21. Function Algebras

22. Banach Algebras with an Involution

23. B‡ -Algebras and the Gelfand-Naimark Theorem

24. Positive Functionals and M(A)

25. Tensor Products of Banach Algebras

If IJ and are finite-dimensional vector spaces with dimensions 78 and , respectively, then we can think of their tensor product IŒJ as " I vectors whose entries are from J78." Thus the tensor product is an -dimensional vector space. To extend this concept to Banach spaces and algebras we need a way of extending the tensor product to infinite dimensional vector spaces, as well as a way of defining an appropriate norm on the tensor product. Two understand the general situation we first consider the important special case of Banach algebras which are "function algebras". This case will serve as a concrete example for the definition of tensor products on a general Banach algebra.

Tensor Products for Function Algebras

Recall that if \GÐ\Ñ is a compact Hausdorff space then is the algebra of complex- valued continuous functions on \ .

Definition: A function algebra is a subalgebra of GÐ\Ñ which (1) contains the constants, (2) separates points of \ , and (3) is uniformly closed.

Note that a function algebra which is also self-adjoint is all of GÐ\Ñ by the Stone-Weierstrass theorem.

Examples: 1. GÐ\Ñ itself is a function algebra 2. The algebra EÐHÑ can be considered a closed subalgebra of GÐHÑ , so it's a function algebra. We can also consider EÐHÑ as a subalgebra of GГ Ñ , so it's a function algebra in a different way.

Definition: Let EF and be function algebra is a subalgebra of GÐ\Ñ which (1) contains the

Tensor Products of Banach Algebras Let EF and be vector spaces. A map "Š : EF‚Ä is called bilinear if it is linear in each variable separately; that is,

(a) for yF−ÈÐÑ , the map x" xy , is linear

(b) for xE−ÈÐÑ , the map y" xy , is linear

Let BLÐÑEF , ;ŠŠ denote the vector space of all bilinear maps from E‚ F into .

Definition: Given xE−−Œ and yFxy , let be the element of BL Ð EFww , ;Š Ñ given by

xyfgfxgyŒÐÐÑœÑÐÑ, () fE, −−ww gF

The algebraic tensor product EFŒŒ of EF and is the linear span of the set Ö xy : xE,−−×Ð yF in BL EFww , ;Š Ñ . The canonical bilinear map is the mapping

, :EF‚ŒÒ EF ÐÑ x,. yØ xŒ y

The algebraic tensor product satisfies the following universal property: If " : EF‚ÄŠ is any bilinear mapping then there exists a unique linear map -"-, such that œ‰ .

If EF and are algebras then their tensor product is also an algebra under according to the following definition.

Definition: If AB and are algebras then ABΠthe algebraic tensor product of AB and is endowed the multiplication

ÐÑÐÑœxŒŒ y z w xz Œ yw

With this multiplication EŒF is an algebra.

Examples: 1. If Iœ‘‘78 and Jœ then IŒJ is isomorphic to ‘ 78 .

2. If E œ GÐ\Ñ and F œ GÐ]Ñ then EŒF is isomorphic to the subalgebra of GÐ\ ‚]Ñ spanned by functions of the form

B Œ CÐ=ß >Ñ œ BÐ=ÑCÐ=Ñ

The identity element of EŒF is "Œ" . Note that EŒF separates points of \‚] .

Tensor Products for Banach Algebras f EF and are locally convex spaces we endow EFŒ with the finest locally convex topology for which the canonical bilinear map , :EF‚ÄŒ EF is continuous. This topology, called the projective tensor product topology (or simply projective topology ) is described as follows. Let hi and be neighborhood bases at 0 for E and F , respectively. Then a neighborhood base at 0 for the projective topology on EFŒ is given by the family ÖÐ>hi>UV:UŒÑ − , V − × , where Ð UV ŒÑ denotes the convex, circled hull of UVŒ . If the topologies of E and F are generated by the directed families of seminorms cd and , respectively, then the projective topology on EFŒ is generated by the directed family of seminorms Ö×pqpŒ− :cd , q − . Here pq Œ is the seminorm defined by pquinf pxqyu : x y ŒœÐÑœ Ö!ii ÐÑÐÑii! ii Œ × where the infimum is taken over all representations of u of the form uxy œŒ!i ii If p is the gauge of U and q is the gauge of V , then the seminorm pŒ q is the gauge of >ÐÑUVŒŒ. The seminorm pq has the property pŒŒ qxÐÑœÐÑÐÑ y px qy for all xE,−− yF . If E and F are locally convex spaces then the projective tensor product EŒ 1 F is the completion of EFŒ in the projective tensor product topology. If EF and are metrizeable locally convex spaces, then EFŒ 1 is a Frechet´ space.

