# Ford lemma for topological *-algebras/Fordi lemma topoloogiliste *-algebrate korral.

At several places in the development of the theory of topological
*-algebras (especially, of the theory of Banach *-algebras and locally
m-convex *-algebras) a self-adjoint square root for a self-adjoint
element with a positive spectrum is needed.

In 1966, James W. M. Ford proved in his doctoral dissertation [20] (see also [21]; [13], Proposition 8.13 and 12.11; [33], Theorem 3.4.5; [34], Proposition 11.1.7; [41], Lemmas 9.8, 9.10 and Corollary 9.9) a general square root lemma for Banach algebras and Banach *-algebras. This result was generalized for complete locally m-convex *-algebras in [26], Lemma 1 and Corollary (see also [38], Theorems 3.9 and 3.10; [22], Theorems 5.5.4, 5.5.8 and Corollary 5.5.5; and [15], Proposition 1.12 and Corollary 1.13); for p-Banach *-algebras in [18], Proposition 3.1; for pseudocomplete locally convex *-algebras in [35], Lemma 1; for complete locally m-convex *-algebras with not necessarily bounded spectrum in [39], Theorem 2.2; for complete locally m-pseudoconvex Hausdorff *-algebras in [11], Proposition 5.3.4 and Corollary 5.3.5 (for several kinds of locally pseudoconvex algebras see [14], Proposition 5.1; [15], Proposition 4.6; [16], Proposition 2.4; and [17], Proposition 3.1), and for fundamental Frechet algebras in [10], Theorems 3.2 and 3.3.

In the present paper all these results are generalized (without using projective limits) to the case of topological algebras and topological *-algebras with continuous involution or not necessarily continuous involution.

1. INTRODUCTION

Let A be a topological algebra over the field of complex numbers C with separately continuous multiplication (in short, a topological algebra), homA the set of all nontrivial characters of A, and m(A) the set of all closed regular (or modular) two-sided ideals in A which are maximal as left ideals or as right ideals. Then A/M (in the quotient topology) is a division Hausdorff algebra for all M [member] m(A) (see [23], Theorem 24.9.6, and [33], Theorem 2.4.12). Here A/M could be topologically isomorphic to C or not. It is known that there exist topological division algebras which are not topologically isomorphic to C (see, for example, [42], pp. 83 and 85, or [40], pp. 731 and 732). When A/M is topologically isomorphic to C for each M [member of] m(A), then A is called a Gelfand-Mazur algebra. Hence, if m(A) is empty, then A is always a Gelfand-Mazur algebra, but when m(A) is nonempty, then most of topological algebras are Gelfand-Mazur algebras.

It is well known (see, for example, [1], Lemma 1.11; [2], Corollary 2; or [8], Theorem 3.3) that all p-normed algebras with p [member of] (0,1], all locally m-convex algebras, all locally convex Frechet algebras, all locally m-pseudoconvex algebras and many more general topological algebras are Gelfand-Mazur algebras. Indeed, Gelfand-Mazur algebras are exactly the class of topological algebra for which the Gelfand theory, well known in the case of commutative Banach algebras, works. If A is a Gelfand-Mazur algebra and m(A) is not empty, then every M [member of] m(A) has the form M = ker [phi] for some [phi] [member of] homA. In this case every commutative Gelfand-Mazur algebra A is homomorphic/isomorphic with a subalgebra of C(homA), similarly to the case of commutative Banach algebras.

A topological algebra is locally pseudoconvex (locally m-pseudoconvex) if it has a base of neighbourhoods of zero consisting of balanced and pseudoconvex (1) (respectively, balanced, idempotent (2), and pseudoconvex) sets. It is well known that the topology of a locally pseudoconvex (locally m-pseudoconvex) algebra can be given by a family of nonhomogeneous (respectively, nonhomogeneous and submultiplicative) seminorms (3). In the particular case when the power of homogeneity k [member of] (0,1] does not depend on the seminorms of this family, one speaks about locally k-convex and locally m-(k-convex) algebras and when k = 1, then about locally convex and locally m-convex algebras. It is well known that all locally convex and all locally bounded algebras4 are locally pseudoconvex algebras and all locally m-convex algebras and all p-normed algebras with p [member of] (0,1] are locally m-pseudoconvex algebras.

A topological algebra A is a simplicial algebra or a normal algebra (in the sense of Michael) if every closed regular two-sided ideal of A is contained in some closed maximal regular two-sided ideal of A. It is known that all commutative locally m-pseudoconvex (in particular, commutative locally m-convex) algebras are simplicial (see [5], Corollary 5; for the case of complete algebras see [4], Proposition 2, and [11], Corollary 7.1.14; and for the case of locally m-convex algebras see [42], p. 110, or [12], pp. 321 and 322).

An element a of a topological algebra A is called topologically quasi-invertible in A if there exist nets [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] and [([b.sub.[mu]]).sub.[mu][member of]M] in A such that [([a.sub.[lambda]] [omicron] a).sub.[lambda][member of][LAMBDA]] and [(a [omicron] [b.sub.[mu]]).sub.[mu][member of]M] converge to the zero element [[theta].sub.A] of A (here a [omicron] b = a + b - ab for every a, b [member of] A) and an element a of a unital topological algebra A is called topologically invertible in A if there exist nets [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] and [([b.sub.[mu]]).sub.[mu][member of]M] in A such that [([a.sub.[lambda]]a).sub.[lambda][member of][LAMBDA]] and [([ab.sub.[mu]]).sub.[mu][member of]M] converge to the unit element [e.sub.A] of A.

Let TqinvA denote the set of all topologically quasi-invertible elements in A, QinvA the set of all quasi-invertible elements in A and, for a unital topological algebra A, let TinvA denote the set of all topologically invertible elements in A and InvA the set of all invertible elements in A. A topological algebra A is called an advertive topological algebra if TqinvA = QinvA and an invertive topological algebra if TinvA = InvA. It is known (see [3], Proposition 2 and Corollary 2, or [28], p. 73) that all Q-algebras (that is, topological algebras in which QinvA, in the unital case InvA, is open) and all complete locally m-pseudoconvex algebras are advertive (in the unital case invertive).

Let A be a topological algebra. A Cauchy sequence ([a.sub.n]) is called a Mackey-Cauchy sequence in A if there exist a balanced and bounded subset B of A and for every [epsilon] > 0 a number [n.sub.[epsilon]] [member of] N such that [a.sub.n+m] - [a.sub.m] [member of] [epsilon]B whenever n > [n.sub.[epsilon]] and m > 0. A topological algebra A is sequentially Mackey complete if every Mackey-Cauchy sequence of A converges in A. Hence, all complete topological algebras are sequentially Mackey complete (because every Mackey-Cauchy sequence (5) is a Cauchy sequence).

Let A be a topological *-algebra, that is, a topological algebra on which an involution a [right arrow] [a.sup.*] has been given. An element a [member of] A is self-adjoint or hermitian if [a.sup.*] = a.

Let again A be a topological algebra. If A has the unit element [e.sub.A], then

[[sigma].sub.A](a) = {[lambda] [member of] c : a - [lambda][e.sub.A] [not member of] InvA},

and if A is an algebra without unit, then

[[sigma].sub.A](a) = {[lambda] [member of] C\{0} : a/[lambda] [not member of] QinvA} [union] {0}

is the (algebraic) spectrum of a [member of] A. In both cases

[[rho].sup.t.sub.A](a) = sup{[absolute value of [lambda]] : [lambda] [member of] [[sigma].sup.t.sub.A](a)}

is the (algebraic) spectral radius of A.

For noninvertive algebras with the unit element

[[sigma].sup.t.sub.A](a) = {[lambda] [member of] c : a - [lambda][e.sub.A] [member of] TinvA}

and for nonadvertive algebras

[[sigma].sup.t.sub.A](a) = {[lambda] [member of] C\{0} : a/[lambda] [not member of] TqinvA} [union] {0}

is the topological spectrum of a [member of] A. In both cases

[[rho].sup.t.sub.A](a) = sup{[absolute value of [lambda]] : [lambda] [member of] [[sigma].sub.A](a)}

is the topological spectral radius of A.

Herewith, we take [[rho].sup.t.sub.A](a) = 0 if [[sigma].sup.t.sub.A](a) = 0, and [[rho].sup.t.sub.A](a) = [infinity] if [[sigma].sup.t.sub.A](a) is an unbounded set in C, similarly as in the case of the algebraic spectrum. It is easy to see that [[sigma]].sup.t.sub.A](a) [subset or equal to] [[sigma].sub.A](a) and [[rho].sup.t.sub.A](a) [less than or equal to] [[rho].sub.A](a) for each a [member of] A.

Moreover, A is an advertive algebra if and only if [[sigma].sup.t.sub.A](a) = [[sigma].sub.A](a) for each a [not member of] QinvA. Indeed, if A is an advertive algebra, then [[sigma].sup.t.sub.A](a) = [[sigma].sub.A](a) for each a [member of] A. Let now a [member of] A \ QinvA. Then 1 [member of] [Q.sub.A](a). If now [[sigma].sub.A](a) = [[sigma].sup.t.sub.A](a), then a [not member of] TqinvA. Hence, QinvA = TqinvA in this case. Therefore, A is an advertive algebra. Similarly, a unital topological algebra is an invertive algebra if and only if [[sigma].sub.A](a) = [[sigma].sup.t.sub.A](a) for each a [not member of] InvA (see [9], p. 258).

