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 sub- algebra whose topology is defined by a sequence of differential seminorms of order 1 is investigated. This includes differential Arens–Michael decomposition, spectral invari- ance, 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 differ- ential 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 dis- cuss 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 gener- alizations 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 induc- tively 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 inequal- ity 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]).