Tensor products of Frechet´ algebras

If AB and are algebras then ABŒ the algebraic tensor product of AB and , is an algebra with the multiplication ÐÑÐÑœxŒŒ y z w xz Œ yw Let AB and be locally m -convex algebras with defining families of seminorms cd and , respectively. As we have shown in Section 1.1, the system of seminorms ÖpqpŒ− : c , qABpq−ŒŒd×, defines a locally convex topology on . The seminorm is defined by pquinf pxqyu : x y ŒÐÑœ Ö!!ii ÐÑÐÑii œ ii Œ × where the infimum is taken over all representations of u of the from uxy œ !i iiŒ In the case where AB and are algebras and pq and are submultiplicative, the following calculation shows that the seminorm pqŒŒ is a submultiplicative seminorm on AB . Let r p q, and u x y , v z w be elements of A B . Then œŒ œ!!ijiiŒŒ œ j j Œ

ruvÐÑœŒ pŒ q xz yw Š!ij ij i j‹

pxz qyw ŸÐÑÐÑ!ij ij i j

px pz qy qw Ÿ!ij ÐÑÐÑÐÑÐiji j Ñ œpx ÐÑÐÑ qy pz ÐÑÐ qw Ñ Š‹Š‹!!ijii j j

Taking the infimum with respect to all representations of uv and as above gives

ruvÐÑŸÐÑÐÑ ru rv

It follows that ABŒŒ is a locally m -convex algebra and its completion A1 B is a complete locally m -convex algebra.

1.3.28 Theorem Let A and B be locally m -convex Q -algebras. Then ```ÐŒAB1 Ѷ ÐÑ‚ÐÑ A B That is, the maximal ideal space of ABŒ 1 is homeomorphic with the product of the maximal ideal spaces of AB and . Proof. See Mallios [Ml" ].

Since a Banach algebra is always a Q-algebra we have the following notable corollary.

1.3.29 Corollary If A and B are Banach algebras then ```ÐѶÐÑ‚ÐÑABŒ 1 A B 26. Introduction to Topological Algebras

A topological algebra is an algebra which is also a topological space such that the algebra operations are continuous. The most useful topological algebras are called locally m- convex algebras; these are natural generalizations of Banach algebras. In fact, every complete locally m- convex algebra can be represented as the inverse limit of Banach algebras. This representation proves useful in transferring many results from Banach algebras to locally m- convex algebras. We begin by giving the definition of a topological algebra.

Definition: A topological algebra is an algebra A over Š together with a Hausdorff topology 77 such that ÐÑA , is a topological vector space and such that the algebra multiplication is jointly continuous; that is,

ÐÑÈx, y xy is continuous from EE‚Ä E .

If h is a base for the topology at 0 in A , then the joint continuity of the multiplication at 0 means that for every UV−−hh there exists such that

VU# §

Locally m-Convex Algebras

To give a useful generalization of Banach algebras we require that a topological algebra have a local base at ! consisting of sets that are "convex" with respect to multiplication, according to the following definition.

Definition: Let E be an algebra. 1. A convex subset UA of is called m -convex (or multiplicatively convex ) if U# § U . 2. A seminorm :E on is called submultiplicative if pxy ÐÑŸÐÑÐÑ px py for x,y− A.

21.1 Theorem: 1. If Z7 is -convex, circled, and absorbing, then the gauge :Z is a submultiplicative seminorm. 2. If :ZœÖB−EÀ:ÐBÑ×7 is submultiplicative seminorm then 1 is -convex, circled, and absorbing.

Proof: 1. We know that :Z is a seminorm (Lemma 4, Appendix 9), it remains to show that it is submultiplicative:

pZZÐBÑ p ÐCÑœ inf Ö--.. !ÀB− Z׆ inf Ö !ÀC− Z×

inf Ö-. !ÀBC− - Z . Z× œinf Ö-. !ÀBC− -. Z# ×

inf Ö-. !ÀBC− -. Z× Z is 7-convex

ÐBCÑ pV 2. Exercise.

Definition: A topological algebra with a neighborhood base for the topology at 0 consisting of m -convex sets is called a locally m-convex algebra .

21.2 Theorem: The topology of a locally m -convex algebra E is generated by a directed family c of submultiplicative seminorms.

Proof: If Y7 is -convex then its circled convex hull is again 7 -convex (exercise). So, if hh is a neighborhood base at !7 consisting of -convex sets, we can replace by the base ihconsisting of the circled convex hull of each element in . The corresponding gauge functionals :Z−Z , i are a family of submultiplicative seminorms (by Theorem 21.1) giving the topology of E . If :"# ß: ßáß: 8 are submultiplicative seminorms, then :ÐBÑ œmax Ö:"# ÐBÑß : ÐBÑß á ß : 8 ÐBÑ× is also submultiplicative. Thus the collection i can be replaced by an equivalent directed family of seminorms as explained in Theorem 6 of Appendix 9.