Let now A be a topological *-algebra. Then

[[sigma].sub.A](a*) = {[bar].[mu]] : [mu] [member of] [[sigma].sub.A](a)}

for each a [member of] A. Therefore, [[sigma].sub.A](a) [subset] R, similarly [[sigma].sup.t.sub.A](a) [subset] R, if a [member of] A is self-adjoint.

Let A be a topological algebra. Then

[[beta].sub.A](a) = inf {[lambda] > 0: {[(a/[lambda]).sup.n] : n [member of] N} is bounded in A}

is the radius of boundedness of a [member of] A. It satisfies the following conditions:

[[beta].sub.A]([mu]a) = [absolute value of [mu]][[beta].sub.A](a) and [[beta].sub.A]([a.sup.k]) = [[beta].sub.A][(a).sup.k]

for each a [member of] A, [mu] [member of] C, and k [member of] N. If a, b [member of] A and the product of any two idempotent bounded subsets of A is bounded, then

[[beta].sub.A](ab) [less than or equal to] [[beta].sub.A](a) + [[beta].sub.A](b),

and if, in addition, the convex hull of an idempotent and bounded set of A is bounded (in particular, A is a locally convex algebra with continuous multiplication), then

[[beta].sub.A](a + b) [less than or equal to] [[beta].sub.A](a) + [B.sub.A](b)

(see [31], p. 281; [32], p. 310; and [19], Lemma II.9).

Herewith, if [[beta].sub.A](a) < [infinity], then a [member of] A is called a bounded element of A, and if all elements in A are bounded, then A is called a topological algebra with bounded elements.

2. FORD LEMMA FOR TOPOLOGICAL ALGEBRAS

Let A be an algebra and a [member of] A. An element b [member of] A is called the quasi-square root of a if b [omicron] b = a. When A is a unital algebra and [b.sup.2] = a, then b is called the square root of a. For each a [member of] A we put S'(a) = {[(a).sup.n] : n [greater than or equal to] 1} and (6) S(a) = [GAMMA](S'(a)).

First, we prove the following generalization of a result of Powell (7) (see [35], Lemma 1).

Theorem 2.1. Let A be a sequentially Mackey complete topological algebra. If a [member of] A and S(a) is bounded in A, then there exists an element b [member of] A such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. In particular, when [[beta].sub.A](a) < 1 and

(a) [[rho].sub.A](x) [less than or equal to] [[beta].sub.A](x) for each x [member of] A,

and

(b) [[beta].sub.A](x + y) [less than or equal to] [[beta].sub.A](x) + [[beta].sub.A](y) if x and y commute in A, hold, then there is only one quasi-square root b of a such that [[beta].sub.A](b) < 1.

Proof. Let a [member of] A be such that S(a) is bounded in A. Then S(a) is an idempotent and bounded subset of A. Since the closure B = cl(S(a)) is a closed, idempotent, bounded and absolutely convex subset in A (see, for example, [27], pp. 103, and [30], pp. 5-6), then the subalgebra [A.sub.B] of A, generated by B, is a normed algebra with respect to the submultiplicative norm [parallel] x [parallel], defined by

[parallel]x[parallel] = inf{[absolute value of [lambda]] : x [member of] [lambda]B}

for each x [member of] [A.sub.B], and the norm topology on [A.sub.B] is not weaker than the topology on [A.sub.B] induced by the topology of A (see [6], Proposition 2.2). Moreover, [A.sub.B] is complete, because A is sequentially Mackey complete, and [parallel]a[parallel] [less than or equal to] 1, because a [member of] S(a) [subset] B.

For each n [member of] n we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

converges (see [33], p. 361), it follows that ([S.sub.n]) is a Cauchy sequence in [A.sub.B]. Hence, ([S.sub.n]) converges in [A.sub.B] to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each [alpha], [beta] [member of] R and n [member of] N (see [36], formula 13, p. 616), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each x [member of] [A.sub.B] and the norm is continuous on [A.sub.B], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let now [[beta].sub.A](a) < 1, A satisfy the conditions (a) and (b), and let c [member of] A be any element such that c [omicron] c = a and [[beta].sub.A]( c) < 1. Then c [omicron] c = a = b [omicron] b, 2c = a + [c.sup.2] and 2b = a + [b.sup.2]. Therefore,

2(b [omicron] c) = (a + [b.sup.2]) + (a + [c.sup.2]) - 2bc = 2a + [(b - c).sup.2] = 2a + [(c - b).sup.2] = 2(c [omicron] b).

Hence, cb = bc. Taking this into account,

[[rho].sub.A](b + c/2) [less than or equal to] [[beta].sub.A](b + c)/2) [less than or equal to] [[beta].sub.A] (b + c/2) [less than or equal to] [[beta].sub.A](b) + [[beta].sub.A](c)/2 < 1

by conditions (a) and (b). It means that

d = b + c/2 [member of] QinvA.

Hence, there exists the quasi-inverse e [member of] A for d. Therefore, from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it follows that b = c.

Corollary 2.2. Let Abe a sequentially complete locally pseudoconvex Hausdorff algebra. If a [member of] A and (8) S'(a) is bounded in A, then there exists an element b [member of] A such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. In particular, when [[beta].sub.A](a) < 1, then there is only one quasi-square root b of a such that [[beta].sub.A](b) < 1.

Proof. In the present case, A is sequentially Mackey complete and [[rho].sub.A](a) [less than or equal to] [[beta].sub.A](a) for each a [member of] A (see [6], Corollary 4.3). Moreover, the convex hull of any idempotent and bounded set is bounded in A, because A is locally pseudoconvex and the convex hull of a set U is a subset of [GAMMA](U). Hence A satisfies condition (b) of Theorem 2.1 (see [31], p. 281). Consequently, Corollary 2.2 holds by Theorem 2.1.

Corollary 2.3. Let A be a sequentially complete locally m-convex Hausdorff algebra. If a [member of] A and [[beta].sub.A](a) < 1, then in A there exists only one quasi-square root b of a such that [[beta].sub.A](b) < 1.

Proof. In the present case A satisfies the conditions (a) and (b) of Theorem 2.1 (see [6], Proposition 4.1, and [19], Lemma II.9). Therefore, Corollary 2.2 completes the proof.

For topological unital algebras we have

Corollary 2.4. Let A be a unital sequentially Mackey complete topological algebra. If a [member of] A and S([e.sub.A] - a) (S([e.sub.A] - a/M) for some M > 1) is bounded in A, then there exists an element b [member of] A such that [b.sup.2] = a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1 (respectively, [[beta].sub.A]([e.sub.A] - b/[square root of M]) [less than or equal to] 1). In particular, when [[beta].sub.A]([e.sub.A] - a) < 1 ([[beta].sub.A]([e.sub.A] - 1/M) < 1 for some M > 1) and A satisfies (a) and (b) of Theorem 2.1, then there is only one square root b for a with [[beta].sub.A]([e.sub.A] - b) < 1 (respectively, [[beta].sub.A]([e.sub.A] - b/[square root of M]) < 1).

Proof. Since S([e.sub.A] - a) is bounded in A, then there exists an element c [member of] A such that c [omicron] c = [e.sub.A] - a or [([e.sub.A] - c).sup.2] = a and [[beta].sub.A](c) [less than or equal to] 1. Hence b = [e.sub.A] - c is a square root of a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1. If now [[beta].sub.A]([e.sub.A] - a) < 1 and A satisfies (a) and (b) of Theorem 2.1, then there is only one square root for a by Theorem 2.1.

If S([e.sub.A] - a/m) is bounded in A for some M > 1, then the proof is similar.

Similarly to Corollaries 2.2 and 2.3 the following corollaries hold.

Corollary 2.5. Let A be a unital sequentially complete locally pseudoconvex Hausdorff algebra. If a [member of] A and S'([e.sub.A] - a) is bounded in A, then there exists an element b [member of] A such that [b.sup.2] = a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1. In particular, when [[beta].sub.A] ([e.sub.A] - a) < 1, then there is only one square root b of a such that [[beta].sub.A]([e.sub.A] - b) < 1.

Corollary 2.6. Let A be a unital sequentially complete locally m-convex Hausdorff algebra. If a [member of] A and [[beta].sub.A]([e.sub.A] - a) < 1, then in A there exists only one square root b of a such that [[beta].sub.A]([e.sub.A] - b) < 1.

Now we consider the case when A is a topological *-algebra.

Theorem 2.7. Let A be a sequentially Mackey complete topological *-algebra. If a [member of] A and S(a) is bounded in A, then there exists an element b [member of] A such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. In particular, when

(c) the involution a [right arrow] [a.sup.*] in A is continuous

or

(d) a has only one quasi-square root in A, then b is self-adjoint if a is self-adjoint.

Proof. By hypothesis and Theorem 2.1, there exists an element

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. Let now [a.sup.*] = a. If A satisfies (c), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and if A satisfies condition (d), then [b.sup.*] = b, because [b.sup.*] [omicron] [b.sup.*] = [a.sup.*] = a.

