ON CERTAIN P-ADIC BANACH LIMITS of P-ADIC TRIANGULAR MATRIX ALGEBRAS

International Journal of Pure and Applied Mathematics Volume 85 No. 3 2013, 435-455 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v85i3.1 ijpam.eu

ON CERTAIN p-ADIC BANACH LIMITS OF p-ADIC TRIANGULAR MATRIX ALGEBRAS

R.L. Baker University of Iowa Iowa City, Iowa 52242, USA

Abstract: In this paper we investigate the class of p-adic triangular UHF (TUHF) Banach algebras.A p-adic TUHF Banach algebra is any unital p-adic Banach algebra of the form = n, where ( n) is an increasing sequence T T T T of p-adic Banach subalgebras of such that each n contains the identity of T S T T and is isomorphic as an Ωp-algebra to Tpn (Ωp) for some pn, where Tpn (Ωp) is the algebra of upper triangular pn pn matrices over the p-adic ﬁeld Ωp. The main × result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra, provided that the algebra satisﬁes certain local dimensionality conditions.

AMS Subject Classiﬁcation: 12J25, 12J99, 46L99, 46S10 Key Words: p-adic Banach algebras, p-adic Banach limits, p-adic UHF algebras, p-adic Triangular UHF algebras

1. Introduction

In the article, Triangular UHF algebras [2], the author of the present paper introduced a very concrete class of non-selfadjoint operator algebras which are currently called standard triangular UHF operator algebras. Brieﬂy: a standard triangular UHF (TUHF) operator algebra is any unital Banach algebra that is isometrically isomorphic to a Banach algebra inductive limit of the form = T lim(Tp ; σp p ). Here (pn) is a sequence of positive integers such that pm pn → n n m | c 2013 Academic Publications, Ltd. Received: December 27, 2011 url: www.acadpubl.eu 436 R.L. Baker

whenever m n, Tp is the algebra of pn pn upper triangular complex matrices ≤ n × .(and for m n, σp p : Tp Tp is the mapping x ֒ 1 x = diag(x, . . . , x ≤ n m m → n → ⊗ The main result in [2] is that two standard triangular UHF operator algebras are isometrically isomorphic if, and only if they have the same supernatural number. This result has been extended in at least two different directions. One direction is to view standard triangular UHF operator algebras as special cases of Banach algebra inductive limits of upper triangular matrix algebras, and the and the the goal is to extend the main result in [2] to these inductive limits by using purely Banach-algebraic methods. Such an extension is presented in the paper [2] where it is proved that the supernatural number associated to an arbitrary triangular UHF (TUHF) Banach algebra is an invariant of the algebra, provided that the algebra satisfies certain “local dimensionality conditions.” In particular, [2] presents a purely Banach-algebraic formulation of the main result in [2]. A second direction in which the main result in [2] can be extended is to view standard triangular UHF algebras as triangular subalgebras of UHF C∗-algebras, such that the diagonal of the triangular subalgebra is a canonical masa in the ambient C∗-algebra. In this vein, the results in [2] have been substantially extended by J.R. Peters, Y.T. Poon and B.H. Wagner [13], and by S.C. Power [11], [12]. The principal results of [3] can be extended in yet a third direction, namely to convert these results into a pure piece of p-adic functional analysis (to borrow a turn of phrase used by I Kaplansky [1]). The main result of [3], although relying only on classical complex Banach algebra techniques, makes essential use of the classical spectral theorem (the Riesz functional calculus) for complex Banach algebras. For any prime number p, let Ωp be the p-adic counterpart of C. Replacing C by Ω in the definition of complex TUHF Banach algebras, we ¯ ¯ p obtain the definition of p-adic TUHF Banach algebras. The results in [3] for complex TUHF algebras can be duplicated for p-adic TUHF algebras, provided that a sufficiently general p-adic version of the Riesz functional calculus can be developed for p-adic Banach algebras. In the preprint [4] we proved an Extended Spectral Theorem for p-adic Banach Algebras (Theorem 2.10 of [4]). In the present article we use this spectral theorem for p-adic Banach algebras to extend the results of [3] to p-adic TUHF algebras:—this is the content of Theorem 4.2 of the present paper, which states that the supernatural number associated to an arbitrary p-adic TUHF Banach algebra is an isomorphism invariant of the algebra, provided that the algebra satisfies certain “local dimensionality conditions.” In [8] J. Glimm proved that the supernatural number of an arbitrary UHF C∗-algebra is a complete C∗-invariant of the algebra. In light of the results on ON CERTAIN p-ADIC BANACH LIMITS OF... 437 p-adic TUHF algebras in the present paper, it seems natural to replace C in the ¯ definition of complex UHF C∗-algebras, thereby obtaining the definition of p- adic UHF Banach algebras. It then seems natural to ask whether or not the the supernatural number associated to an arbitrary p-adic UHF Banach algebra is an isomorphism invariant of the algebra. In the preprint [5] we use the Extended Spectral Theorem for p-adic Banach Algebras obtained in [4] to prove that the supernatural number associated to an arbitrary p-adic UHF Banach algebra is an isomorphism invariant of the algebra.