A Locally m-Convex Algebras is a Sub algebra of a Products of Banach Algebras

Complete locally m -convex algebras share several key properties with Banach algebras. The proofs of many of these follow from the representation of a locally m -convex algebra as the inverse limit of Banach algebras, which we now describe. Let Am be a locally -convex algebra and let c œÖ p! :!A − × be a directed family of submultiplicative seminorms generating the topology of A . Let

KxApx!!œÖ− :0 ÐÑœ ×

be the kernel of pK!! . is a closed ideal of A . Define the quotient algebra A !œ A/K ! and let 1!!: AAÄ

be the natural homomorphism. Clearly A is a normed algebra under the norm 1 ÐÑx ~ ! ll! ! œÐÑB−Epx!!, . Let A be the completion of A ! with respect to this norm. Clearly 1 ! has a natural extension to a continuous homomorphism ~~ 1~ : AAÄ ~ !! where AA is the completion of . 21.3 Theorem: The topology of a locally me -convex algebra with identity can be w generated by a family c œÖp! :!A − × of submultiplicative seminorms for which w pe!ÐÑœ1, !A −

Proof. For each !A1− , !!! ÐÑ e is an identity in the normed algebra Ð A , ²†²Ñ . Thus Ae can be renormed by the norm ²†²w so that 1 ÐÑw œ 1. The !!! ll! submultiplicative seminorm on A defined by

pxw ÐÑœ1 ÐÑ xw ! ll! ! w is equivalent to pp! . Thus the family Ö−×! :!A of submultiplicative seminorms generates the original topology of A .

21.4 Theorem: A (complete) locally mA -convex algebra is isomorphic to a (closed) subalgebra of the Cartesian product of Banach algebras. ~ Proof. Let A A . The map p œ # ! !&A 1 :AAÄ p xxÈÐ1! ÐÑÑ

is an embedding since it is clearly one-to-one and the topology of A is the weak topology induced by 11 . So is an isomorphism onto its image. If AA is complete then 1ÐÑ is closed in Ap .

A Locally m-Convex Algebras is an Inverse Limit of Banach Algebras

Let Am be a locally -convex algebra whose topology is given by a directed family of seminorms c œÖp! :!A − × . We define a partial order on the index set A as follows: For !",− A set !"¡ÐÑŸÐÑ iff px!" px ( x− A )

The set Ac is directed under this partial order because the family is a directed set under Ÿ¡. For !" define the map 1!", :AA "Ä ! 11"!ÐÑÈxx ÐÑ The maps 1!", are homomorphisms since 11!",,ÐÐÑÐÑÑ "xy 1 " œÐÐÑÑ 11 !" " xy œ111!!! Ðxy Ñœ ÐÑ x ÐÑ y œ11!",, Ð " ÐÑÑxy 11 !" Ð " ÐÑÑ

As a consequence of the inequality pxÐÑŸ px ÐÑ we get 11 Ð ÐÑÑ x Ÿ 1 ÐÑ x !"llll !""", !" and thus each 11!",, is continuous. It follows that each !" can be extended by continuity to a continuous homomorphism ~ ~~ 1!", : AA "Ä ! Unless otherwise stated, for the rest of this section we assume that E is a complete locally 7-convex topological algebras.

~ ~ 21.5 Theorem: The family ÐßA!!"1 , Ñ!"&A, of Banach algebras and maps is an inverse system and ~ E¶lim ÐA , 1~ Ñ Ã !!",

(where ¶ denotes an algebraic and topological isomorphism.)

~ ~ Proof. First we note that limÐÑ A!!" ,1 , is a complete locally m -convex algebra. By à ~ Theorem 21.5, AAAA is isomorphic to the closed subalgebra 11ÐÑ of ! . Clearly Ðѧ ~~~~~# limÐÑAAAAA!!" ,11,, . Moreover, ÐÑ is dense in lim ÐÑ !!" , 1 because ! is dense in ! for ÃÃ~ !A−ÐÑœÐÑ. So 1AA lim , 1~ . This completes the proof. à !!", ~ ~ 21.6 Corollary: If ÐÑxAxx!!!""!!%A − with 1!"A, ÐÑœ for all , − , then there ~ # exists xA− such that 1!!ÐÑœxx.

The next two theorems give some immediate consequences of the inverse limit representation. Recall that KÐEÑ denotes denote the group of invertible elements of A .

~ ~ 21.7 Theorem: xGA− Ð Ñ if and only if 1!A!! ÐÑ− x GA Ð Ñ for all − .

Proof. If x−ÐÑ G A then

~~" ~~ " 11!!ÐÑœexxxx Ð Ñœ 11 !! ÐÑ Ð Ñ ~ ~ so 1!!ÐÑ−xGA Ð Ñ. ~ ~ Conversely, if 1!A!!ÐÑ−xGAÐ Ñ for every − , then

~~~~~~" 111111!!""!"""ÐÑeeœ,, ÐÐÑÑœ ÐÐÑÐÑ xx

~~ ~~" œÐÐÑÑÐÐÑÑ11!",, "xx 11 !" !