Corollary 2.8. Let A be a unital sequentially Mackey complete topological *-algebra. If a [member of] A and S([e.sub.A] - a) is bounded in A, then there exists an element b [member of] A such that [b.sup.2] = a and [[beta].sub.A] ([e.sub.A] - b) [less than or equal to] 1. In particular, when A satisfies condition (c) of Theorem 2.7 or condition

(e) a has only one square root in A,

then b is self-adjoint if a is self-adjoint.

Corollary 2.9. Let A be a unital sequentially Mackey complete topological *-algebra. If a [member of] A is self-adjoint and S([e.sub.A] - a) is bounded in A, then there exists a self-adjoint element b [member of] A such that [b.sup.2] = a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1.

3. SOME RESULTS FOR TOPOLOGICAL *-ALGEBRAS

In the sequel we need the following result.

Proposition 3.1. Let A be a commutative simplicial Gelfand-Mazur algebra with nonempty set m(A). Then

(i) [sp.sup.t.sub.A](a)\{0} [subset] {[phi](a) : [phi] [member of] homA} [subset] [sp.sup.t.sub.A](a) for each a [member of] A

and

(ii) [sp.sup.t.sub.A](a) = {[phi](a) : [phi] [member of] homA} for each a [member of] A if A is a unital algebra.

Proof. (i) Take a [member of] A and [mu] [member of] [sp.sup.t.sub.A](a)\{0}. Then a/[mu] [not member of] TqinvA. Therefore, the set

I = cl{a/[mu] b - b : b [member of] A}

cannot contain a/[mu], otherwise there is a net [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] in A such that [(a/[mu] [a.sub.[lambda]] - [a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] converges to a/[mu] in A or [(a/[mu] [omicron] [a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] converges to [[theta].sub.A]. This means that a/[mu] [member of] TqinvA. Hence, I [not equal to] A. Therefore, I is a closed regular ideal in A. Since A is simplicial, there exists a closed maximal regular ideal M in A such that I [subset] M and, since A is a Gelfand-Mazur algebra, then M = ker [phi] for a [phi] [member of] homA. Consequently, [phi](a/[mu] b - b) = 0 for each b [member of] A. Hence, [phi](a) = [mu] (because [phi] is not trivial). This shows that

[sp.sup.t.sub.A](a)\{0} [subset] {[phi](a) : [phi] [member of] homA}.

Let now a [member of] A and [mu] = [phi](a) for some [phi] [member of] homA. We must show that [mu] [member of] [sp.sup.t.sub.A](a). We suppose that [mu] [not equal to] 0 and [mu] [not member of] [sp.sup.t.sub.A](a). Then a/[mu] [member of] TqinvA and so there is a net [([c.sub.[alpha]]).sub.[alpha][member of A]] such that [(a/[mu] [omicron] [c.sub.[alpha]]).sub.[alpha][member of]A] converges to [[theta].sub.A] in A. Since [phi] is continuous, [([phi](a)/[mu] [omicron] [phi][([c.sub.[alpha]])).sub.[alpha][member of A]] converges to 0, but it is not possible, since [phi](a) = [mu]. Consequently, [mu] [member of] [sp.sup.t.sub.A](a) if [mu] [not equal to] 0.

Let now [phi](a) = 0. If A does not have a unit, then automatically 0 [member of] [sp.sup.t.sub.A](a) and so [phi](a) [member of] [sp.sup.t.sub.A](a). If A has a unit and 0 [not member of] [sp.sup.t.sub.A](a), then a [member of] TqinvA. Therefore, there exists a net [(a[beta])[beta]).sub.[beta][member of]B] such that [([aa.sub.[beta]]).sub.[beta][member of]B] converges to [e.sub.A] in A. Then [([phi](a)[phi]([a.sub.[beta]])).sub.[beta][member of]B] converges to 1. But this is impossible, since [phi](a) = 0. (ii) Let now A be a unital algebra. By statement (i), it is sufficient to show that 0 [member of] [sp.sup.t.sub.A](a) if and only if [phi](a) = 0 for some [phi] [member of] homA.

Suppose first that 0 [member of] [sp.sup.t.sub.A](a). Then a [not member of] TinvA and, similarly as above,

I = cl{ab : b [member of] A}

is a closed ideal in A. Since A is a commutative unital simplicial Gelfand-Mazur algebra, there exists a M [member of] m(A) such that I [subset] M = ker [phi] for some [phi] homA. Therefore, [phi](a)[phi](b) = 0 for each b [member of] A. Again, since [phi] is not trivial, [phi](a) = 0.

Suppose next that [phi](a) = 0 for some [phi] [member of] homA. Then a [not equal to] TinvA. Otherwise, there exists a net [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] such that [(a[a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] converges to [e.sub.A] in A. Then [([phi](a)[phi]([a.sub.[lambda]])).sub.[lambda][member of][LAMBDA] converges to 1 contrary to [phi](a) = 0. Consequently, in this case 0 [member of] [sp.sup.t.sub.A](a).

Corollary 3.2. Let A be a commutative advertive simplicial Gelfand-Mazur algebra with nonempty set m(A). Then

(i) [sp.sub.A](a)\{0} [subset] {[phi](a) : [phi] [member of] homA} [subset] [sp.sub.A](a) for each a [member of] A

and

(ii) [sp.sub.A](a) = {[phi](a) : [phi] [member of] homA} for each a [member of] A if A is an invertive algebra.

Proof. In the present case [sp.sub.A](a) = [sp.sup.t.sub.A](a) for each a [member of] A. Therefore, the statements hold by Corollary 3.2.

Corollary 3.2 (ii) was proved in [3], Proposition 5. Moreover, it was shown in [3], Proposition 6, that every topological algebra for which

[sp.sub.A](a) = {[phi](a) : [phi] [member of] homA}

for each a [member of] A is an advertive algebra.

Proposition 3.3. Let A be a unital sequentially Mackey complete locally pseudoconvex Hausdorff algebra for which [[beta].sub.A](a) = [[rho].sub.A](a) for each a [member of] A. If, in addition, A satisfies the condition (9)

(f) [sp.sub.A](a) is a closed subset in c for each a [member of] A with [sp.sub.A](a) C (0,1), then for every element a [member of] A with [sp.sub.A](a) [subset] (0, [infinity]), there exists an element (10) b [member of] A such that [b.sup.2] = a. In particular, when every maximal commutative unital subalgebra B of A is an invertive simplicial Gelfand-Mazur algebra, then [sp.sub.A](b) C (0, [infinity]).

Proof. Let a [member of] A be such that [sp.sub.A](a) [subset] (0, [infinity]). If [[rho].sub.A]([e.sub.A] - a) < 1, then [[beta].sub.A]([e.sub.A] - a) < 1 by assumption. Therefore (see the proof of Theorem 2.1 and Corollary 2.5), there exists an element

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that [b.sub.2] = a and [[rho]/sub.A]- b) < 1.

Let B be a maximal commutative unital subalgebra of A, containing a. If now B is an invertive simplicial Gelfand-Mazur algebra, then TinvA = InvA, homB is not empty and

{[phi](a) : [phi] [member of] hom B} = [sp.sub.B](a) = [sp.sub.A](a) [subset] (0,[infinity])

for each a [member of] B by Corollary 3.2. Therefore, [phi](a) > 0 for each [phi] [member of] homB. Hence (by the formula (3), p. 361 from [33])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each [phi] homB. Consequently,

[sp.sub.A](b) = [sp.sub.B](b) = {[phi](b) : [phi] homB} [subset] (0, [infinity]).

Let again a [member of] A be such that [sp.sub.A](a) [subset] (0, [infinity]). Since -1 [not equal to][sp.sub.A](a), then [e.sub.A] + a [member of] InvA. Let c = [([e.sub.A] + a).sup.-1], v = ac and B be a maximal commutative unital subalgebra of A, containing [e.sub.A], a, and c. Then v = [e.sub.A] - c [member of] B and [sp.sub.B](x) = [sp.sub.A](x) for each x [member of] B. Since

[sp.sub.A](c) = {1/[mu] + 1 : [mu] [member of] [sp.sub.A](a)} [subset] (0,1)

(see, for example, the equality (4.14) in [22]), then

[sp.sub.A](v) = [sp.sub.A]([e.sub.A] - c) = 1 - [sp.sub.A](c) [subset] (0, 1)

and

[sp.sub.A]([e.sub.A] - v) = 1 - [sp.sub.A](v) [subset] (0, 1).

Therefore [[rho].sub.A]([e.sub.A] - c) < 1 and [[rho].sub.A]([e.sub.A] - v) < 1 by condition (f). Thus, by the first part of the proof, there exist z, w [member of] A such that [z.sup.2] = c and [w.sup.2] = v. Moreover, if B is a commutative invertive simplicial Gelfand-Mazur algebra, then

[sp.sub.A](z) = [sp.sub.B](z) = {[phi](z) : [phi] [member of] homB} [subset] (0,[infinity])

and

[sp.sub.A](w) = [sp.sub.B](w) = {[phi](w) : [phi] [member of] homB} [subset] (0,[infinity]).