2. Preliminaries

In the present section of the paper we put forth the preliminary material on p- adic analysis and p-adic Banach algebras that is necessary for proving Theorem 4.2 in Section 4. Let p be a prime number, and let Q be the ﬁeld of rational numbers. Let ¯ p be the function deﬁned on Q by | · | ¯

a ordpb−ordpa = p , 0 p = 0. b | | p

Here ordp of a non-zero integer is the highest power of p dividing the integer. Then p is a norm on Q. The field Qp is defined to be the completion of Q | · | ¯ ¯ ¯ under the norm p. Unlike the case of the real numbers R, whose algebraic | · | ¯ closure C is only a quadratic extension of R, the algebraic closure Q of Qp has ¯ ¯ ¯ p ¯ infinite degree over Qp. However, the norm p on Qp can be extended to a ¯ | · | ¯ norm p on Q . But it turns out that Q is not complete under this extended | · | ¯ p ¯ p norm. Thus, in order to do analysis, we must take a larger field than Q . We ¯ p denote the completion of Q under the norm p by Ωp, that is, ¯ p | · | ∧ Ωp =Q , ¯ p ∧ where means completion with respect to p.(Note: The symbol “Cp” is | · | ¯ sometimes used to denote Ωp (see [9]: page 13). We define Ωp p = x p : x | | { | | ∈ Ωp . } Definition 2.1. Let r 0 be an nonnegative real number, let a Ωp and ≥ ∈ let σ Ωp. We have the following definitions. ⊆

Da(r) = x Ωp x a p r ; { ∈ | | − | ≤ } 438 R.L. Baker

− Da(r ) = x Ωp x a p < r ; { ∈ | | − | } Dσ(r) = x Ωp dist(x, σ) r ; − { ∈ | ≤ } Dσ(r ) = x Ωp dist(x, σ) < r . { ∈ | } − − − If b Da(r), then Db(r) = Da(r), and if b Da(r ), then Db(r ) = Da(r ). ∈ ∈ − Thus, any point in a disc is its center. Hence Da(r),Da(r ) are both open and closed in the topological sense. It is conventional to take inf = + , hence for ∅ ∞ r > 0 we have D (r−) = and D (r) = (see [7], page 4). ∅ ∅ ∅ ∅ Lemma 2.2. Let a Ωp, and let 0 < r Ωp p. Let f : Da(r) Ωp. ∈ ∈ | | k → Suppose that there exists a sequence (ck) in Ωp be such that lim r ck p = 0, k→∞ | | and for all x Da(r), ∈ ∞ k f(x) = ck(x a) . − k X=0 Then max f(x) p is attained when x a p = r, and we have x∈Da(r) | | | − | k max f(x) p = max r ck p . x∈Da(r) | | k | |

Proof. See [9], Lemma 3, page 130.

Definition 2.3. Let a Ωp, and let 0 < r Ωp p. Let = σ Ωp.A ∈ ∈ | | ∅ 6 ⊆ function f : Da(r) Ωp is said to be Krasner analytic on Da(r) iff f can be → ∞ k represented by a power series on Da(r) of the form f(x) = k=0 ck(x a) , k − where lim r ck p = 0. Define Br(σ) to be the set of all functions k→∞ | | P

f : Dσ(r) Ωp → such that f is Krasner analytic on Da(r) whenever a Ωp and Da(r) Dσ(r). ∈ ⊆ If σ is compact, we define L(σ) = Br(σ) 0 < r Ωp p , and we call L(σ) { | ∈ | | } the set of locally analytic functions on σ (see [9], page 136). S Definition 2.4. Let = σ Ωp. For 0 < r, s Ωp p and a , . . . , aN , b ,..., ∅ 6 ⊆ ∈ | | 1 1 bM Ωp, define the order relation (s, b , . . . , bM ) (r, a , . . . , aN ) by ∈ 1 ≤ 1 M N (s, b , . . . , bM ) (r, a , . . . , aN ) s r and Db (s) Da (r). 1 ≤ 1 ⇐⇒ ≤ i ⊆ i i i [=1 [=1 For 0 < r Ωp p and a , . . . , aN Ωp, define ∈ | | 1 ∈ N Br,a ,...,a = f : Da (r) Ωp f is Krasner analytic on each Da (r) . 1 N i → i i=1 [

ON CERTAIN p-ADIC BANACH LIMITS OF... 439

Let I(σ) to be the set of all (r, a1, . . . , aN ) such that

(1) 0 < r Ωp p and a , . . . , aN Ωp; ∈ | | 1 ∈ (2) Dai (r) Daj (r) = for i = j. N ∩ ∅ 6 − (3) σ Dai (r ). ⊆ i=1 Finally, deﬁneS

(σ) = Br,a ,...,a (r, a , . . . , aN ) I(σ) . L { 1 N | 1 ∈ } [

Lemma 2.5. Let = σ Ωp. Then the following statements hold. ∅ 6 ⊆ (a) The set I(σ) is decreasingly ﬁltered under the order relation . ≤ (b) Let (r, a1, . . . , aN ) I(σ) and f Br,a1,...,aN . Then f is bounded on N ∈ ∈

U = Dai (r), and we deﬁne the uniform norm f u of f by i=1 k k S f u = max f(x) p. k k x∈U | |

(c) Let (s, b1, . . . , bM ), (r, a1, . . . , aN ) I(σ), with (s, b1, . . . , bN ) (r, a1,..., M ∈ ≤ aN ). Define V = Dbi (s), then f V Bs,b1,...,bM and the mapping f f V is i=1 | ∈ 7→ | continuous on Br,aS1,...,aN in the uniform norm. (d) (σ) is an algebra over Ωp. L (e) For α = (r, a1, . . . , aN ) I(σ), set Bα = Br,a1,...,aN . For β = (s, b1,..., ∈ M α bM ) in I(σ), with β α, let V = Dbi (s) and define ϕβ : Bα Bβ by ≤ i=1 → S α ϕ (f) = f V , all f Bα. β | ∈ Then for α β γ I(σ) following conditions are satisfied ≥ ≥ ∈ α ϕ (f) = f, for all f Bα, α ∈ α β α ϕβ ϕγ = ϕγ .

It is worth noting here that we can deﬁne the inverse limit lim Bα of the system ←− Bα α I(σ) and place the projective topology on lim Bα. But we will not { | ∈ } ←− make use of this topology in the present paper. 440 R.L. Baker

(f) If σ is compact, then (σ) = L(σ), i.e., L (σ) = Br(σ) 0 < r Ωp p . (1) L { | ∈ | | } [ (g) For all (r, a , . . . , aN ) I(σ), if a , . . . , aN σ, then Br,a ,...,a = Br(σ). 1 ∈ 1 ∈ 1 N

Proof. See [4], Lemma 1.8.