~~~" œÐÑÐÐÑÑ111!!""xx,

~~" ~ " ~ Thus 11!", ÐÐ "xx ÑÑœ 1 ! ÐÑ . By Corollary 1.3.6 there exists yA− such that 1! ÐÑy ~" ~ ~~ ~~ " ~ œÐÑ1!x. For !A1 − , ! ÐÑœÐÑÐÑœÐÑÐÑœÐÑ xyxyxxexy 11 !! 11 !! 1 ! . Thus œ e and the result follows.

In general GAÐÑ is not a topological group since the operation of taking inverses is not necessarily continuous. For 7 -convex algebras we have the following result. 21.8 Theorem: The map xÈ x-" is continuous on GAÐÑ . In particular, GA ÐÑ is a topological group.

~ Proof. Let ÖקÐÑ x- G A be a net converging to x! . For every !A1 − , !- ÐÑÄ x ~~~" " 111!ÐÑxxx! . By the continuity of inversion in Banach algebras !- ÐÑ Ä ! ÐÑ! . But ~~" " " " 11!!ÐÑœÐÑxxxx so - Ä! .

A fundamental property of Banach algebras is the Gelfand-Mazur Theorem. The following generalization of this theorem to Frechet algebras is essential in what follows.

21.9 Theorem: (Gelfand-Mazur) Let A be a complete locally 7 -convex algebra over ‚ . If every nonzero element in AA is invertible then is isomorphic to the field of complex numbers.

Proof: The proof is the same as the one for Banach algebras (Theorem 2.7). Recall that the proof required two things: that there exists 0−Ew , and that the operation of taking an inverse be continuous. Since E is a locally convex space there are a lot of continuous linear functionals on E (Appendix 9) and inversion is continuous by the preceding theorem, so the Gelfand-Mazur theorem holds. Note that, any topological algebra satisfying these two conditions has the Gelfand-Mazur property.

Frechet Algebras

A locally convex topological algebra is metrizable if and only if its topology can be defined by a countable family of submultiplicative seminorms, and hence by an increasing sequence of submultiplicative seminorms

:Ÿ:Ÿ:Ÿá"#$ (See Appendix 9)

Definition: A Frechet´ algebra is a complete metrizable locally m -convex algebra.