Taking this into account, z, w [member of] InvA, from [z.sup.2] = c it follows that [([z.sup.-1]).sup.2] = [e.sub.A] + a and b = [z.sup.-1] w is the square root of a. Since

[sp.sub.B](b) = {[phi](b): [phi] [member of] homB} = {[phi]([z.sup.-1])[phi](w):[phi] [member of] homB} = {[phi](w)/[phi](z) : [phi] [member of] homB},

[sp.sub.B](z) [subset] (0,[infinity]) and [sp.sub.B](w) [subset] (0,[infinity]), then

[phi](w)/[phi](z) > 0

for each [phi] [member of] homB. Consequently, [sp.sub.A](b) = [sp.sub.B](b) [subset] (0,[infinity]).

Corollary 3.4. Let A be a unital complete locally m-(k-convex) Hausdorff algebra with bounded elements and k [member of] (0, 1]. Then for every element a [member of] A with [sp.sub.A](a) [subset] (0, [infinity]) there exists an element (11) b [member of] A such that [b.sup.2] = a and [sp.sub.A](b) [subset] (0,[infinity]).

Proof. Take a [member of] and B a maximal commutative unital closed subalgebra of A, containing a. Then B is a commutative unital complete Hausdorff locally m-(k-convex) algebra with bounded elements. Therefore, [[beta].sub.A](a) = [[rho].sub.A](b) for each b [member of] B (see [6], Corollary 4.4) and [sp.sub.B](b) is a closed subset in C (see the proof of Proposition 3.2 in [7], pp. 203-204). Since [sp.sub.A](b) = [sp.sub.B](b) and [[beta].sub.A](b) = [[beta].sub.B](b) for each b [member of] B, then [[beta].sub.A](a) = [[rho].sub.A](a) and condition (f) holds. Moreover, every maximal commutative unital (not necessarily closed) subalgebra of A is an invertive (by Corollary 2 in [3]) simplicial (by Corollary 5 in [5]) Gelfand-Mazur algebra (see, for example, [2], Corollary 2, or [8], Theorem 3.3). Hence, the result follows from Proposition 3.3.

Theorem 3.5. Let A be a unital sequentially Mackey complete topological *-algebra with continuous involution, for which [[beta].sub.A](a) = [[rho].sub.A](a) for each a [member of] A. If, moreover, A satisfies the condition (12)

(g) [sp.sub.A](a) is a closed subset in c for each self-adjoint a [member of] A with [sp.sub.A](a) [subset] (0,1),

then for every self-adjoint element h [member of] A with [sp.sub.A](h) [subset] (0, [infinity]) there exists a self- adjoint element (13) u [member of] A such that [u.sup.2] = h. In particular, when every maximal commutative unital *-subalgebra B of A is an invertive simplicial Gelfand-Mazur *-algebra, then [sp.sub.A](u) [subset] (0, [infinity]).

Proof. The proof is similar to that of Proposition 3.3. Herewith, u is self-adjoint by Corollary 2.8.

Corollary 3.6. Let A be a unital complete locally m-(k-convex) Hausdorff *-algebra with continuous involution. If all elements of A are bounded, then for each self-adjoint element h [member of] A with [sp.sub.A](h) [subset] (0,[infinity]) there exists a self-adjoint element (14) u [member of] A such that [u.sup.2] = h and [sp.sub.A](u) [subset] (0, [infinity]).

Proof. The proof is similar to that of Corollary 3.4.

Theorem 3.7. Let A be a topological *-algebra in which every maximal commutative *-subalgebra is an advertive simplicial Gelfand-Mazur *-algebra and let [h.sub.1], ..., [h.sub.n] be self-adjoint elements in A such that [sp.sub.A]([h.sub.k]) [subset] [0,[infinity]) for each k with 1 [less than or equal to] k [less than or equal to] n. Then

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [0,[infinity]).

Proof. Since [h.sub.1] + ... + [h.sub.n] is self-adjoint,

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [0,[infinity]).

Let B be a maximal commutative *-subalgebra of A which contains all elements [h.sub.1], ..., [h.sub.n]. Since B is an advertive simplicial Gelfand-Mazur algebra, by assumption, hom B is not empty. Let [lambda] be an arbitrary negative real number. It is known that

[sp.sub.A]([h.sub.k]) [union] {0} = [sp.sub.B]([h.sub.k)] [union] {0}

by Lemma 4.11 from [41], p. 47. As [sp.sub.B]([h.sub.k]) [subset] [0,[infinity]) for each k, it follows that [h.sub.k]/[lambda] [member of] QinvB for each k. Therefore [phi]([h.sub.k]/[lambda]) [not equal to] 1 or [phi]([h.sub.k]) [not equal to] [lambda] for each k. It means that [phi]([h.sub.k]) [greater than or equal to] 0 for each k. Hence

[phi]([h.sub.1] + ... + [h.sub.n]/[lambda]) = [phi]([h.sub.1]) + ... + [phi]([h.sub.n])/[lambda] [less than or equal to] 0

for each [phi] [member of] hom B. Namely by [3], p. 20,

[h.sub.1] + ... + [h.sub.n]/[lambda] [member of] TqinvB = QinvB

for each [lambda] < 0, since B is advertive. Consequently,

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [sp.sub.B]([h.sub.1] + ... + [h.sub.n]) [union] {0} [subset] [0, [infinity]).

Corollary 3.8. Let A be a complete locally m-pseudoconvex Hausdorff *-algebra and [h.sub.1, ...,][h.sub.n] be self- adjoint elements in A such that [sp.sub.A]([h.sub.k]) [subset] [0, [infinity]) for each k with 1 [less than or equal to] k [less than or equal to] n. Then

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [0, [infinity]).

Proof. Let B be a maximal commutative closed *-subalgebra of A. Then B is a commutative complete locally m-pseudoconvex Hausdorff *-algebra. Since, as above, B is an advertive simplicial Gelfand-Mazur *-algebra, by Theorem 3.7 the proof is complete.

Remark 3.9. Notice that Theorem 2.1 for fundamental Frechet algebras was proved partly in [10], Theorem 3.2, and for pseudocomplete locally convex algebras partly in [35], Lemma 1; Corollary 2.2 for complete locally m-pseudoconvex algebras (using projective limits of p-Banach algebras) was proved partly in [11], Theorem 5.3.4; Corollary 2.3 for complete locally m-convex algebras (using projective limits of Banach algebras) was proved in [38], Theorem 3.9, and in [22], Theorem 5.5.4; Corollary 2.4 for fundamental Frechet algebras was proved partly in [10], Theorem 3.3; Corollary 2.5 for complete unital locally m-pseudoconvex algebras was proved partly in [11], Corollary 5.3.5, and for unital complete locally m-convex algebras in [22], Corollary 5.5.5, and partly in [25], Corollary 1.13; Corollary 3.2 for Banach algebras was proved in [41], Theorem 3.12, and for commutative locally m-convex Q-algebras in [29], pp. 74-76; Corollary 3.4 for complete unital locally m-convex algebras was proved in [39], Theorem 2.2, and in [22], Theorem 5.5.8; and Corollary 3.8 has been proved mostly for [C.sup.*]-algebras (see, for example, [37], Lemma 4.7.4, or [41], Lemma 6.4).

On toestatud Fordi lemma analoog teatud liiki topoloogiliste algebrate (erijuhul topoloogiliste *-algebrate) jaoks ja saadud tulemusi on kasutatud topoloogiliste *-algebrate omaduste kirjeldamisel.

doi: 10.3176/proc.2011.2.01

ACKNOWLEDGEMENTS

We gratefully acknowledge the financial support of the Estonian Science Foundation (grant 7320) and of the Estonian Targeted Financing Project (SF0180039s08).

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Mart Abel and Mati Abel *

Institute of Pure Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia

Received 9 March 2010, accepted 12 October 2010

* Corresponding author, mati.abel@ut.ee

(1) A set U in A is pseudoconvex if there is a [mu] > 0 such that U + U [subset] [mu]U.

(2) A set U in A is idempotent if UU [subset] U.

(3) A seminorm p on A is nonhomogeneous if p([lambda]a) = [[absolute value of [lambda]].sup.k]p(a) for each a [member of] A and [lambda] [member of] C, where the power of homogeneity is k = k(p) [member of] (0,1], and p is submultiplicative if p(ab) [less than or equal to] p(a)p(b) for each a,b [member of] A.

(4) A topological algebra is locally bounded if it has a bounded neighbourhood of zero.

(5) It is known (see [24], p. 122) that there exist Cauchy sequences which are not Mackey-Cauchy sequences.

(6) The absolutely convex hull of S [subset] A is the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(7) He considered the case when A is a pseudocomplete locally convex *-algebra.

(8) Since A is locally pseudoconvex, then S(a) is bounded by the boundedness of S' (a).

(9) If A is a Q-algebra, then condition (f) is superfluous, because in this case the spectrum of every element of A is closed (see, for example, [29], Proposition 4.2).

(10) When [[beta].sub.A]([e.sub.A] - a) < 1, then a has only one square root by Theorem 2.1.

(11) See footnote 8.

(12) See footnote 7.

(13) See footnote 8.

(14) See footnote 8.