Deﬁnition 2.6. Let = σ Ωp be compact, let H (σ) denote the set of ∅ 6 ⊆ 0 functions ϕ : σ Ωp which are Krasner analytic on the complement σ of σ, → i.e., (1) ϕ is the limit of rational functions whose poles are contained in σ, the limit being uniform in any set of the form

− Dσ(r) = z Ωp dist(z, σ) r , σ Dσ(r ), r > 0. { ∈ | ≥ } ⊆ (2) lim ϕ(z) = 0. |z|p→∞ The following deﬁnition gives the p-adic analogue of the classical line integral. Shnirelman introduced this deﬁnition in 1938 [16]. The Shnirelman integral can be used to prove p-adic analogues of the Cauchy integral theorem, the residue theorem, and the maximum modulus principle of classical complex analysis.

Deﬁnition 2.7. (The Shnirelman Integral) Let r > 0, with r Ωp p. Let ∈ | | a Ωp, and let f be an Ωp-valued function whose domain contains all x Ωp ∈ ∈ such that x a p = r. Let Γ Ωp, with Γ p = r. Then the Shnirelman | − | ∈ | | integral of f over the circle

x Ωp : x a p = r { ∈ | − | } is deﬁned to be the following limit, provided the limit exists. 1 f(x) dx = lim f(a + ξΓ). n→∞ n p6 |n n a,ZΓ ξX=1

Theorem 2.8. (p-adic Cauchy Integral Formula) Let a Ωp and 0 < r ∈ ∈ Ωp p. Let f be Krasner analytic on Da(r), and let Γ Ωp, with Γ p = r. Then | | ∈ | | for ﬁxed z Ωp, we have, for m = 0, 1, 2,..., ∈ 1 (m) f (z), if z a p < r; f(x)(x a) m! | − | − dx = (x z)m+1  a,ZΓ − 0, if z a p > r. | − |  ON CERTAIN p-ADIC BANACH LIMITS OF... 441

Proof. See [9], Lemma 4, page 131.

3. p-Adic Triangular UHF Algebras

The following lemma is a key lemma (Lemma 2.1 of [3]) used in proving the main result of [3]. Lemma 3.1. Let be a complex Banach algebra. Let ǫ > 0 be positive A number, and let I = ei 1 i n be an orthogonal family of idempotents in { | ≤ ≤ } . Then there exists a positive number δ(ǫ, I) > 0 with the following property. A Let be a unital Banach subalgebra of , and suppose that ai 1 i n B A { | ≤ ≤ } is a family of elements in such that ei ai δ(ǫ, I) for 1 i n. Then B k − k ≤ ≤ ≤ there exists an orthogonal family fi 1 i n of idempotents in such { | ≤ ≤ } B that ei fi < ǫ for 1 i n. k − k ≤ ≤ The basic tool used in the proof of Lemma 3.1 is the classical Riesz functional calculus (see [6], prop. 4.5.1, p. 27). In this section we prove Lemma 3.18, which is the p-adic counterpart of Lemma 3.1 for p-adic Banach algebras. Theorem 4.2 of the present paper is the exact p-adic counterpart of the principal theorem (Theorem 1.1) of [3]. Using the remarks after Deﬁnition 3.2 and Theorem 3.15 of this section, and then Lemma 3.18 of this section, the proof of Theorem 1.1 in [3] can be easily adapted to produce a proof of Theorem 4.2. Theorem 3.11 of the present section (Extended Spectral Theorem for p-adic Banach Algebras) is Theorem 2.10 of [4].

Definition 3.2. A p-adic Banach space over Ωp is a vector space over Ωp X together with a norm p from to the nonnegative real numbers such that for k·k X all x, y : (a) x p = 0 if and only if x = 0; (b) x+y p max x p, y p ; ∈ X k k k k ≤ { k k k k } (c) ax p = a p x p; (d) is complete under p. We shall assume that k k | | k k X k · k p = Ωp p, i.e., for every 0 = x there exists a Ωp such that ax p = 1. kX k | |∗ 6 ∈ X ∈ k k The dual of a p-adic Banach space over Ωp is defined in the usual way. If X and are p-adic Banach spaces over Ωp, we define to mean that X Y X ≃ Y and are isometrically isomorphic as p-adic Banach spaces over Ωp.A p- X Y adic Banach algebra over Ωp is a p-adic Banach space over Ωp such that for A x, y , we have xy p x p y p. We shall assume that has a unit. For ∈ A k k ≤ k k k k A x , the spectrum σx of x has the usual meaning, and the resolvent of x is ∈ A −1 defined by R(z; x) = (z x) , z σx. If and are p-adic Banach spaces − 6∈ X Y over Ωp, then B( , ) is the vector space of Ωp-linear continuous maps form X Y to . B( , ) is a p-adic Banach space under the usual operator norm, and X Y X Y 442 R.L. Baker

B( ) = B( , ) is a p-adic Banach algebra under this operator norm. If X X X X is a p-adic Banach space over Ωp and A B( ) is an operator with compact ∈ X spectrum σA, then A is analytic iff the resolvent R(z; A) is Krasner analytic in the sense that for all x and for all h ∗, function z h(R(z; A)x) is ∈ X ∈ X 7→ in H0(σA). Let J be any nonempty indexing set, and define Ωp(J) to be the set of all “sequences” c = (cj)j∈J in Ωp such that for every ǫ > 0 only finitely many cj p are > ǫ. Define c p = max cj p. Then Ωp(J) is a p-adic Banach | | || || j | | space over Ωp (see [14], Corollary 1, page 185). The notation Ωp(J) is used in [9], page 143. However the usual notation for Ωp(J) is c0(J;Ωp), which is used in [14], page 185. Let be a p-adic Banach algebra and let be a p-adic A X Banach space. Let A have compact spectrum σA = . Then we say that ∈ A 6 ∅ A is -analytic iff there exists an Ωp-monomorphism θ : B( ) such that X A → X θ( ) is a p-adic Banach subalgebra of B( ), θ−1 : θ( ) is bounded and A X A → A θ(A) is analytic in B( ). Such a monomorphism θ : B( ) is said to be an X A → X embedding of into B( ). Let B . Then B is -analytic iff the following A X ∈ A A two conditions hold:

(a) For every r > 0, R(x; B) p is bounded on the complement DσB (r) of − k k DσB (r ). (b) There exists a sequence (Bk) in and a sequence ( k) of p-adic Banach A X spaces k of the form k Ωp(Jk), with Jk = , such that σB = is compact, X X ≃ 6 ∅ k 6 ∅ Bk is k-analytic and lim B Bk p = 0. X k→∞ k − k

Remark. Let Ωp(J), where J = . Let be a p-adic Banach algebra X ≃ 6 ∅ A over Ωp. If A has nonempty compact spectrum σA, and if A is -analytic, ∈ A X then A is -analytic (see Lemma 2.13 of [4]). If J is nonempty and finite, A and there exists an embedding of into B( ), then every element of is A X A -analytic, hence every element of is -analytic (see Lemma 3.2 of [4]). X A A Definition 3.3. Let be a p-adic Banach algebra over Ωp. Let A A ∈ A have spectrum σA = . Define (A) = (σA). 6 ∅ F L Definition 3.4. (Operator-Valued Shnirelman Integral) Let be a p-adic A Banach algebra over Ωp. Let a Ωp, and let 0 < r Ωp p. Let Γ Ωp, with ∈ ∈ | | ∈ Γ p = r. Let F be a function defined on the circle z Ωp : z a p = r into | | { ∈ | − | } . Then we define A 1 F (x) dx = lim F (a + ξΓ), n→∞ n p6 |n ξn a,ZΓ X=1 provided that this limit exists. ON CERTAIN p-ADIC BANACH LIMITS OF... 443

Lemma 3.5. Let be a p-adic Banach algebra. Let A be -analytic. A ∈ A A Let (Ak) be a sequence in and ( k) a sequence of p-adic Banach spaces k A X X of the form k Ωp(Jk), with Jk = , such that σA = is compact, Ak is X ≃ 6 ∅ k 6 ∅ k-analytic and lim A Ak p = 0. Let A have spectrum σA. Let 0 < r2 r1 X k→∞ k − k ≤ be in Ωp p, and let Γ , Γ be in Ωp, with Γ p = r , Γ p = r . Assume that | | 1 2 | 1| 1 | 2| 2 a , . . . , aM ; b , . . . , bN Ωp are given, with 1 1 ∈ M N − − σA Da (r ) and σA Db (r ), (1) ⊆ i 1 ⊆ i 2 i i [=1 [=1 where the Dai (r1) are disjoint and the Dbi (r2) are disjoint. Let f be Krasner analytic on the Dai (r1) and on the Dbi (r2). Then the following limits exist and are equal:

M N lim f(x)(x ai)R(x; Ak) dx = lim f(x)(x bi)R(x; Ak) dx. k→∞ − k→∞ − i=1 Z i=1 Z X ai,Γ1 X bi,Γ2

Proof. See [4], Lemma 2.5. Lemma 3.6. Let be a p-adic Banach algebra. Let A be -analytic A ∈ A A with spectrum σA. Let (Ak) be a sequence in and ( k) a sequence of p-adic A X Banach spaces k of the form k Ωp(Jk), with Jk = , such that σA = is X X ≃ 6 ∅ k 6 ∅ compact, Ak is k-analytic and lim A Ak p = 0. Let 0 < r Ωp p, and let X k→∞ k − k ∈ | | Γ Ωp, with Γ p = r. Assume that a , . . . , aN Ωp are given, with ∈ | | 1 ∈ N − σA Da (r ), (1) ⊆ i i [=1 where the Dai (r) are disjoint. Assume that f is Krasner analytic on the Dai (r). Then the following limits exist and are equal.

N N 1 i i i lim f(xξ)(xξ ai)R(xξ; A) = lim f(x)(x ai)R(x; Ak) dx, n→∞ n − k→∞ − p6 |n i=1 ξn=1 i=1 Z X X X ai,Γ i where for each 1 i N and ξ Ωp, we set x = ai + ξΓ. Therefore the sum ≤ ≤ ∈ ξ N f(x)(x ai)R(x; A) dx − i=1 Z X ai,Γ 444 R.L. Baker exits and we have

N N f(x)(x ai)R(x; A) dx = lim f(x)(x ai)R(x; Ak) dx. − k→∞ − i=1 Z i=1 Z X ai,Γ X ai,Γ

Proof. See [4], Lemma 2.6. Deﬁnition 3.7. Let be a p-adic Banach algebra. Let A be - A ∈ A A analytic with spectrum σA. Let (Ak) be a sequence in and ( k) a sequence A X of p-adic Banach spaces k of the form k Ωp(Jk), with Jk = , such that X X ≃ 6 ∅ σAk = is compact, Ak is k-analytic and lim A Ak p = 0. Let f (A). 6 ∅ X k→∞ k − k ∈ F Let 0 < r Ωp p, and let Γ Ωp, with Γ p = r. Assume that a , . . . , aN Ωp ∈ | | ∈ | | 1 ∈ are given, with N − σA Da (r ), (1) ⊆ i i [=1 where the Dai (r) are disjoint. Assume that f is Krasner analytic on the Dai (r). By Lemma 3.6, (1) implies that

N N f(x)(x ai)R(x; A) dx = lim f(x)(x ai)R(x; Ak) dx. − k→∞ − i=1 Z i=1 Z X ai,Γ X ai,Γ Lemma 3.5 implies that