So a Frechet algebra is an inverse limit of a sequence of Banach algebras:

~~~:::"#$ AAA"#$ÑÑÑâ ~ where AAA/K555 is the completion œ87 in the quotient norm and for the map ~ ~~ 1187, ÀÄAA 7 8 is the extension of the natural maps 8ß7 ÀEÄE 7 8 .

Examples: (Frechet Algebras) 1. Let W be the algebra of complex sequences with pointwise operations equipped with the seminorms :85 ÐBÑœmax ÖlBlÀ"Ÿ5Ÿ8×

where BœÐBÑ−W5 . Then W is a Frechet algebra. 2. Let GБ‘ Ñ be the algebra of continuous complex-valued functions on with pointwise operations and the seminorms

:8 ÐBÑ œsup ÖlBÐ>Ñl À 8 Ÿ > Ÿ 8×

Then GБ Ñ is a Frechet algebra.

3. Let G_ Ò!ß "Ó be the algebra of infinitely differentiable functions on Ò!ß "Ó with pointwise operations and the seminorms

Ð8Ñ :8 ÐBÑ œsup ÖlB Ð>Ñl À 8 Ÿ > Ÿ 8×

where B8Ð8Ñ denotes the th derivative of BGÒ!ß"Ó . Then _ is a Frechet.

4. Let X be the algebra of entire functions with pointwise operations and the seminorms

:8 ÐBÑœsup ÖlBÐ>ÑlÀl>lŸ8×

Then X is a Frechet algebra.

Multiplicative Linear Functionals

Let AA be a topological algebra. Recall that a multiplicative linear functional on is a linear functional fA :ÄŠ that satisfies fxy Ð Ñœ fxfy ÐÑ ÐÑ for all x,y− A . We make the following definitions:

`#ÐÑAA is the set of nonzero multiplicative linear functionals on .

``ÐÑAA is the set of continuous elements of # ÐÑ .

# ``!ÐÑAA is the set of members of ÐÑ that are continuous with respect to p! .

Clearly we have the following relationship between these sets:

# ```!ÐѧAA Ðѧ ÐÑ A

(See the definition of a continuous linear map in Appendix 9). It is well known that for a Banach algebra AA , `ÐÑ is nonvoid. The same holds for complete locally 7 -convex algebras.

21.10 Theorem: (a) AA ``ÐÑœ- ! ÐÑ !&A (b) `ÐÑÁgA . Proof. (a) Clear from the definition. ~ ~ (b) For each FA−ÐÑ`1!! define the functional fAfxFx on by ÐÑœÐÐÑÑ . Evidently fA−ÐÑ`! and the correspondence FfFÇœ‰1~ ~~~! is one-one between ``ÐÑAA!! and ÐÑ . Since each ` ÐÑÁg A ! (because A ! is a Banach algebra) the result follows from (a).

21.11 Theorem: 1. There is a one-to-one correspondence between the elements of `ÐÑA and the closed maximal ideals of A . 2. There is a one-to-one correspondence between the elements of `#ÐÑA and the maximal ideals of codimension 1 of A .

Proof. In each case the correspondence is given by fÈœÐÑ M ker f .

Because of this correspondence we will treat the elements of `ÐÑA as either closed maximal ideals or as continuous linear functionals, whichever is more convenient. Note that `#ÐÑAA does not in general exhaust all the maximal ideals of . In other words, it may happen that a maximal ideal is of infinite codimension. The next theorem shows that RadÐÑAA is always a closed ideal of . Recall that RadÐEÑ is the intersection of all the maximal ideals of E .

21.12 Theorem: (a) Rad AA ÐÑœ+ ` ÐÑ (b) Rad AA# ÐÑœ+ ` ÐÑ (c) RadÐÑœÖAxAx− :3 ÐÑœ 0 ×

Proof. (a) Clearly RadÐѧ A` ÐÑ A . To show the other inclusion let x− ` ÐÑ A and MxMxM be a maximal ideal. We show that − . To this end suppose that . Then the ideal generated by Mx and is all of A . So there exists yA−− and mM such that xy œme. Now for every f−ÐÑ` A ,

fmÐÑœ fxy Ð m Ñœ fe ÐÑœ1

By Theorem 21.6, mGA−ÐÑ , contradicting mM − .

(b) This follows from part (a) and the obvious inclusion

Rad AAA# Ðѧ++`` Ðѧ ÐÑ (c) This follows from part (a) together with Theorem 1.3.13(a).

The topology of `ÐEÑ We provide ``#*ÐÑAA and ÐÑ with the weak -topology they inherit as subsets of the dual space AAw . With this topology, `ÐÑ is called the maximal ideal space of A . Clearly the topology of `ÐÑA is the weak topology generated by the Gelfand transforms; in particular, each ^x is continuous.

21.13 Theorem: If AA is a Frechet´ algebra then every compact subset of `ÐÑ is equicontinuous.

Proof. First observe that since E is a complete metric space it must be of category II (by the Baire Category Theorem). It follows that any any closed, convex, and circled set in E!Y8YœE must be a neighborhood of . Because if is such a set then , so at least one of the 8Y 's must have interior, and so they all must have interior. It follows that Y is a neighborhood of ! . Now, let OÐÑ be a compact subset of ` A . The set

Z œÖB−EÀsup l0ÐBÑlŸ"× 0−O

is a closed, 7E -convex, and circled subset of (exercise), so by the above it must be a neighborhood of !: . It follows that the guage Z is a continuous submultiplicative seminorm on EB−Eß . Now for any