In 1966, James W. M. Ford proved in his doctoral dissertation [20] (see also [21]; [13], Proposition 8.13 and 12.11; [33], Theorem 3.4.5; [34], Proposition 11.1.7; [41], Lemmas 9.8, 9.10 and Corollary 9.9) a general square root lemma for Banach algebras and Banach *-algebras. This result was generalized for complete locally m-convex *-algebras in [26], Lemma 1 and Corollary (see also [38], Theorems 3.9 and 3.10; [22], Theorems 5.5.4, 5.5.8 and Corollary 5.5.5; and [15], Proposition 1.12 and Corollary 1.13); for p-Banach *-algebras in [18], Proposition 3.1; for pseudocomplete locally convex *-algebras in [35], Lemma 1; for complete locally m-convex *-algebras with not necessarily bounded spectrum in [39], Theorem 2.2; for complete locally m-pseudoconvex Hausdorff *-algebras in [11], Proposition 5.3.4 and Corollary 5.3.5 (for several kinds of locally pseudoconvex algebras see [14], Proposition 5.1; [15], Proposition 4.6; [16], Proposition 2.4; and [17], Proposition 3.1), and for fundamental Frechet algebras in [10], Theorems 3.2 and 3.3.

In the present paper all these results are generalized (without using projective limits) to the case of topological algebras and topological *-algebras with continuous involution or not necessarily continuous involution.

1. INTRODUCTION

Let A be a topological algebra over the field of complex numbers C with separately continuous multiplication (in short, a topological algebra), homA the set of all nontrivial characters of A, and m(A) the set of all closed regular (or modular) two-sided ideals in A which are maximal as left ideals or as right ideals. Then A/M (in the quotient topology) is a division Hausdorff algebra for all M [member] m(A) (see [23], Theorem 24.9.6, and [33], Theorem 2.4.12). Here A/M could be topologically isomorphic to C or not. It is known that there exist topological division algebras which are not topologically isomorphic to C (see, for example, [42], pp. 83 and 85, or [40], pp. 731 and 732). When A/M is topologically isomorphic to C for each M [member of] m(A), then A is called a Gelfand-Mazur algebra. Hence, if m(A) is empty, then A is always a Gelfand-Mazur algebra, but when m(A) is nonempty, then most of topological algebras are Gelfand-Mazur algebras.

It is well known (see, for example, [1], Lemma 1.11; [2], Corollary 2; or [8], Theorem 3.3) that all p-normed algebras with p [member of] (0,1], all locally m-convex algebras, all locally convex Frechet algebras, all locally m-pseudoconvex algebras and many more general topological algebras are Gelfand-Mazur algebras. Indeed, Gelfand-Mazur algebras are exactly the class of topological algebra for which the Gelfand theory, well known in the case of commutative Banach algebras, works. If A is a Gelfand-Mazur algebra and m(A) is not empty, then every M [member of] m(A) has the form M = ker [phi] for some [phi] [member of] homA. In this case every commutative Gelfand-Mazur algebra A is homomorphic/isomorphic with a subalgebra of C(homA), similarly to the case of commutative Banach algebras.

A topological algebra is locally pseudoconvex (locally m-pseudoconvex) if it has a base of neighbourhoods of zero consisting of balanced and pseudoconvex (1) (respectively, balanced, idempotent (2), and pseudoconvex) sets. It is well known that the topology of a locally pseudoconvex (locally m-pseudoconvex) algebra can be given by a family of nonhomogeneous (respectively, nonhomogeneous and submultiplicative) seminorms (3). In the particular case when the power of homogeneity k [member of] (0,1] does not depend on the seminorms of this family, one speaks about locally k-convex and locally m-(k-convex) algebras and when k = 1, then about locally convex and locally m-convex algebras. It is well known that all locally convex and all locally bounded algebras4 are locally pseudoconvex algebras and all locally m-convex algebras and all p-normed algebras with p [member of] (0,1] are locally m-pseudoconvex algebras.

A topological algebra A is a simplicial algebra or a normal algebra (in the sense of Michael) if every closed regular two-sided ideal of A is contained in some closed maximal regular two-sided ideal of A. It is known that all commutative locally m-pseudoconvex (in particular, commutative locally m-convex) algebras are simplicial (see [5], Corollary 5; for the case of complete algebras see [4], Proposition 2, and [11], Corollary 7.1.14; and for the case of locally m-convex algebras see [42], p. 110, or [12], pp. 321 and 322).

An element a of a topological algebra A is called topologically quasi-invertible in A if there exist nets [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] and [([b.sub.[mu]]).sub.[mu][member of]M] in A such that [([a.sub.[lambda]] [omicron] a).sub.[lambda][member of][LAMBDA]] and [(a [omicron] [b.sub.[mu]]).sub.[mu][member of]M] converge to the zero element [[theta].sub.A] of A (here a [omicron] b = a + b - ab for every a, b [member of] A) and an element a of a unital topological algebra A is called topologically invertible in A if there exist nets [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] and [([b.sub.[mu]]).sub.[mu][member of]M] in A such that [([a.sub.[lambda]]a).sub.[lambda][member of][LAMBDA]] and [([ab.sub.[mu]]).sub.[mu][member of]M] converge to the unit element [e.sub.A] of A.

Let TqinvA denote the set of all topologically quasi-invertible elements in A, QinvA the set of all quasi-invertible elements in A and, for a unital topological algebra A, let TinvA denote the set of all topologically invertible elements in A and InvA the set of all invertible elements in A. A topological algebra A is called an advertive topological algebra if TqinvA = QinvA and an invertive topological algebra if TinvA = InvA. It is known (see [3], Proposition 2 and Corollary 2, or [28], p. 73) that all Q-algebras (that is, topological algebras in which QinvA, in the unital case InvA, is open) and all complete locally m-pseudoconvex algebras are advertive (in the unital case invertive).

Let A be a topological algebra. A Cauchy sequence ([a.sub.n]) is called a Mackey-Cauchy sequence in A if there exist a balanced and bounded subset B of A and for every [epsilon] > 0 a number [n.sub.[epsilon]] [member of] N such that [a.sub.n+m] - [a.sub.m] [member of] [epsilon]B whenever n > [n.sub.[epsilon]] and m > 0. A topological algebra A is sequentially Mackey complete if every Mackey-Cauchy sequence of A converges in A. Hence, all complete topological algebras are sequentially Mackey complete (because every Mackey-Cauchy sequence (5) is a Cauchy sequence).

Let A be a topological *-algebra, that is, a topological algebra on which an involution a [right arrow] [a.sup.*] has been given. An element a [member of] A is self-adjoint or hermitian if [a.sup.*] = a.

Let again A be a topological algebra. If A has the unit element [e.sub.A], then

[[sigma].sub.A](a) = {[lambda] [member of] c : a - [lambda][e.sub.A] [not member of] InvA},

and if A is an algebra without unit, then

[[sigma].sub.A](a) = {[lambda] [member of] C\{0} : a/[lambda] [not member of] QinvA} [union] {0}

is the (algebraic) spectrum of a [member of] A. In both cases

[[rho].sup.t.sub.A](a) = sup{[absolute value of [lambda]] : [lambda] [member of] [[sigma].sup.t.sub.A](a)}

is the (algebraic) spectral radius of A.

For noninvertive algebras with the unit element

[[sigma].sup.t.sub.A](a) = {[lambda] [member of] c : a - [lambda][e.sub.A] [member of] TinvA}

and for nonadvertive algebras

[[sigma].sup.t.sub.A](a) = {[lambda] [member of] C\{0} : a/[lambda] [not member of] TqinvA} [union] {0}

is the topological spectrum of a [member of] A. In both cases

[[rho].sup.t.sub.A](a) = sup{[absolute value of [lambda]] : [lambda] [member of] [[sigma].sub.A](a)}

is the topological spectral radius of A.

Herewith, we take [[rho].sup.t.sub.A](a) = 0 if [[sigma].sup.t.sub.A](a) = 0, and [[rho].sup.t.sub.A](a) = [infinity] if [[sigma].sup.t.sub.A](a) is an unbounded set in C, similarly as in the case of the algebraic spectrum. It is easy to see that [[sigma]].sup.t.sub.A](a) [subset or equal to] [[sigma].sub.A](a) and [[rho].sup.t.sub.A](a) [less than or equal to] [[rho].sub.A](a) for each a [member of] A.

Moreover, A is an advertive algebra if and only if [[sigma].sup.t.sub.A](a) = [[sigma].sub.A](a) for each a [not member of] QinvA. Indeed, if A is an advertive algebra, then [[sigma].sup.t.sub.A](a) = [[sigma].sub.A](a) for each a [member of] A. Let now a [member of] A \ QinvA. Then 1 [member of] [Q.sub.A](a). If now [[sigma].sub.A](a) = [[sigma].sup.t.sub.A](a), then a [not member of] TqinvA. Hence, QinvA = TqinvA in this case. Therefore, A is an advertive algebra. Similarly, a unital topological algebra is an invertive algebra if and only if [[sigma].sub.A](a) = [[sigma].sup.t.sub.A](a) for each a [not member of] InvA (see [9], p. 258).

Let now A be a topological *-algebra. Then

[[sigma].sub.A](a*) = {[bar].[mu]] : [mu] [member of] [[sigma].sub.A](a)}

for each a [member of] A. Therefore, [[sigma].sub.A](a) [subset] R, similarly [[sigma].sup.t.sub.A](a) [subset] R, if a [member of] A is self-adjoint.