N lim f(x)(x ai)R(x; Ak) dx k→∞ − i=1 Z X ai,Γ does not depend on a1, . . . , aN , r, Γ, hence we may deﬁne f(A) by

N f(A) = f(x)(x ai)R(x; A) dx. − i=1 Z X ai,Γ Definition 3.8. Let K be an arbitrary field. Let be a unital algebra over A K. A family of elements eij 1 i j n in is said to be a triangular { | ≤ ≤ ≤ } A system of matrix units in if for 1 i j n the following conditions are A ≤ ≤ ≤ satisfied. (i) eij = 0; 6 ON CERTAIN p-ADIC BANACH LIMITS OF... 445

(ii) eii 1 i n is an orthogonal family of idempotents such that n { | ≤ ≤ } eii = 1; i=1 P (iii) for 1 i p n and 1 q j n, we have ≤ ≤ ≤ ≤ ≤ ≤

eij, ifp=q; eipeqj = (0, otherwise.

A family of elements fij 1 i, j n in is said to be a system of matrix { | ≤ ≤ } A units in if A (iv) fij = 0; n 6 (v) fii = 1; i=1 (vi)P for 1 i, j, p, q n, we have fipfqj = δpqfij. ≤ ≤ Definition 3.9. A p-adic Banach algebra over Ωp is said to be a p- T adic triangular (TUHF) Banach algebra if there exists an increasing sequence ( n) of finite-dimensional p-adic Banach subalgebras of such that each n is T T T(n) generated as a p-adic Banach algebra over Ωp by a triangular system e { ij | 1 i j pn of matrix units in , and = n. ≤ ≤ ≤ } T T T Definition 3.10. Let (pn) be a sequenceS of positive integers. The supernatural number of (pn) is the function N[(pn)], defined on the set of prime numbers, such that for any prime number q,

m N[(pn)](q) = sup m n : q pn . { | ∃ | }

Theorem 3.11. (Extended Spectral Theorem for p-adic Banach Algebras) Let be a p-adic Banach algebra over Ωp. Let A be -analytic, with A ∈ A A spectrum σA. Let f, g (σA). Let 0 < r Ωp p, and let Γ be in Ωp, with ∈ F ∈ | | Γ p = r. Assume that a , . . . , aN in Ωp are given, with | | 1 N − σA Da (r ), (1) ⊆ i i [=1 where the Dai (r) are disjoint. Finally, suppose that f and g are Krasner analytic on each Dai (r). By (1) and Deﬁnition 3.7, f(A) and g(A) are well-deﬁned, with

N f(A) = f(x)(x ai)R(x; A) dx, − i=1 Z Xai,Γ 446 R.L. Baker

N g(A) = g(x)(x ai)R(x; A) dx. − i=1 Z Xai,Γ

Let g (A), and let α, β Ωp. Then αf + βg, f g (A), and ∈ F ∈ · ∈ F (2) (αf + βg)(A) = αf(A) + βg(A); (3) (f g)(A) = f(A)g(A); · (4) The mapping h h(A) is continuous on Br,a ,...,a . 7→ 1 N Proof. See [4], Theorem 2.10. Theorem 3.12. Let be a p-adic p-adic Banach algebra. Let A have A ∈ A spectrum σA, and assume that A is -analytic. Let f (A). Let 0 < r be A ∈ F in Ωp p, and let Γ be in Ωp, with Γ p = r. Assume that a , . . . , aN in Ωp are | | | | 1 given, with N − σA Da (r ), ⊆ i i [=1 where the Dai (r) are disjoint and f is Krasner analytic on the Dai (r). Then for any ǫ > 0 there exists a δ > 0 such that if B is -analytic, with ∈ A A A B p < δ, then k − k N σB Da (r), ⊆ i i [=1 f (B) and f(A) f(B) p < ǫ. ∈ F k − k Proof. See [4], Theorem 2.8. Lemma 3.13. Let K be a ﬁeld and let e K be an idempotent. If ∈ 0, 1 = z K, then 6 ∈ (z 1) + e (z e)−1 = − . − z(z 1) − Proof. A calculation shows that (z e)[(z 1) + e] = z(z 1) = 0. − − − 6 Lemma 3.14. Let K be a nonarchimedean ﬁeld, i.e., for all x, y K, ∈ x + y max x , y . Let 0 < r < 1. Then | | ≤ {| | | |} (a) D (r) D (r) = . 0 ∩ 1 ∅ (b) If z K with z(z 1) < r, then z D (r−) D (r−). ∈ | − | ∈ 0 ∪ 1 Proof. Because K is nonarchimedean, if λ, µ K, with λ > µ , then ∈ | | | | λ + µ = λ (see [10], Theorem, p. 5). To prove (a), suppose that λ D (r) | | ∈ 0 ∩ D (r). Then λ r < 1, hence λ 1 = 1, which contradicts λ D (r), i.e., 1 | | ≤ | − | ∈ 1 ON CERTAIN p-ADIC BANACH LIMITS OF... 447