l0ÐBÑl Ÿ :Z ÐBÑ for all 0 − O

This shows that O is equicontinuous.

A topological space XC is hemicompact if it contains a sequence Ö×n of compact subsets such that each compact subset of XC is contained in some k .

21.14 Theorem: If AA is a Frechet´ algebra, then `ÐÑ is hemicompact.

Proof: Let Öœá× pn : n 1, 2, be a sequence of seminorms generating the topology of AAA. By Theorem 21.10(a), . We show that the sequence of subsets ``ÐÑœ- n ÐÑ ÖÐÑ×`n A has the desired properties. First, each `8ÐEÑ is compact. To see this, note that the correspondence in the proof of Theorem 21.10 ~ FFlj1n ~ is a homeomorphism between ``nnÐÑAA and Ð Ñ because the topologies of ` n ÐÑ A (as a ~ subspace of ``ÐÑAA ) and Ðn Ñ are the weak topologies generated by the Gelfand ~ transforms. Thus each ``8ÐEÑ is compact (because ÐAn Ñ is compact). Second, each compact subset of ``ÐÑA is contained in some 8 ÐEÑ . To see this, let OÐEÑO be a compact subset of ` . by Theorem 21.12, is equicontinuous. So there exists a seminorm :57 and Q! such that l0ÐBÑlŸQ: ÐBÑ for all 0−OßB−E . Thus O is contained in `kÐÑA .

The Gelfand Transform

Let Am be a complete locally -convex algebra with identity. The Gelfand transform of xA− is the evaluation map ^xA:`‚#ÐÑÄ

where BÐ^^ffx Ñ œ Ð Ñ . We will often consider B restricted to ` ÐEÑ .

21.15 Theorem: (Wiener ) xGA−ÐÑ if and only if ^xf ÐÑÁ 0 for all f −` ÐÑ A .

Proof. Suppose ^ xÐÑÁ f 0 for all f −` Ð A Ñ . By (2) in the proof of Theorem 21.10(b), ~~~ for each !A− , fxFx ÐÑœ Ð 1!!! ÐÑÑ where F − ` Ð A Ñ . Thus Fx Ð 1 ÐÑÑÁ 0. By the ~ ~ Wiener property for Banach algebras, 1!!ÐÑ−xGA Ð Ñ . Thus xGA − Ð Ñ by Theorem 21.7.

Let A^ be the algebra of Gelfand transforms:

AxxA^ œÖ^ : − ×

Since each ^xAACA is continuous on ``ÐÑ , we see that ^ § Ð ÐÑÑ , the algebra of continuous complex-valued functions on the space `ÐÑA equipped with the compact- open topology. AC^ is a topological algebra with the topology it inherits from ÐÐÑÑ` A . Let > be the Gelfand map >`:AAp§ÐÐÑÑ^ C A

x È ^x

The map > is an algebra homomorphism from AA onto ^ .

21.16 Theorem: The map > is one-to-one if and only if A is semisimple.

Proof. If >`ÐÑœ x 0, then ^ x ÐÑœ f 0 for f − Ð A Ñ . Thus x − Rad Ð A Ñ by Theorem 21.12, and so x œ 0.

21.17 Theorem: If A is a Frechet´ algebra, then > is a continuous algebra homomorphism into CAÐÐÑÑ` .

Proof. Since A is a locally convex space, its topology is the topology of uniform convergence on equicontinuous subsets of its dual. Since A is a Frechet´ algebra, each compact subset of `ÐÑA is equicontinuous (Theorem 21.13). It follows that the topology of AAAA is finer than the topology of ^^ if we identify and via > .

The Spectrum and Spectral Radius

Recall that the spectrum of xA−ÐÑœÖ− is 5-‚A x : xÂ- eGA ÐÑ× and the spectral radius of xx is 3--55AÐÑœ sup Ö± ± : − ÐÑ× x . We sometimes write ÐÑ x and 353551ÐÑxxxxx instead of the more complete ÐÑ and ÐÑ . We also write ÐÑœ~ Ð~ ÐÑÑ AA !!A! and 331ÐÑœxx~ Ð~ ÐÑÑ. !!A!

21.18 Theorem: Let Am be a locally -convex algebra with identity. (a) 5`ÐÑœxxA^ Ð Ð ÑÑ (b) 5`ÐÑœxxA^ Ð# Ð ÑÑ (c) xx 55ÐÑœ- ! ÐÑ !&A Proof. (a) and (c) follow immediately from Theorems 21.15 and 21.10. For (b), trivially ^^xAÐÐÑѪÐÐÑÑ``# xA. For the other containment, let - œ FxF ÐÑ for some − `--#ÐÑA. Then Fx Ð e Ñœ 0, so x  e is not invertible. Thus -5−ÐÑ x .

21.19 Theorem: (a) 3`ÐÑœxxffA sup Ö±^ Ðѱ : − Ð Ñ× (b) 3`ÐÑœxxffA sup Ö±^ Ðѱ : −# Ð Ñ× (c) 33!AÐÑœxx sup Ö! ÐÑ : − × nn"Î (d) 33ÐÑœxx sup lim [! Ð Ñ ] ! n

Proof. (a)–(c) follow from the definition and 21.15. Part (d) is an exercise.

The Operational Calculus

Let's recall the operational calculus for a Banach algebra E . If U is an open subset of ‚ containing 5:ÐÑxHU and Ð Ñ is the algebra of holomorphic functions on U then for every −ÐÑHU there exists y− A such that

^^yfÐÑœ:` ÐÐÑÑ xf f − Ð A Ñ (3)

where y is given by the Cauchy integral formula:

1 :'ÐÑ ydœ 21ixe ' ' 'C where CUx is a contour (a finite system of smooth Jordan curves) in enclosing 5ÐÑ and the integral is the Riemann integral. We use the operational calculus for Banach algebras to show that the same holds for locally m -convex algebras. 21.20 Theorem: Let Am be a locally -convex algebra and let xAU− . If is an open subset of ‚5 , Ux¨ÐÑ , and : −ÐÑ HU , then there exists yA − such that

^^yfÐÑœ:` ÐÐÑÑ xf f − Ð A Ñ (*)