Let A be a topological algebra. Then

[[beta].sub.A](a) = inf {[lambda] > 0: {[(a/[lambda]).sup.n] : n [member of] N} is bounded in A}

is the radius of boundedness of a [member of] A. It satisfies the following conditions:

[[beta].sub.A]([mu]a) = [absolute value of [mu]][[beta].sub.A](a) and [[beta].sub.A]([a.sup.k]) = [[beta].sub.A][(a).sup.k]

for each a [member of] A, [mu] [member of] C, and k [member of] N. If a, b [member of] A and the product of any two idempotent bounded subsets of A is bounded, then

[[beta].sub.A](ab) [less than or equal to] [[beta].sub.A](a) + [[beta].sub.A](b),

and if, in addition, the convex hull of an idempotent and bounded set of A is bounded (in particular, A is a locally convex algebra with continuous multiplication), then

[[beta].sub.A](a + b) [less than or equal to] [[beta].sub.A](a) + [B.sub.A](b)

(see [31], p. 281; [32], p. 310; and [19], Lemma II.9).

Herewith, if [[beta].sub.A](a) < [infinity], then a [member of] A is called a bounded element of A, and if all elements in A are bounded, then A is called a topological algebra with bounded elements.

2. FORD LEMMA FOR TOPOLOGICAL ALGEBRAS

Let A be an algebra and a [member of] A. An element b [member of] A is called the quasi-square root of a if b [omicron] b = a. When A is a unital algebra and [b.sup.2] = a, then b is called the square root of a. For each a [member of] A we put S'(a) = {[(a).sup.n] : n [greater than or equal to] 1} and (6) S(a) = [GAMMA](S'(a)).

First, we prove the following generalization of a result of Powell (7) (see [35], Lemma 1).

Theorem 2.1. Let A be a sequentially Mackey complete topological algebra. If a [member of] A and S(a) is bounded in A, then there exists an element b [member of] A such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. In particular, when [[beta].sub.A](a) < 1 and

(a) [[rho].sub.A](x) [less than or equal to] [[beta].sub.A](x) for each x [member of] A,

and

(b) [[beta].sub.A](x + y) [less than or equal to] [[beta].sub.A](x) + [[beta].sub.A](y) if x and y commute in A, hold, then there is only one quasi-square root b of a such that [[beta].sub.A](b) < 1.

Proof. Let a [member of] A be such that S(a) is bounded in A. Then S(a) is an idempotent and bounded subset of A. Since the closure B = cl(S(a)) is a closed, idempotent, bounded and absolutely convex subset in A (see, for example, [27], pp. 103, and [30], pp. 5-6), then the subalgebra [A.sub.B] of A, generated by B, is a normed algebra with respect to the submultiplicative norm [parallel] x [parallel], defined by

[parallel]x[parallel] = inf{[absolute value of [lambda]] : x [member of] [lambda]B}

for each x [member of] [A.sub.B], and the norm topology on [A.sub.B] is not weaker than the topology on [A.sub.B] induced by the topology of A (see [6], Proposition 2.2). Moreover, [A.sub.B] is complete, because A is sequentially Mackey complete, and [parallel]a[parallel] [less than or equal to] 1, because a [member of] S(a) [subset] B.

For each n [member of] n we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

converges (see [33], p. 361), it follows that ([S.sub.n]) is a Cauchy sequence in [A.sub.B]. Hence, ([S.sub.n]) converges in [A.sub.B] to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each [alpha], [beta] [member of] R and n [member of] N (see [36], formula 13, p. 616), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each x [member of] [A.sub.B] and the norm is continuous on [A.sub.B], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let now [[beta].sub.A](a) < 1, A satisfy the conditions (a) and (b), and let c [member of] A be any element such that c [omicron] c = a and [[beta].sub.A]( c) < 1. Then c [omicron] c = a = b [omicron] b, 2c = a + [c.sup.2] and 2b = a + [b.sup.2]. Therefore,

2(b [omicron] c) = (a + [b.sup.2]) + (a + [c.sup.2]) - 2bc = 2a + [(b - c).sup.2] = 2a + [(c - b).sup.2] = 2(c [omicron] b).

Hence, cb = bc. Taking this into account,

[[rho].sub.A](b + c/2) [less than or equal to] [[beta].sub.A](b + c)/2) [less than or equal to] [[beta].sub.A] (b + c/2) [less than or equal to] [[beta].sub.A](b) + [[beta].sub.A](c)/2 < 1

by conditions (a) and (b). It means that

d = b + c/2 [member of] QinvA.

Hence, there exists the quasi-inverse e [member of] A for d. Therefore, from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it follows that b = c.

Corollary 2.2. Let Abe a sequentially complete locally pseudoconvex Hausdorff algebra. If a [member of] A and (8) S'(a) is bounded in A, then there exists an element b [member of] A such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. In particular, when [[beta].sub.A](a) < 1, then there is only one quasi-square root b of a such that [[beta].sub.A](b) < 1.

Proof. In the present case, A is sequentially Mackey complete and [[rho].sub.A](a) [less than or equal to] [[beta].sub.A](a) for each a [member of] A (see [6], Corollary 4.3). Moreover, the convex hull of any idempotent and bounded set is bounded in A, because A is locally pseudoconvex and the convex hull of a set U is a subset of [GAMMA](U). Hence A satisfies condition (b) of Theorem 2.1 (see [31], p. 281). Consequently, Corollary 2.2 holds by Theorem 2.1.

Corollary 2.3. Let A be a sequentially complete locally m-convex Hausdorff algebra. If a [member of] A and [[beta].sub.A](a) < 1, then in A there exists only one quasi-square root b of a such that [[beta].sub.A](b) < 1.

Proof. In the present case A satisfies the conditions (a) and (b) of Theorem 2.1 (see [6], Proposition 4.1, and [19], Lemma II.9). Therefore, Corollary 2.2 completes the proof.

For topological unital algebras we have

Corollary 2.4. Let A be a unital sequentially Mackey complete topological algebra. If a [member of] A and S([e.sub.A] - a) (S([e.sub.A] - a/M) for some M > 1) is bounded in A, then there exists an element b [member of] A such that [b.sup.2] = a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1 (respectively, [[beta].sub.A]([e.sub.A] - b/[square root of M]) [less than or equal to] 1). In particular, when [[beta].sub.A]([e.sub.A] - a) < 1 ([[beta].sub.A]([e.sub.A] - 1/M) < 1 for some M > 1) and A satisfies (a) and (b) of Theorem 2.1, then there is only one square root b for a with [[beta].sub.A]([e.sub.A] - b) < 1 (respectively, [[beta].sub.A]([e.sub.A] - b/[square root of M]) < 1).

Proof. Since S([e.sub.A] - a) is bounded in A, then there exists an element c [member of] A such that c [omicron] c = [e.sub.A] - a or [([e.sub.A] - c).sup.2] = a and [[beta].sub.A](c) [less than or equal to] 1. Hence b = [e.sub.A] - c is a square root of a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1. If now [[beta].sub.A]([e.sub.A] - a) < 1 and A satisfies (a) and (b) of Theorem 2.1, then there is only one square root for a by Theorem 2.1.

If S([e.sub.A] - a/m) is bounded in A for some M > 1, then the proof is similar.

Similarly to Corollaries 2.2 and 2.3 the following corollaries hold.

Corollary 2.5. Let A be a unital sequentially complete locally pseudoconvex Hausdorff algebra. If a [member of] A and S'([e.sub.A] - a) is bounded in A, then there exists an element b [member of] A such that [b.sup.2] = a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1. In particular, when [[beta].sub.A] ([e.sub.A] - a) < 1, then there is only one square root b of a such that [[beta].sub.A]([e.sub.A] - b) < 1.

Corollary 2.6. Let A be a unital sequentially complete locally m-convex Hausdorff algebra. If a [member of] A and [[beta].sub.A]([e.sub.A] - a) < 1, then in A there exists only one square root b of a such that [[beta].sub.A]([e.sub.A] - b) < 1.

Now we consider the case when A is a topological *-algebra.

Theorem 2.7. Let A be a sequentially Mackey complete topological *-algebra. If a [member of] A and S(a) is bounded in A, then there exists an element b [member of] A such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. In particular, when

(c) the involution a [right arrow] [a.sup.*] in A is continuous

or

(d) a has only one quasi-square root in A, then b is self-adjoint if a is self-adjoint.

Proof. By hypothesis and Theorem 2.1, there exists an element

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that b [omicron] b = a and [[beta].sub.A](b) [less than or equal to] 1. Let now [a.sup.*] = a. If A satisfies (c), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and if A satisfies condition (d), then [b.sup.*] = b, because [b.sup.*] [omicron] [b.sup.*] = [a.sup.*] = a.

Corollary 2.8. Let A be a unital sequentially Mackey complete topological *-algebra. If a [member of] A and S([e.sub.A] - a) is bounded in A, then there exists an element b [member of] A such that [b.sup.2] = a and [[beta].sub.A] ([e.sub.A] - b) [less than or equal to] 1. In particular, when A satisfies condition (c) of Theorem 2.7 or condition

(e) a has only one square root in A,

then b is self-adjoint if a is self-adjoint.