λ 1 r < 1. This proves (a). To prove (b), suppose that λ(λ 1) < r, | − | ≤ | − | i.e., λ λ 1 < r. There are three cases to consider. In the ﬁrst Case, λ > 1. | || − | | | Then λ 1 = λ , and hence r > λ λ 1 = λ 2, which gives r > λ 2, i.e., | − | | | | || − | | | | | 1 > √r > λ . This contradiction shows that the ﬁrst case can not hold. In | | the second case, λ = 1. Then r > λ λ 1 = λ 1 , and hence λ D (r−). | | | || − | | − | ∈ 1 Finally, in the third case, λ < 1. Then we have λ 1 = 1, and hence we get | | | − | r > λ λ 1 = λ , consequently, λ D (r−). This proves (b). | || − | | | ∈ 0 Lemma 3.15. Let be a p-adic Banach algebra over Ωp. Let E be A ∈ A an idempotent. Then for all r > 0, R(x; E) p is bounded on Dσ (r). k k E Proof. Let r > 0 be arbitrary. Suppose that E = 0, then σE = 0 . If { } −1 1 x Dσ (r), then x p r, and hence R(x; E) p = x . This shows that ∈ E | | ≥ k k | |p ≤ r R(x; E) p is bounded on Dσ (r). Now suppose that E = 1, then σE = 1 . If k k E { } −1 1 x Dσ (r), then x 1 p r, and hence R(x; E) p = x 1 . We ∈ E | − | ≥ k k | − |p ≤ r conclude that R(x; E) p is bounded on Dσ (r). Now assume that E = k k E 6 0, 1, then σE = 0, 1 . Let x Ωp, with x p > E p. Then by The- { } ∈ | | k k ∞ E −1 E −1 E n orem 2.1(a) of [4], 1 exists and 1 = . Hence − x − x x n=0 X E −1 (x E)−1 = x−1 1 exists and − − x −1 ∞ n −1 −1 E −1 E (x E) p = x 1 = x k − k | |p − x | |p x p n=0 p X ∞ n −1 E p x p k k ≤ | | x p n=0 | | X −1 −1 E p = x p 1 k k . | | − x p | | 2 It follows that lim R(x; E) p = 0. Let M 1, r , E p be so large that |x|p→∞ k k ≥ k k

R(x; E) p 1 for x p M. k k ≤ | | ≥

Let x Dσ (r), with x p M. Then x p = x 0 p r and x 1 p r, ∈ E | | ≤ | | | − | ≥ | − | ≥ hence we have 1 1 1 2 . x p · x 1 p ≤ r | | | − | 448 R.L. Baker

Because E = 0, 1, Lemma 3.13 implies that for x = 0, 1, 6 6 (x 1) + E R(x; E) = (x E)−1 = − . − x(x 1) − Therefore we get

(x 1) + E p R(x; E) p = k − k k k x p x 1 p | | | − | max x 1 p, E p {| − | k k } ≤ r2 max 1, x p, E p { | | k k } ≤ r2 max 1,M, E p { k k } ≤ r2 M = . r2 M Thus, if we deﬁne N = , we see that r r2

R(x; E) p Nr for all x Dσ (r). k k ≤ ∈ E

Hence for all r > 0, R(x; E) p is bounded on Dσ (r). k k E Remark. Let be a p-adic TUHF algebra over Ωp. Let A have spec- T ∈ T trum σA. Assume that for all r > 0, R(x; A) p is bounded on the complement − k k Dσ (r) of Dσ (r ). According to Theorem 3.3 of [4], A is then -analytic. A A T Hence by Theorem 3.15, every idempotent E is -analytic. ∈ T T Theorem 3.16. Let be a p-adic Banach algebra over Ωp. Let 0, 1 = e A 6 ∈ be an idempotent such that e is -analytic. Let r Ωp p with 0 < r < 1, A A ∈ | | and deﬁne 0, if x D (r); f(x) = ∈ 0 1, if x D (r). ( ∈ 1 Then f (e) and f(e) = e. ∈ F Proof. Since 0, 1 = e and e is an idempotent, we have σe = 0, 1 . Because 6 { } 0 < r < 1, Lemma 3.14 implies that

− − σe D (r ) D (r ),D (r) D (r) = . (1) ⊆ 0 ∪ 1 0 ∩ 1 ∅ ON CERTAIN p-ADIC BANACH LIMITS OF... 449

Now, f is Krasner analytic on D (r−) and on D (r−), and hence f (e). Let 0 1 ∈ F Γ Ωp, with Γ p = r. By Theorem 3.11 and Lemma 3.13, (1) implies that ∈ | |

f(e) = f(x)(e a)−1 dx + f(x)(e a)−1 dx − − 0Z,Γ 1Z,Γ

= (0)(e a)−1 dx + (1)(e a)−1 dx − − 0Z,Γ 1Z,Γ (x 1) + e x 1 dx dx = − dx = − + e. x(x 1) x x 1 x(x 1) 1Z,Γ − 1Z,Γ − 1Z,Γ −

By Theorem 2.8 we have

x 1 dx x 1 − = − = 0 x x 1 x Z x=1 1,Γ − and dx x 1 dx d x 1 = − = − = 1. x(x 1) x (x 1)2 dx x Z Z x=1 1,Γ − 1,Γ −

Therefore we get that f(e) = (0) + (1)e = e. This completes the proof of the theorem.

Theorem 3.17. Let be a p-adic Banach algebra over Ωp. Let e be A ∈ A an idempotent such that e is -analytic. Let ǫ > 0 be arbitrary. Then there A exists a positive number γ(ǫ, e) > 0 with the following property. Let be a B unital p-adic Banach subalgebra of , and let a be -analytic such that A ∈ B A e a p < γ(ǫ, e). Then there exists an idempotent f such that k − k ∈ B e f < ǫ. k − k

Proof. We may assume that e = 0, 1. Let 0 < r < 1 be arbitrary. By 6 Lemma 3.14, D (r) D (r) = . Deﬁne 0 ∩ 1 ∅ 0, if x D (r); g(x) = ∈ 0 (1, if x D1(r). ∈

Then by Theorem 3.16, g (e) and g(e) = e. For z Ωp deﬁne p(z) = ∈ F ∈ z(z 1). By the p-adic spectral mapping theorem ( [10], p. 111) for all a − ∈ A 450 R.L. Baker

