2ǫ + 2ǫ (A + ǫ1) n. Now as in the proof of Theorem 2.5 in [16],

1/2 −1 −1/2 (A + ǫ) n ≤ (π) [8 + 2Cǫ + (An + ǫ)(6 + C + 2ǫ)]; 1/2 with the result supǫ(ǫ (A + ǫ1))n = Mn < ∞. This gives (1) from which (2) fol- ∞ lows immediately. If A ∈ (b(AS ))+, then supn An < ∞; and by above estimate 2 ∞ sq n sq supn supǫ Bǫ n < ∞. This gives (3). Notice that (AS )+ =∩(AS)+ . ∞ (e) The Frechet algebra AS is a Schawrtz space (respectively a nuclear space) if and n+1 n only if for each n, the inclusion AS → AS is compact (respectively nuclear) operator. This is a straightforward consequence of the deﬁnitions. ∞ ∞ n (f) We show that the Frechet space AS is a dual space. Notice that AS =∩AS = lim An. First we show that each An is a dual Banach space. Indeed the Banach space ←− S S 1 AS is isometrically isomorphic to the graph of δS by the map A →{A, δS(A)}; and the M B H graph of δS is an ultra weakly closed (and hence norm closed) subspace of S ( ), M U M B K where S is the von Newman algebra generated by S.Now S ( ) is the dual of M C1 H M M C1 H the direct sum Banach space ∗ ( ), where ∗ is the predual of S and ( ) is the Banach space of trace class operators on H whose dual is B(H). Hence it follows A1 M C1 H L L that S is a dual space. In fact, it is the dual of ( ∗ ( ))/ 1, where 1 is the M C1 H A1 annihilator of the graph of δS in ∗ ( ), identifying the graph of δS with S.For 140 Subhash J Bhatt

A2 A1 B H n = 2, notice that S is isometrically isomorphic to a closed subspace of S ( ); A2 A1 C1 H L L and the latter is a dual space. Thus S is the dual of {( S)∗ ( )}/ 2, where 2 is A2 A1 C1 H ∗ An+1 the annihilator of S in {( S)∗ ( )}. In general the Banach -algebra S is the K An C1 H L L An+1 dual of n+1 := {( S)∗ ( )}/ n+1}, n+1 being the annihilator of S . Further, n n−1 n−1 n for each n, the inclusion AS ⊂ AS gives the continuous inclusion (AS )∗ ⊂ (AS)∗ by restriction; and this in turn induces a continuous linear map φn : Kn → Kn+1.Nowthe system {(Kn,φn)} forms an inductive system of Banach spaces. Let K := limn→∞ Kn be ∞ the inductive limit space. Then the Frechet space AS is the strong dual of K. ∞ (g) We say that a linear functional φ on AS is strongly positive (respectively positive) ∞ ∞ + ∞ + ∞ if φ((AS )+) ≥ 0 (respectively φ((AS ) ) ≥ 0). Here (AS ) ={T ∈ AS : T = ﬁnite ∗ ∞ ∞ ∞ sum of elements of the form K K with K in AS } the positive cone of AS . Since AS ∗ ∞ ∞ is spectrally invariant in the C -algebra US,wehaveAS ∩ (US)+ = (AS )+ contains ∞ + ∗ + {(AS ) ); and US being a C -algebra, US = (US)+. Thus every strongly positive linear ∞ ∞ functional on AS is positive; and is continuous in the Frechet topology on AS [13]. Then by the well known Krein extension theorem, every strongly positive linear functional on ∞ ∗ AS extends as a positive linear functional on the C -algebra US. This corresponds to the well known result that a positive distribution on a manifold is a measure.

Acknowledgements The author thanks the referee for several suggestions that help reorganize the paper in the present form. The author gratefully acknowledge the UGC support under UGC-SAP- DRS programme F-510/3/DRS/2009 (SAP-II) to the Department of Mathematics, Sardar Patel University; as well as DST support under DST-PURSE Programme to Sardar Patel University.

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