~~ Proof. By Theorem 21.18(c), 55151ÐÑœ x~~ Ð!! ÐÑÑ x . Thus Ð ÐÑѧ x U . For - AA!! !&A ~ :!A−ÐÑHU, and for each − there exists y!! in the Banach algebra A satisfying (*) given by 1 ~~" yxed!!!œÐÑÐÐÑÐÑÑ21i :' 1 '1 ' 'C! where CU is a contour in enclosing 51~ ÐÐÑÑ~ x . Clearly the definition of y is !!!A! independent of the choice of the contour CCU!" . So, for !"¡ let be a contour in that encloses 51~~Ð~~ ÐxC ÑÑ . The contour also encloses 51 Ð Ð x ÑÑ since clearly AA" ""!! 51~~ÐÐÑѧ~~xx 51 ÐÐÑÑ. We have AA! !"" 11:'1'1'~~ÐÑyxedœÐÑÐÐÑÐÑÑ1 ~~" !",, " !"Š‹21i " " 'C" 1 ~~ ~~ " œ21i :' Ð Ñ Ò 1"!,, Ð 1 " Ðxed ÑÑ  '1 "! Ð 1 " Ð ÑÑÓ ' 'C" 1 ~~" œÐÑÐÐÑÐÑÑ21i :' 1!!xed '1 ' 'C" œ y! ~ By Corollary 21.6 there exists yA−ÐÑœ such that 1!! y y . The element y has the desired properties. 27. Multiplicative Linear Functinals and Michael's Problem

We have seen that many results that are true for Banach algebras continue to hold for Frechet algebras. Surprisingly, however, the following is still an open question:

Michael's Problem: Is every multiplicative linear functional on a Frechet algebra continuous?

This question is interesting because the continuity of multiplicative linear functionals is perhaps the most basic and useful tool in the study of Banach algebras. In this section we identify certain classes of topological algebras that have the property that every multiplicative linear functional is continuous. At the end of the section we give an intriguing equivalent form of Michael's problem, due to Dixon and Esterle, which is stated entirely in terms of holomorphic maps on ‚8 .

Continuity of Multiplicative Linear Functionals

For Banach algebras it is well known and easy to prove that every maximal ideal is closed and so every multiplicative linear functional is continuous. In other words

``#ÐÑœAA ÐÑ

A topological algebra with this property is called functionally continuous . The next two results are due to Arens. From parts (a) and (b) of Theorem 21.18, we see that if FxÐÑœ - for some F − ``#ÐÑA, then there exists f − ÐÑ A such that fx ÐÑœ - . Theorem 22.1 extends this property to a finite set of elements Ö×x," á ,xn .

# 22.1 Theorem: Let Ax,,xAFA be a Frechet algebra and " á 8 −−ÐÑ . Then, for ` there exists fA−ÐÑ` such that

FxÐÑœÐÑ55 f x 5 œ 1, á , n

Proof. Suppose there exists J0−ÐEÑ such there is no ` for which the above equations holdÞœÐÑCœB/ß5œ"ßáß8CßáßC Let --55Fx , and 555 . Then ss"8 have no zero in common on `ÐEÑ . It follows (by a hard exercise) that there exists ?"8 ßáß? −E such that 8 ?C55 œ/ 5œ"!

But this is a contradiction because CßáßC"8 are in the kernel of J , which is a proper ideal (because J is discontinuous).

22.2 Theorem: A finitely generated Frechet algebra is functionally continuous. # Proof. Suppose that A is generated by Öá x" , , xn × . Let F −` Ð AÑ− , and let x A . By Theorem 22.1, there exists f−ÐÑ` A such that F ÐÑœÐÑ x55 f x for 5 œá 1, , n, and FxÐÑœ f ÐÑ x. Since any continuous multiplicative linear functional is completely determined by its values on a generating set, fx must be independent of the choice of . Thus FxÐÑœ f ÐÑ x for all x − A , and hence F −` Ð A. Ñ

The following result shows that it is often sufficient to discuss the problem of functional continuity for semisimple algebras with identity. This lemma is due to A. E. Michael.

22.3 Theorem: Let Am be a complete locally -convex algebra. (i) If AAÎÐÑ Rad is functionally continuous, then so is A . (ii) If AmB is an ideal in a functionally continuous locally -convex algebra , then A is also functionally continuous. (iii) If AAe is functionally continuous, then so is .

Proof. (i) If f−ÐÑ`# A then f vanishes on Rad ÐÑ A . Thus there exists gAA−`((# ÐÎÐÑÑRad such that fgœ ‰ , where : AAA ÄÎÐÑ Rad is the canonical map. Since gf is continuous by hypothesis and ( is continuous it follows that is continuous. ~ (ii) Let fA−Ð`Š# Ñ and pick xAfx − such that ÐÑœ 1. Define fB : Ä by !!~ fxÐÑœ fxx Ð Ñ ~~! Clearly fBfAf is a multiplicative linear functional on which agrees with on . But is continuous by hypothesis, and so f is continuous. (iii) Since AA is an ideal in e the result follows by part (ii).

U-algebras

A class of topological algebras that is functionally continuous is the class of Q -algebras. A topological algebra A- is called a Q algebra if its set of invertible elements GAÐÑ is an open set in A .