Corollary 2.9. Let A be a unital sequentially Mackey complete topological *-algebra. If a [member of] A is self-adjoint and S([e.sub.A] - a) is bounded in A, then there exists a self-adjoint element b [member of] A such that [b.sup.2] = a and [[beta].sub.A]([e.sub.A] - b) [less than or equal to] 1.

3. SOME RESULTS FOR TOPOLOGICAL *-ALGEBRAS

In the sequel we need the following result.

Proposition 3.1. Let A be a commutative simplicial Gelfand-Mazur algebra with nonempty set m(A). Then

(i) [sp.sup.t.sub.A](a)\{0} [subset] {[phi](a) : [phi] [member of] homA} [subset] [sp.sup.t.sub.A](a) for each a [member of] A

and

(ii) [sp.sup.t.sub.A](a) = {[phi](a) : [phi] [member of] homA} for each a [member of] A if A is a unital algebra.

Proof. (i) Take a [member of] A and [mu] [member of] [sp.sup.t.sub.A](a)\{0}. Then a/[mu] [not member of] TqinvA. Therefore, the set

I = cl{a/[mu] b - b : b [member of] A}

cannot contain a/[mu], otherwise there is a net [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] in A such that [(a/[mu] [a.sub.[lambda]] - [a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] converges to a/[mu] in A or [(a/[mu] [omicron] [a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] converges to [[theta].sub.A]. This means that a/[mu] [member of] TqinvA. Hence, I [not equal to] A. Therefore, I is a closed regular ideal in A. Since A is simplicial, there exists a closed maximal regular ideal M in A such that I [subset] M and, since A is a Gelfand-Mazur algebra, then M = ker [phi] for a [phi] [member of] homA. Consequently, [phi](a/[mu] b - b) = 0 for each b [member of] A. Hence, [phi](a) = [mu] (because [phi] is not trivial). This shows that

[sp.sup.t.sub.A](a)\{0} [subset] {[phi](a) : [phi] [member of] homA}.

Let now a [member of] A and [mu] = [phi](a) for some [phi] [member of] homA. We must show that [mu] [member of] [sp.sup.t.sub.A](a). We suppose that [mu] [not equal to] 0 and [mu] [not member of] [sp.sup.t.sub.A](a). Then a/[mu] [member of] TqinvA and so there is a net [([c.sub.[alpha]]).sub.[alpha][member of A]] such that [(a/[mu] [omicron] [c.sub.[alpha]]).sub.[alpha][member of]A] converges to [[theta].sub.A] in A. Since [phi] is continuous, [([phi](a)/[mu] [omicron] [phi][([c.sub.[alpha]])).sub.[alpha][member of A]] converges to 0, but it is not possible, since [phi](a) = [mu]. Consequently, [mu] [member of] [sp.sup.t.sub.A](a) if [mu] [not equal to] 0.

Let now [phi](a) = 0. If A does not have a unit, then automatically 0 [member of] [sp.sup.t.sub.A](a) and so [phi](a) [member of] [sp.sup.t.sub.A](a). If A has a unit and 0 [not member of] [sp.sup.t.sub.A](a), then a [member of] TqinvA. Therefore, there exists a net [(a[beta])[beta]).sub.[beta][member of]B] such that [([aa.sub.[beta]]).sub.[beta][member of]B] converges to [e.sub.A] in A. Then [([phi](a)[phi]([a.sub.[beta]])).sub.[beta][member of]B] converges to 1. But this is impossible, since [phi](a) = 0. (ii) Let now A be a unital algebra. By statement (i), it is sufficient to show that 0 [member of] [sp.sup.t.sub.A](a) if and only if [phi](a) = 0 for some [phi] [member of] homA.

Suppose first that 0 [member of] [sp.sup.t.sub.A](a). Then a [not member of] TinvA and, similarly as above,

I = cl{ab : b [member of] A}

is a closed ideal in A. Since A is a commutative unital simplicial Gelfand-Mazur algebra, there exists a M [member of] m(A) such that I [subset] M = ker [phi] for some [phi] homA. Therefore, [phi](a)[phi](b) = 0 for each b [member of] A. Again, since [phi] is not trivial, [phi](a) = 0.

Suppose next that [phi](a) = 0 for some [phi] [member of] homA. Then a [not equal to] TinvA. Otherwise, there exists a net [([a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] such that [(a[a.sub.[lambda]]).sub.[lambda][member of][LAMBDA]] converges to [e.sub.A] in A. Then [([phi](a)[phi]([a.sub.[lambda]])).sub.[lambda][member of][LAMBDA] converges to 1 contrary to [phi](a) = 0. Consequently, in this case 0 [member of] [sp.sup.t.sub.A](a).

Corollary 3.2. Let A be a commutative advertive simplicial Gelfand-Mazur algebra with nonempty set m(A). Then

(i) [sp.sub.A](a)\{0} [subset] {[phi](a) : [phi] [member of] homA} [subset] [sp.sub.A](a) for each a [member of] A

and

(ii) [sp.sub.A](a) = {[phi](a) : [phi] [member of] homA} for each a [member of] A if A is an invertive algebra.

Proof. In the present case [sp.sub.A](a) = [sp.sup.t.sub.A](a) for each a [member of] A. Therefore, the statements hold by Corollary 3.2.

Corollary 3.2 (ii) was proved in [3], Proposition 5. Moreover, it was shown in [3], Proposition 6, that every topological algebra for which

[sp.sub.A](a) = {[phi](a) : [phi] [member of] homA}

for each a [member of] A is an advertive algebra.

Proposition 3.3. Let A be a unital sequentially Mackey complete locally pseudoconvex Hausdorff algebra for which [[beta].sub.A](a) = [[rho].sub.A](a) for each a [member of] A. If, in addition, A satisfies the condition (9)

(f) [sp.sub.A](a) is a closed subset in c for each a [member of] A with [sp.sub.A](a) C (0,1), then for every element a [member of] A with [sp.sub.A](a) [subset] (0, [infinity]), there exists an element (10) b [member of] A such that [b.sup.2] = a. In particular, when every maximal commutative unital subalgebra B of A is an invertive simplicial Gelfand-Mazur algebra, then [sp.sub.A](b) C (0, [infinity]).

Proof. Let a [member of] A be such that [sp.sub.A](a) [subset] (0, [infinity]). If [[rho].sub.A]([e.sub.A] - a) < 1, then [[beta].sub.A]([e.sub.A] - a) < 1 by assumption. Therefore (see the proof of Theorem 2.1 and Corollary 2.5), there exists an element

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that [b.sub.2] = a and [[rho]/sub.A]- b) < 1.

Let B be a maximal commutative unital subalgebra of A, containing a. If now B is an invertive simplicial Gelfand-Mazur algebra, then TinvA = InvA, homB is not empty and

{[phi](a) : [phi] [member of] hom B} = [sp.sub.B](a) = [sp.sub.A](a) [subset] (0,[infinity])

for each a [member of] B by Corollary 3.2. Therefore, [phi](a) > 0 for each [phi] [member of] homB. Hence (by the formula (3), p. 361 from [33])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each [phi] homB. Consequently,

[sp.sub.A](b) = [sp.sub.B](b) = {[phi](b) : [phi] homB} [subset] (0, [infinity]).

Let again a [member of] A be such that [sp.sub.A](a) [subset] (0, [infinity]). Since -1 [not equal to][sp.sub.A](a), then [e.sub.A] + a [member of] InvA. Let c = [([e.sub.A] + a).sup.-1], v = ac and B be a maximal commutative unital subalgebra of A, containing [e.sub.A], a, and c. Then v = [e.sub.A] - c [member of] B and [sp.sub.B](x) = [sp.sub.A](x) for each x [member of] B. Since

[sp.sub.A](c) = {1/[mu] + 1 : [mu] [member of] [sp.sub.A](a)} [subset] (0,1)

(see, for example, the equality (4.14) in [22]), then

[sp.sub.A](v) = [sp.sub.A]([e.sub.A] - c) = 1 - [sp.sub.A](c) [subset] (0, 1)

and

[sp.sub.A]([e.sub.A] - v) = 1 - [sp.sub.A](v) [subset] (0, 1).

Therefore [[rho].sub.A]([e.sub.A] - c) < 1 and [[rho].sub.A]([e.sub.A] - v) < 1 by condition (f). Thus, by the first part of the proof, there exist z, w [member of] A such that [z.sup.2] = c and [w.sup.2] = v. Moreover, if B is a commutative invertive simplicial Gelfand-Mazur algebra, then

[sp.sub.A](z) = [sp.sub.B](z) = {[phi](z) : [phi] [member of] homB} [subset] (0,[infinity])

and

[sp.sub.A](w) = [sp.sub.B](w) = {[phi](w) : [phi] [member of] homB} [subset] (0,[infinity]).