2 we have σp(a) = z(z 1) z σa . Because e = e, there exists a δ1 > 0 such { − | ∈ } 2 that if a with a e p < δ , then a a p < r. Then for a e p < δ , ∈ A k − k 1 k − k k − k 1 we have (see [10], Theorem 6, p. 114 and Theorem 2, p. 107)

max z p = rσp(a) ν(p(a)) p(a) p < r, z∈σp(a) | | ≤ ≤ k k and hence for w σa, ∈

w p w 1 p max z(z 1) = max z p < r. | | | − | ≤ z∈σa | − | z∈σp(a) | |

Therefore, by Lemma 3.14, if a e p < δ , then k − k 1 − − σa D (r ) D (r ). ⊆ 0 ∪ 1 By Theorem 3.12 there exists a δ > 0 such that if a is -analytic and 2 ∈ A A a e p < δ , then g (a) and g(e) g(a) p < ǫ, i.e., e g(a) p < ǫ. k − k 2 ∈ F k − k k − k Define γ(ǫ, e) = min δ , δ , . { 1 2 } Let a be -analytic, with a e p < γ(ǫ, e). Because a is -analytic, a ∈ B A k − k A satisfies the hypothesis of Theorem 3.11 and hence we may define f = g(a). 2 2 Then by Theorem 3.11 we get f = g (a) = g(a) = f and e f p < ǫ. Because k − k f , this completes the proof of the theorem. ∈ B Lemma 3.18. Let be a p-adic Banach algebra over Ωp. Let ǫ > 0 be A positive number, and let I = ei 1 i n be an orthogonal family of { | ≤ ≤ } -analytic idempotents in . Then there exists a positive number δ(ǫ, I) > 0 A A with the following property. Let be a unital p-adic Banach subalgebra of B A such that each member of is -analytic, and suppose that ai 1 i n B A { | ≤ ≤ } is a family of elements in such that ei ai δ(ǫ, I) for 1 i n. Then B k − k ≤ ≤ ≤ there exists an orthogonal family fi 1 i n of idempotents in such { | ≤ ≤ } B that ei fi < ǫ for 1 i n. k − k ≤ ≤ Proof. The proof that we give of this lemma is modeled on the proof of Lemma 1.7 of [8]. We prove the lemma by induction on n. Each member of is -analytic, hence the case n = 1 follows from Lemma 3.17. Assume that B A the lemma is true for n 1, and let ǫ > 0 be a given positive number. Let ≥ J = ei 1 i n + 1 be an orthogonal family of -analytic idempotents in { | ≤ ≤ } A . Without loss, we may assume that ǫ < 1. Define µ = max ei : 1 A { k k ≤ i n + 1 + 1. Let γ(ǫ, en+1) = γ > 0 be the positive number that Theorem ≤ } ′ 2 2 3.17 assigns to the 3-tuple , ǫ, en , and define ǫ = γ/7n (1 + µ) . We may A +1 assume that γ < 1. Define δ(ǫ, J) > 0 by δ(ǫ, J) = min ǫ′, δ(ǫ′,I) , where { } ON CERTAIN p-ADIC BANACH LIMITS OF... 451

′ I = ei 1 i n and δ(ǫ ,I) > 0 is the positive number given by the { | ≤ ≤ } induction hypothesis. Let be a unital p-adic Banach subalgebra of such B A that each member of is -analytic, and let ai 1 i n + 1 be a family of B A { | ≤ ≤ } elements in such that ei ai δ(ǫ, J) for 1 i n + 1. By the induction B k − k ≤ ≤ ≤ hypothesis, we can ﬁnd an orthogonal family fi 1 i n of idempotents { | ≤ n≤ } ′ in such that fi ei < ǫ for 1 i n. Deﬁne f = fi, then we have B k − k ≤ ≤ i=1 P en (1 f)an (1 f) k +1 − − +1 − k = en (1 f)en (1 f) + (1 f)(en an )(1 f) k +1 − − +1 − − +1 − +1 − k en (1 f)en (1 f) + (1 f)(en an )(1 f) ≤ k +1 − − +1 − k k − +1 − +1 − k 2 (1 + f ) fen + en f + (1 + f ) en an ≤ k k · k +1k k +1 k k k k +1 − +1k < 7n2(1 + µ)2ǫ′ < γ.

Let be the unital commutative p-adic Banach subalgebra of generated by C B the family 1, (1 f)an (1 f) fi 1 i n . { − +1 − } ∪ { | ≤ ≤ } Then each member of is -analytic. By our choice of γ, Theorem 3.17 im- C A plies that there exists an idempotent fn such that fn en < ǫ. +1 ∈ C k +1 − +1k Moreover, for 1 i n, we have ≤ ≤

fn fi ( fi ei + ei ) fn en + en fi ei k +1 k ≤ k − k k k · k +1 − +1k k +1k · k − k < (δ(ǫ, J) + µ)δ(ǫ, J) + µδ(ǫ, J) < 7n2(1 + µ)2ǫ′ < γ < 1.

Because is commutative, for 1 i n, fn fi is an idempotent, hence we C ≤ ≤ +1 must have fn fi = 0. Thus fi 1 i n + 1 is an orthogonal family of +1 { | ≤ ≤ } idempotents in such that ei fi < ǫ for 1 i n + 1. This completes the B k − k ≤ ≤ proof of the lemma.

4. On Classifying p-Adic Triangular UHF Algebras

We are now ready to state the the main result of the the present paper, namely, Theorem 4.2. We will use the following terminology: Let be a unital p- A adic Banach algebra over Ωp and let be a p-adic Banach subalgebra of B A 452 R.L. Baker which contains the identity of . Then is said to be a full matrix algebra A B if there exists a positive integer n such that is isomorphic to Mn(Ωp) as an B Ωp-algebra. Throughout the remainder of this paper, all p-adic Banach algebras will be unital.