22.4 Theorem: Let A be a locally 7 -convex algebra identity. The following are equivalent. (a) E is a Q -algebra. (b) H œ ÖB − E À3 ÐBÑ Ÿ "× is a neighborhood of ! in E . (c) 3 is a continuous seminorm on E . (d) ``ÐEÑ œ! ÐEÑ for some !A − .

Proof. (a)ÊEU (b) Since is a -algebra there exists a circled convex neighborhood Y of !/Y such that consists entirely of invertible elements. We show that Y§H and it will follow that H!YH is a neighborhood of . If is not contained in then there exists B−Y with 3--- ÐBÑ" . Thus there is a scalar with l l" such that B / is not invertible. But l--" l" so " B−Y (because Y is circled) from which it follows that /--" B is invertible or B / is invertible. This contradiction shows that Y §HÞ (b)ÊH (c) is a convex and circled, and by hypothesis, H is also a neighborhood of ! . The gauge of H is :H ÐBÑ œ33 ÐBÑ , so is a continuous seminorm.

(c)Ê: (d) The statement means that there is a continuous seminorm ! such that ``ÐEÑ œ! ÐEÑ. The required seminorm is 3 .

(d)Ê (a) If such a :!! exists then l0ÐBÑl Ÿ : ÐBÑ for each 0 −`3 ÐEÑ . So ÐBÑ Ÿ : ! ÐBÑÞ So 33 is a continuous seminorm. But then ÖB − E À ÐB  /Ñ  "× is a neighborhood of ! consisting entirely of invertible elements. Thus EU is a -algebra.

22.5 Theorem: If AmQA is a complete locally -convex -algebra then is functionally continuous.

Proof. In a U -algebra every maximal ideal is closed (Why?) and by the Gelfand-Mazur theorem its codimension is 1. Thus every multiplicative linear functional is continuous.

The Result of Dixon and Esterle

The theorem of Dixon and Esterle equates Michael's problem with the existence of a holomorphic function from some ‚7 into itself with a certain property. We say that a map 2À ‚‚78 Ä is holomorphic if its composition with each of the coordinate functions on ‚‚‚87 is a holomorphic map from into .

22.6 Theorem: (Dixon-Esterle) If there is a discontinuous multiplicative linear functional on the Frechet algebra A then the inverse limit of every inverse system

::::JJJ 7 ‚‚‚‚ÑÑÑââ J−5 Hol Ð ‚ Ñ

is not empty.

T<990Þ Let 0 be a discontinuous multiplicative linear functional on E , so O œ ker Ð0Ñ is dense in EO . Endow with the discrete topology. Then

: Ð8"Ñ: IœE‚O8 8−

is a complete metrizable space. Consider the maps XÀI88"8Ò I defined by

Ð+":"8: ßáß+ ßD ßáßD Ñ Ø ÐJÐ+ ": ßáß+ ÑÐDÐ8"Ñ:" ßáßD 8:" ÑßÐD ßáßD Ð8"Ñ: Ñ

Each XXÐIÑI888"8 is continuous and is dense in . By the Mittag-Leffler Theorem (Appendix 8) each projection map 1 À lim ÐI ß X Ñ Ò I 5 Ñ 88 5

has dense range. In particular, lim ÐI ß X Ñ Á g, s o we can pick an element Ñ 88 AœÐAÑ− lim ÐI ß X Ñ where 8 Ñ 88

: Ð8"Ñ: A888 œÐ,ßB Ñ , 8 −E ßB 8 −O

and XÐ,88"8" ßB ÑœÐ,ßBÑ 8 8 . Now by the definition of the map X 8 it follows that

: JÐ,8" Ñ, 8 − O for all 8 − 

It follows that 0ÐJÐ,8" ÑÑ0Ð, 8 Ñ œ ! or

0ÐJÐ,8" ÑÑœ 0Ð, 8 Ñ

: Note that ,8 œÐ, 8ß" ß, 8ß# ßâß, 8ß: Ñ−E and by 0Ð, 8 Ñ we mean (0Ð,8ß" Ñß 0Ð, 8ß# Ñß âß 0Ð, 8ß: ÑÑ . Now consider the subalgebra of E generated by ,ß,ßâß,8ß" 8ß# 8ß:. Since this subalgebra is finitely generated it is functionally continuous (Theorem 22.2), so 0 is continuous when restricted to this subalgebra. So we can apply the operational calculus to the last displayed equality to get

JÐ0Ð,8" ÑÑœ 0Ð, 8 Ñ

Now the spaces ‚: and the function J form an inverse system. The last equality shows that the sequence Ð0Ð, ÑÑ belongs to lim ÐßJÑ‚: . 8 Ñ Theorem 22.6 is originally stated more generally with the inverse system ==8" 8 consisting of holomorphic functions JÀ88‚‚ Ä where Ð= ) is any sequence of positive integers. The proof of the more general result is basically the same as that given here with the necessary modifications.

22.7 Corollary: Let :−‚‚ Þ If there is a holomorphic function JÀ:: Ä such that _ 8: + JЂ Ñœg 8œ"

then every Frechet Algebra is functionally continuous.

Proof. This is merely a restatement of Theorem 22.7 with the observation that the inverse limit is isomorphic to the given intersection.

It may not be easy to find such a function J: . For one thing must be greater than 1. For :œ" the map J is entire and a theorem of Picard says that every entire fundtion takes on each complex number as value with possibly one single exception. But then J8: Ђ ÑÁg by the Baire Category Theorem. For :" it is known that there are holomorphic functions from ‚‚:: to whose range is smaller than ‚ : . So a function J as described in Corollary 22.7 might exist.