Taking this into account, z, w [member of] InvA, from [z.sup.2] = c it follows that [([z.sup.-1]).sup.2] = [e.sub.A] + a and b = [z.sup.-1] w is the square root of a. Since

[sp.sub.B](b) = {[phi](b): [phi] [member of] homB} = {[phi]([z.sup.-1])[phi](w):[phi] [member of] homB} = {[phi](w)/[phi](z) : [phi] [member of] homB},

[sp.sub.B](z) [subset] (0,[infinity]) and [sp.sub.B](w) [subset] (0,[infinity]), then

[phi](w)/[phi](z) > 0

for each [phi] [member of] homB. Consequently, [sp.sub.A](b) = [sp.sub.B](b) [subset] (0,[infinity]).

Corollary 3.4. Let A be a unital complete locally m-(k-convex) Hausdorff algebra with bounded elements and k [member of] (0, 1]. Then for every element a [member of] A with [sp.sub.A](a) [subset] (0, [infinity]) there exists an element (11) b [member of] A such that [b.sup.2] = a and [sp.sub.A](b) [subset] (0,[infinity]).

Proof. Take a [member of] and B a maximal commutative unital closed subalgebra of A, containing a. Then B is a commutative unital complete Hausdorff locally m-(k-convex) algebra with bounded elements. Therefore, [[beta].sub.A](a) = [[rho].sub.A](b) for each b [member of] B (see [6], Corollary 4.4) and [sp.sub.B](b) is a closed subset in C (see the proof of Proposition 3.2 in [7], pp. 203-204). Since [sp.sub.A](b) = [sp.sub.B](b) and [[beta].sub.A](b) = [[beta].sub.B](b) for each b [member of] B, then [[beta].sub.A](a) = [[rho].sub.A](a) and condition (f) holds. Moreover, every maximal commutative unital (not necessarily closed) subalgebra of A is an invertive (by Corollary 2 in [3]) simplicial (by Corollary 5 in [5]) Gelfand-Mazur algebra (see, for example, [2], Corollary 2, or [8], Theorem 3.3). Hence, the result follows from Proposition 3.3.

Theorem 3.5. Let A be a unital sequentially Mackey complete topological *-algebra with continuous involution, for which [[beta].sub.A](a) = [[rho].sub.A](a) for each a [member of] A. If, moreover, A satisfies the condition (12)

(g) [sp.sub.A](a) is a closed subset in c for each self-adjoint a [member of] A with [sp.sub.A](a) [subset] (0,1),

then for every self-adjoint element h [member of] A with [sp.sub.A](h) [subset] (0, [infinity]) there exists a self- adjoint element (13) u [member of] A such that [u.sup.2] = h. In particular, when every maximal commutative unital *-subalgebra B of A is an invertive simplicial Gelfand-Mazur *-algebra, then [sp.sub.A](u) [subset] (0, [infinity]).

Proof. The proof is similar to that of Proposition 3.3. Herewith, u is self-adjoint by Corollary 2.8.

Corollary 3.6. Let A be a unital complete locally m-(k-convex) Hausdorff *-algebra with continuous involution. If all elements of A are bounded, then for each self-adjoint element h [member of] A with [sp.sub.A](h) [subset] (0,[infinity]) there exists a self-adjoint element (14) u [member of] A such that [u.sup.2] = h and [sp.sub.A](u) [subset] (0, [infinity]).

Proof. The proof is similar to that of Corollary 3.4.

Theorem 3.7. Let A be a topological *-algebra in which every maximal commutative *-subalgebra is an advertive simplicial Gelfand-Mazur *-algebra and let [h.sub.1], ..., [h.sub.n] be self-adjoint elements in A such that [sp.sub.A]([h.sub.k]) [subset] [0,[infinity]) for each k with 1 [less than or equal to] k [less than or equal to] n. Then

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [0,[infinity]).

Proof. Since [h.sub.1] + ... + [h.sub.n] is self-adjoint,

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [0,[infinity]).

Let B be a maximal commutative *-subalgebra of A which contains all elements [h.sub.1], ..., [h.sub.n]. Since B is an advertive simplicial Gelfand-Mazur algebra, by assumption, hom B is not empty. Let [lambda] be an arbitrary negative real number. It is known that

[sp.sub.A]([h.sub.k]) [union] {0} = [sp.sub.B]([h.sub.k)] [union] {0}

by Lemma 4.11 from [41], p. 47. As [sp.sub.B]([h.sub.k]) [subset] [0,[infinity]) for each k, it follows that [h.sub.k]/[lambda] [member of] QinvB for each k. Therefore [phi]([h.sub.k]/[lambda]) [not equal to] 1 or [phi]([h.sub.k]) [not equal to] [lambda] for each k. It means that [phi]([h.sub.k]) [greater than or equal to] 0 for each k. Hence

[phi]([h.sub.1] + ... + [h.sub.n]/[lambda]) = [phi]([h.sub.1]) + ... + [phi]([h.sub.n])/[lambda] [less than or equal to] 0

for each [phi] [member of] hom B. Namely by [3], p. 20,

[h.sub.1] + ... + [h.sub.n]/[lambda] [member of] TqinvB = QinvB

for each [lambda] < 0, since B is advertive. Consequently,

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [sp.sub.B]([h.sub.1] + ... + [h.sub.n]) [union] {0} [subset] [0, [infinity]).

Corollary 3.8. Let A be a complete locally m-pseudoconvex Hausdorff *-algebra and [h.sub.1, ...,][h.sub.n] be self- adjoint elements in A such that [sp.sub.A]([h.sub.k]) [subset] [0, [infinity]) for each k with 1 [less than or equal to] k [less than or equal to] n. Then

[sp.sub.A]([h.sub.1] + ... + [h.sub.n]) [subset] [0, [infinity]).

Proof. Let B be a maximal commutative closed *-subalgebra of A. Then B is a commutative complete locally m-pseudoconvex Hausdorff *-algebra. Since, as above, B is an advertive simplicial Gelfand-Mazur *-algebra, by Theorem 3.7 the proof is complete.

Remark 3.9. Notice that Theorem 2.1 for fundamental Frechet algebras was proved partly in [10], Theorem 3.2, and for pseudocomplete locally convex algebras partly in [35], Lemma 1; Corollary 2.2 for complete locally m-pseudoconvex algebras (using projective limits of p-Banach algebras) was proved partly in [11], Theorem 5.3.4; Corollary 2.3 for complete locally m-convex algebras (using projective limits of Banach algebras) was proved in [38], Theorem 3.9, and in [22], Theorem 5.5.4; Corollary 2.4 for fundamental Frechet algebras was proved partly in [10], Theorem 3.3; Corollary 2.5 for complete unital locally m-pseudoconvex algebras was proved partly in [11], Corollary 5.3.5, and for unital complete locally m-convex algebras in [22], Corollary 5.5.5, and partly in [25], Corollary 1.13; Corollary 3.2 for Banach algebras was proved in [41], Theorem 3.12, and for commutative locally m-convex Q-algebras in [29], pp. 74-76; Corollary 3.4 for complete unital locally m-convex algebras was proved in [39], Theorem 2.2, and in [22], Theorem 5.5.8; and Corollary 3.8 has been proved mostly for [C.sup.*]-algebras (see, for example, [37], Lemma 4.7.4, or [41], Lemma 6.4).

On toestatud Fordi lemma analoog teatud liiki topoloogiliste algebrate (erijuhul topoloogiliste *-algebrate) jaoks ja saadud tulemusi on kasutatud topoloogiliste *-algebrate omaduste kirjeldamisel.

doi: 10.3176/proc.2011.2.01

ACKNOWLEDGEMENTS

We gratefully acknowledge the financial support of the Estonian Science Foundation (grant 7320) and of the Estonian Targeted Financing Project (SF0180039s08).

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Mart Abel and Mati Abel *

Institute of Pure Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia

Received 9 March 2010, accepted 12 October 2010

* Corresponding author, mati.abel@ut.ee

(1) A set U in A is pseudoconvex if there is a [mu] > 0 such that U + U [subset] [mu]U.

(2) A set U in A is idempotent if UU [subset] U.

(3) A seminorm p on A is nonhomogeneous if p([lambda]a) = [[absolute value of [lambda]].sup.k]p(a) for each a [member of] A and [lambda] [member of] C, where the power of homogeneity is k = k(p) [member of] (0,1], and p is submultiplicative if p(ab) [less than or equal to] p(a)p(b) for each a,b [member of] A.

(4) A topological algebra is locally bounded if it has a bounded neighbourhood of zero.

(5) It is known (see [24], p. 122) that there exist Cauchy sequences which are not Mackey-Cauchy sequences.

(6) The absolutely convex hull of S [subset] A is the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(7) He considered the case when A is a pseudocomplete locally convex *-algebra.

(8) Since A is locally pseudoconvex, then S(a) is bounded by the boundedness of S' (a).

(9) If A is a Q-algebra, then condition (f) is superfluous, because in this case the spectrum of every element of A is closed (see, for example, [29], Proposition 4.2).

(10) When [[beta].sub.A]([e.sub.A] - a) < 1, then a has only one square root by Theorem 2.1.

(11) See footnote 8.

(12) See footnote 7.

(13) See footnote 8.

(14) See footnote 8.

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Author: | Abel, Mart; Abel, Mati |
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Publication: | Proceedings of the Estonian Academy of Sciences |

Article Type: | Report |

Geographic Code: | 4EXES |

Date: | Jun 1, 2011 |

Words: | 7370 |

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