Deﬁnition 4.1. Let be a p-adic Banach algebra over Ωp, and let B T ⊆ B be a p-adic TUHF Banach algebra given by

(n) = n, n = f 1 i j qn , ( n) increasing. T T T h ij | ≤ ≤ ≤ i T Let be a family[ of finite-dimensional unital p-adic Banach subalgebras of G B such that each member of contains the identity of and is a full matrix G B algebra. We say that the family satisfies the local dimensionality condition G with respect to if the following conditions hold. T (i) For each n, there exists such that n and the triangular (n) N ∈ G T ⊆ N system of matrix units fij 1 i j qn can be extended to a system of (n) { | ≤ ≤ ≤ } matrix units f 1 i, j qn in . { ij | ≤ ≤ } N (ii) Let m be a positive integer such that m , where . Then T ⊆ M M ∈ G for any n m, there exists such that , n . ≥ N ∈ G M T ⊆ N Theorem 4.2. Let be a unital p-adic Banach algebra over Ωp, and let B be a p-adic TUHF Banach subalgebra of given by T ⊆ B B (n) = n, n = f 1 i j qn , ( n) increasing. T T T h ij | ≤ ≤ ≤ i T Let be a family[ of finite-dimensional p-adic Banach subalgebras of such G B that each member of contains the identity of and is a full matrix algebra. G B Assume that satisfies the local dimensionality condition with respect to . G T Let be a p-adic TUHF Banach algebra given by S (n) = n, n = e 1 i j pn , ( n) increasing. S S S h ij | ≤ ≤ ≤ i S Suppose that[ there exists an algebraic isomorphism Φ: such that S → T Φ( n) = . Then N[(qn)] N[(pn)]. Moreover, if there exists a unital p-adic S T ≤ Banach algebra such that and a family of finite-dimensional p-adic S A S ⊆ A F Banach subalgebras of such that each member of is a full matrix algebra A F containing the identity of and satisfies the local dimensionality condition A F with respect to , then there exists an Ωp-algebra isomorphism Ψ: such S T → S that Ψ( n) = , only if N[(pn)] = N[(qn)]. T S UsingS the p-adic Banach-Steinhaus theorem (see [15], Theorem 3.12), p-adic Banach algebra inductive limits are constructed in the same way that they are ON CERTAIN p-ADIC BANACH LIMITS OF... 453 for complex Banach algebras ([Bl], Section 3.3). The next lemma provides a means of constructing a large class of p-adic triangular UHF algebras such that Theorem 4.2 applies to each member of this class. The proof of Lemma 4.3 is routine.

Lemma 4.3. Let (pn) be a sequence of positive integers, and let ( n) be k · k a sequence of p-adic Banach algebra norms on Mp (Ωp). Let Φnm m n n { | ≤ } be a family of continuous, unital monomorphisms Φnm :(Mp (Ωp); m) m k · k → (Mp (Ωp); n), such that the following conditions hold. n k · k (i) Φnm(Tp (Ωp)) Tp (Ωp), for m n; m ⊆ n ≤ (ii) ΦnpΦpm = Φnm, for m p n; ≤ ≤ (iii) lim sup Φnm < + for all m. n≥m k k ∞

Let be the p-adic Banach algebra inductive limit lim(Mp (Ωp); n;Φnm), B −→ n k·k and write = n, where n is the canonical image of Mp (Ωp) in . Define B N N n B to be the family n n 1 . Finally, let = n, where n is the G S {N | ≥ } T T T canonical image of Tp (Ωp) in . Then is a p-adic TUHF Banach algebra n B T S over Ωp such that the triple , , satisfies conditions (i) and (ii) of Theorem B T G 4.2. We are now ready to present a purely p-adic Banach-algebraic formulation of n the main result in [2]. In for any positive integer n, set Ωp = Ωp( 1, 2, . . . , n ). n { } Then by Definition 3.2, Ωp is a p-adic Banach algebra over Ωp under the norm

(x1, . . . , xn) p = max xi p. k k 1≤i≤n | |

n We identify the Ωp-algebra Mn(Ωp) of n n matrices over Ωp with B(Ω ). Then × p the Ωp-algebra Tn(Ωp) of n n upper triangular matrices over Ωp is identified × with a p-adic Banach subalgebra of Mn(Ωp). Proposition 4.4. For each pair of positive integers m, n such that m n, | let there correspond a unital monomorphism Θm n : Mm(Ωp) Mn(Ωp) such → that the following conditions are satisfied. (i) Θm n(Tm(Ωp)) Tn(Ωp); ⊆ (ii) if m r n, then Θm rΘr,n = Θm n; | | −1 (iii) sup Θm n p, Θ p : m n < . { k k k m nk | } ∞ Let (pn) and (qn) be sequences of positive integers such that pm pn and | qm qn whenever m n. Define and to be the TUHF Banach algebras | ≤ S T

= lim(Tp (Ωp); Θp p ), = lim(Tq (Ωp); Θq q ). S −→ n n m T −→ n n m 454 R.L. Baker

For each n let n = Φn(Tp (Ωp)) and n = Ψn(Tq (Ωp)), where Φn (resp., S n T n Ψn) is the canonical mapping of Tp (Ωp) (resp., Tq (Ωp)) into (resp., ). n n S T Then there exist Ωp-algebra isomorphisms Φ : and Ψ : such that S → T T → S Φ( n) = and Ψ( n) = , if, and only if N[(pn)] = N[(qn)]. S T T S S Proof. The proofS of Proposition 4.4 is a simple adaptation of the proof of Proposition 3.5 in [3].

Let p, q be positive integers such that q p; define ρpq, σpq : Mq(Ωp) Mp(Ωp) | → by ρpq(x) = x 1d, σpq(x) = 1d x, x Mq(Ωp) and d = p/q. Then Propo- ⊗ ⊗ ∈ sition 4.4 applied to the mappings σpq yields the exact p-adic counterpart of the main result in [2]. On the other hand, Proposition 4.4 applied to the mappings ρpq yields a classification of p-adic TUHF Banach algebras of the form = lim(Tp (Ωp); ρp q ), where (pn) is a sequence of positive integers such that T −→ n n m pm pn whenever m n; algebras of this form are also classified in [13] and in | ≤ [11], [12].

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